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Mathspace for Florida
Algebra 1
B.E.S.T. 2022 edition
© 2022 Mathspace Pty Ltd
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ISBN 978-1-7369153-8-7 (print)
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Contents
1
Equations and Inequalities in
One Variable 2
1.01 Interpreting algebraic expressions 4
1.02 Equations with variables on one side 9
1.03 Equations with variables on both sides 14
1.04 Literal equations 19
1.05 Absolute value equations 22
1.06 Multi-step inequalities 28
1.07 Compound inequalities 35
1.08 Honors: Absolute value inequalities - solidifylesson 40
2
Equations and Inequalities in
Two Variables 46
2.01 Slope-intercept form 48
2.02 Standard form 60
2.03 Point-slope and connecting forms 68
2.04 Parallel and perpendicular lines 78
2.05 Linear inequalities in two variables 84
3
Systems of Linear Equations and
Inequalities96
3.01 Solving systems of equations by graphing 98
3.02 Solving systems of equations by substitution 106
3.03 Solving systems of equations by elimination 111
3.04 Systems of linear inequalities 116
Contents
iii
4
Functions and Linear relationships 126
4.01 Evaluating functions 128
4.02 Domain and range 132
4.03 Average rate of change 140
4.04 Characteristics of functions from a graph 146
4.05 Linear relationships 153
4.06 Transforming linear functions 162
4.07 Characteristics of linear functions 172
4.08 Comparing linear and nonlinear functions 183
4.09 Linear absolute value functions 189
4.10 Transforming absolute value functions 194
5
Exponential Functions 204
5.01 Operations with numerical radicals 206
5.02 Rational exponents 213
5.03 Exponential relationships 220
5.04 Characteristics of exponential functions 228
5.05 Exponential growth and decay 237
5.06 Percent growth and decay 249
5.07 Simple interest 259
5.08 Compound interest 264
6
Polynomials and Factoring 270
6.01 Adding and subtracting polynomials 272
6.02 Multiplying polynomials 276
6.03 Multiplying special products 280
6.04 Dividing polynomials by a monomial 284
6.05 Factoring GCF 287
6.06 Factoring by grouping 291
6.07 Factoring trinomials where a = 1 295
6.08 Factoring trinomials where a > 1 300
6.09 Factoring special products 305
iv
Mathspace Florida B.E.S.T - Algebra 1
7
Quadratic Functions 310
7.01 Quadratic relationships 312
7.02 Key features of quadratic graphs 321
7.03 Quadratic functions in factored form 334
7.04 Quadratic functions in vertex form 344
7.05 Quadratic functions in standard form 355
7.06 Modeling with quadratic functions 366
7.07 Linear, quadratic, and exponential models 374
7.08 Honors: Combining linear and quadratic functions 384
8
Solving Quadratic Equations 392
8.01 Solving quadratic equations using graphs and tables 394
8.02 Solving quadratic equations by factoring 403
8.03 Solving quadratic equations using square roots 409
8.04 Solving quadratic equations by completing the square 414
8.05 The quadratic formula and the discriminant 419
8.06 Solving quadratic equations using appropriate methods 425
9
Descriptive Statistics 432
9.01 Populations and sampling 434
9.02 Classifying data 440
9.03 Displaying and interpreting univariate data 445
9.04 Two-way frequency tables 457
9.045 Honors: Two-way relative frequency tables 466
9.05 Associations in bivariate data 478
9.06 Scatter plots and lines of fit 483
9.07 Analyzing lines of fit using residuals 490
Contents
v
1
Equations and
Inequalities in
One Variable
Chapter outline
1.01 Interpreting algebraic expressions 4
1.02 Equations with variables on one side 9
1.03 Equations with variables on both sides 14
1.04 Literal equations 19
1.05 Absolute value equations 22
1.06 Multi-step inequalities 28
1.07 Compound inequalities 35
1.08 Honors: Absolute value inequalities 40
1.01 Interpreting
algebraic expressions
Concept summary
An expression is a mathematical statement that contains one or more numbers and variables
joined together by operators and grouping symbols. An expression does not contain an equal sign
or inequality symbol.
An algebraic expression is an expression which includes at least one variable.
Algebraic operation
A mathematical process, such as addition, subtraction, multiplication, or division
Example: +, −, ×, and ÷
Variable
A symbol, usually a letter, used to represent an unknown value.
Example: Expression: 3x − y + 2 Variables: x and y
Term
One part of an expression. Terms are separated by addition or subtraction.
Example: Expression: 3x − y + 2 Terms: 3x, −y, and 2
Coefficient
The number or constant that multiplies a variable in an algebraic term. If no number is
specified, the coefficient is 1.
Example: Term: 3x Coefficient: 3
Constant term
A term that has a fixed value and does not contain a variable. Sometimes just called a constant.
Example: Expression: 3x − y + 2 Constant: 2
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Mathspace Florida B.E.S.T - Algebra 1
Like terms are terms that have the same variables with the same exponents. An example is shown
below:
Worked examples
Example 1
3x + 5x 2 − 2x − 8
Terms: 3x, 5x 2 , −2x, and −8
Like terms: 3x and −2x
Consider the algebraic expression: 15y 2 + 7y 3 − 3y 2 − 5
Identify:
• The constant term
• The like terms
• The coefficient of y 3
Approach
The constant term is the numeric term without any variable factor, the like terms are the algebraic
terms that have the same algebraic factors, and the coefficient of y 3 is the numeric factor of the
term containing a factor of y 3 .
Solution
The constant term is −5, as it is the term without any variable part.
The like terms are 15y 2 and −3y 2 , as they have the same variables with the same exponents.
The coefficient of y 3 is 7, as this is the numeric factor of the term 7y 3 .
Reflection
Notice that we include the sign with all terms. For example, one of the like terms is −3y 2 , which has
a coefficient of −3.
Example 2
Vincenzo runs a removalist company that charges $37.50 per hour plus a one-off truck hire fee of
$150.00.
Write an expression that models how much he charges for a job that lasts a hours.
Chapter 1 Equations and Inequalities in One Variable 5
Approach
We need to look at the two values that affect the price of the job; the cost per hour of $37.50 and
the truck hire fee of $150.00. Once we determine how to write the cost per job in terms of a hours,
we need to write an expression that is the sum of these two values.
Solution
Cost: 37.5a + 150
Example 3
Bernie and Paula are painting the walls of their house. Bernie can paint x square meters an hour
while Paula can paint y square meters an hour. Bernie spends 4 hours painting while Paula spends
6 hours over the weekend. Together they paint a total of 4x + 6y square meters over the weekend.
Determine what 6y represents in the expression.
Approach
We can look at the units that are included in the term 6y. Paula paints for 6 hours and y is the
number of square meters she can paint per hour. We can use this to help us determine what
6y represents.
Solution
The number of square meters Paula painted over the weekend.
What do you remember?
1 Match the following terms with their definitions:
a Constant i Symbol that represents an unknown number
b Variable ii A purely numeric term in an algebraic expression
c Coefficient iii The value that indicates how many of a variable in a term
d Algebraic term iv Term including a variable
2 Write the expression 9x in words.
3 State whether the following expressions are like terms:
a 11p and 3p b 2p and 15 c 9p and 6q d 12 and 8
e 5h and 5hk f 4h 4 and 3h g 8h 3 k 4 and 9k 4 h 3 h 6h k and 7k h
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Mathspace Florida B.E.S.T - Algebra 1
4 For the following algebraic expressions:
i State the number of terms.
ii What is the numerical coefficient of the first term?
iii What is the constant term?
iv Does it contain like terms?
a 2x + 4 b 7y + 3 + 5x
c 3x + 2y − 8x + 9 d 8p + 5
Let’s practice
5 Consider the algebraic expression 3x + 10 + 4x.
a
b
c
d
State the number of terms.
What is the numerical coefficient of the third term?
What is the constant term?
Does it contain like terms?
6 Dylan purchased 3 pens, 4 pencils and a single note pad. The total cost of all these items
was 3x + 4y + 5 dollars.
a What does the variable y represent? b What does the variable x represent?
c
What does 5 represent?
7 Deborah earns $25 per hour of work and is paid double for every hour worked on the
weekend. At the end of a week, Deborah calculates her pay for that week to be 25x + 50y
dollars.
a
b
What represents the number of hours Deborah worked during the weekdays?
What represents the number of hours Deborah worked during the weekends?
8 Mohamad and Valentina run a clothes business. Mohamad is able to sew 5 shirts an hour,
while Valentina is able to sew 3 shirts an hour. The expression 6( 5 + 3 ) represents the total
number of shirts the couple is able to produce in a six-hour period.
What does 5 + 3 represent?
9 Brad and Patricia are making paper cranes to decorate their room. Brad can make m an hour
while Patricia can make n an hour. Brad spends 6 hours making cranes while Patricia spends
5 hours over the weekend.
Together they make a total of 6m + 5n paper cranes over the weekend.
a What does 6m represent? b What does 5n represent?
10 The width of a rectangular football field is 7 yards more than half of its length. If L represents
the length of the field, write an expression for the width.
Chapter 1 Equations and Inequalities in One Variable 7
11 Roxy is 5 inches taller than Jane, while Katrina is 9 inches shorter than Jane. Let k be Jane’s
height in inches.
a
b
Write an algebraic expression for Roxy’s height.
Write an algebraic expression for Katrina’s height.
12 Valerie is with a mobile phone provider that charges $0.26 per minute plus a connection fee
of $0.45. Write an expression that models how much he will be charged for a call that lasts
a minutes.
Let’s extend our thinking
13 For each of the following, determine if the statement is true or false. Explain your reasoning.
a
b
An expression must include a variable.
Any expression with more than one term can be simplified.
14 Tobias has two times as many books as Isabelle does, and Isabelle has four less than triple
the number of books that William does. If William has n books, explain the method used to
write an expression for the number of books that Tobias has.
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Mathspace Florida B.E.S.T - Algebra 1
1.02 Equations with
variables on one side
Concept summary
An equation is a mathematical relation statement where two equivalent expressions and values
are seperated by an equal sign. The solutions to an equation are the values of the variable( s ) that
make the equation true. Equivalent equations are equations that have the same solutions.
One particular type of equation is a linear equation.
Linear equation
An equation that contains a variable term with an exponent of 1, and no variable terms with
exponents other than 1.
Example: 2x + 3 = 5
Equations are often used to solve mathematical and real world problems. To solve equations we
use a variety of inverse operations and the properties of equality.
Inverse operation
Two operations that, when performed on any value in either order, always result in the original
value; inverse operations “undo” each other.
Example: Operation: Multiply by 2 Inverse: Divide by 2
The following are the properties of equality:
Reflexive property of equality
a = a
Symmetric property of equality
If a = b, then b = a
Transitive property of equality
If a = b and b = c, then a = c
Chapter 1 Equations and Inequalities in One Variable 9
Addition property of equality
If a = b, then a + c = b + c
Subtraction property of equality
If a = b, then a − c = b − c
Multiplication property of equality
If a = b, then ac = bc
Division property of equality
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Substitution property of equality
If a = b then b may be substituted for a in any expression containing a.
The following is another important property:
Distibutive property
a( b + c ) = ab + ac
Worked examples
Example 1
Solve the following equation:
Solution
Subtract 3 from both sides of the equation
x = 4 Multiply both sides of the equation by 2
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Mathspace Florida B.E.S.T - Algebra 1
Example 2
Solve the following equation:
Solution
3x + 4x + 7 = 3 Multiply the entire equation by 3
7x + 7 = 3
Combine like terms
7x = -4 Subtract 7 from both sides of the equation
Divide both sides of the equation by 7
Example 3
Yolanda works at a restaurant 5 nights a week and receives tips. On the first three nights, the total
tips she received was $32, $27, and $26. She earned twice as much in tips on the fourth night
compared to the fifth night. The average amount of tips received per night for the week was $29.
If the amount she received on the fifth night was $f, determine how much she received that night.
Approach
The average is equal to the sum of values divided by the number of values. We can use this to
build an equation in terms of f about the average tips Yolanda received. Then we can solve the
equation for f.
Solution
Writing an equation in terms of f
Combining like terms in numerator
Yolanda received $20 on the fifth night.
Multiply both sides of equation by 5
Subtract 85 from both sides of the equation
Divide both sides by 3
Chapter 1 Equations and Inequalities in One Variable 11
What do you remember?
1 Solve the following equations:
a 4x = 8 b 5m = −15 c 3p = 12 d 2k = −18
2 Solve the following equations:
a 8m + 9 = 65 b −10 + 3k = 5 c 7 − 8t = 15 d −2d + 10 = 24
Let’s practice
3 Solve the following equations:
a b c
4 Solve the following equations:
a −2( −3h − 5 ) = 34 b 9x + 6x + 7 − 3x = 55
c 4x + 2 + 9x − 12 + 6x + 7 = 130
5 Solve the following equations:
a
b
c
6 The product of 6 and the sum of x and 4 is equal to 42. Solve for the value of x.
7 Consider the given triangle which has a
perimeter of 171 cm.
Solve for the value of x.
8x cm
5x + 76 cm
6x + 19 cm
8 Consider the given
quadrilateraL which has a
perimeter of 393 cm.
Solve for the value of x.
8x − 30 cm
2x + 24 cm
9x + 29 cm
5x − 14 cm
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Mathspace Florida B.E.S.T - Algebra 1
9 Ursula is constructing a tabletop from a rectangular piece of board. The tabletop is to have
a perimeter of 68 inches, and its length is to be 5 inches shorter than double the width.
Solve for the width of the tabletop.
10 Emma and Beth do some fundraising for their tennis team and together they raised $664.
Emma raised $292 more than Beth.
Solve for the amount of money raised by Beth.
11 Yvonne is the youngest child in her family. She has three older brothers of ages 23, 20 and
16, and a sister who is eight years older than Yvonne is. The average age of the five children
is 19 years.
If Yvonne is k years old, solve for the value of k.
Let’s extend our thinking
12 Stormie tries to guess how many people are at a concert, but she guesses 800 too many.
Pierce guesses 300 too few. The average of their guesses is 4250.
a
b
Describe a method to construct the equation to solve for the exact number of people at
the concert.
Solve for the exact number of people at the concert.
13 Athena and Emilio want to go karting. It costs 50 cents per lap of the course.
a
b
c
Write an equation to solve for the number of laps Athena and Emilio can afford if they
have $12.
Solve for the number of laps they can afford.
How does that equation change if they have to put a deposit of $1 on each cart used?
14 Bassam is changing km/h into mi/h for a project on peregrine falcons. He knows that the top
speed of the peregrine falcon is 389 km/h. He knows that there is 1.6 km to 1 mi. To convert
389 km/h to mi/h, Bassam multiplies 389 by
What error does Bassam make?
15 Determine whether each statement is true or false. Explain your thinking.
a
b
A known solution can be checked by substituting it into the equation.
No two equations have the same solution.
Chapter 1 Equations and Inequalities in One Variable 13
1.03 Equations with
variables on both sides
Concept summary
The first step in solving equations with variables on both sides is usually to move all variable terms
to one side of the equation, by applying the properties of equality to variable terms.
A fully simplified equation in one variable will take one of the following three forms, corresponding
to how many solutions the equation has:
• x = a, where a is a number ( a unique solution )
• a = a, where a is a number ( infinitely many solutions )
• a = b, where a and b are different numbers ( no solutions )
An equation of the second form, which is true for any possible value of the variable( s ), is
sometimes called an identity.
Worked examples
Example 1
Determine how many solutions the following equations have:
a 3( −8 + x ) = 3( −8 + x )
Solution
3( −8 + x ) = 3( −8 + x )
−24 + 3x = −24 + 3x Distribute the 3
−24 = −24
Subtract 3x from both sides
Since the equation is of the form a = a, where a is a number, there are infinitely many solutions.
14
Mathspace Florida B.E.S.T - Algebra 1
b
Solution
9 + x = x + 5 Multiply both sides by 9
9 = 5 Subtract x from both sides
Since the equation is of the form a = b, where a and b are different numbers, there are
no solutions.
c 5( 7 + x ) = 2x + 85
Solution
5( 7 + x ) = 2x + 85
35 + 5x = 2x + 85 Distribute the 5
35 + 3x = 85 Subtract 2x from both sides
This equation will simplify to one of the form x = a, where a is a number, so there is a unique solution.
Reflection
We did not have to solve the equation to determine that it has a unique solution - we only needed
to check that the variable terms did not cancel each other out, like they did in part ( a ) and part ( b ).
Example 2
Solve the following equation: 4( x − 9 ) = x + 6
Solution
4( x − 9 ) = x + 6
4x − 36 = x + 6 Distribute the 4
3x − 36 = 6
Subtract x from both sides
3x = 42
Add 36 to both sides
x = 14 Divide both sides by 3
Reflection
We can check if x = 14 is correct by substituting 14 in for x into each side and checking that they
are equal:
LHS = 4( 14 − 9 ) = 4 ⋅ 5 = 20
RHS = 14 + 6 = 20
Chapter 1 Equations and Inequalities in One Variable 15
Example 3
Right now, Bianca’s father is 48 years older than Bianca. 2 years ago, her father was 5 times older
than her. Solve for y, Bianca’s current age.
Approach
We want to write expressions that represent Bianca’s age and her father’s age. Then we relate
them with an equation and solve for y.
Bianca’s father is currently y + 48 years old.
Two years ago, Bianca was y − 2 years old. Her father was y + 48 − 2 = y + 46 years old.
Her father’s age was five times her age at this time, which produces our equation:
y + 46 = 5( y − 2 )
Solution
y + 46 = 5( y − 2 )
y + 46 = 5y − 10 Distribute the 5
y + 56 = 5y
Add 10 to both sides
56 = 4y Subtract y from both sides
14 = y Divide both sides by 4
Bianca is currently 14 years old.
What do you remember?
1 Solve the following equations:
a m + m + 3 + 12 = 13 + m + m + m b x + x + x + x + 2 + 5 = x + 4 + x + x
c p + p − 3 + 5 = p − p − p + 1 − 5 d −k − k − k − 6 = −k − k − k − k + 6
2 Determine the number of solutions of the following equations:
a 7( 7 + x ) = 2x + 28 b 10( 5 + x ) = 10( x + 5 )
c d −31 − 3( x − 9 ) = 2( x − 2 ) − 5x
3 For each of the following relations:
i Write an equation for the relation, using n to represent the unknown number.
ii Determine the number of solutions to the equation.
a
b
c
Three more than three times a number is equal to six more than four times the number.
Six more than two times a number is equal to five more than two times the number.
Two more than four times a number is equal to two added to quadruple the number.
16
Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
4 Solve the following equations:
a 3f − 8 = f b 10r + 4 = 6r
c 4w + 24 = w + 15 d −72 − 9p = −32 − p
5 Solve the following equations:
a 4( x − 15 ) = x b 5( x − 16 ) = x − 16
c 3( x − 3 ) = 2( x + 1) d 2( 3x − 4 ) = 2( x + 4 )
e 3y = −13 − 4( 2y + 5 ) f x − 6 − 7( x + 3 ) = 21 − 9( x − 18 )
6 Solve the following equations:
a
b
c
d
7 Consider the given rectangle with a perimeter of 126 + 3y centimeters.
4y + 8
4y + 3
Find the value of y.
8 A Payroll Officer has been told to distribute a bonus to the employees of a company worth
15% of the company’s net income. Since the bonus is an expense to the company, it must be
subtracted from the income to determine the net income. If the company has an income of
$120 000 before the bonus, then the Payroll Officer must solve:
for B. Find the bonus, B.
B = 0.15( 120 000 − B )
9 Right now, Skye’s father is 36 years older than Skye. 4 years ago, her father was 4 times as
old as her. Solve for y, Skye’s current age.
10 Xanthe, a tennis player, has won 69 out of 96 matches in her career. Find x, the number of
matches she must win in a row to raise her win percentage to 75%.
Chapter 1 Equations and Inequalities in One Variable 17
Let’s extend our thinking
11 Three consecutive integers are such that the sum of the first and twice the second is 15
more than twice the third.
a Let x be the smallest of the integers. Find x.
b
c
State the three consecutive integers.
Are there any other sets of three consecutive integers that could fulfill the requirements?
Explain how you know.
12 Determine whether the following statements are true or false. Explain your thinking.
a
b
Any equation with variables on both sides must have multiple solutions.
Two equations will always have the same solution if one is a multiple of the other.
18
Mathspace Florida B.E.S.T - Algebra 1
1.04 Literal equations
Concept summary
Some equations involve more than one variable. These are usually still referred to as just
equations, but sometimes they are given other names.
Literal equation
An equation that involves two or more variables
Example: x = 5y + 3z
We can use the properties of equality to isolate any variable in a literal equation.
Formula
A type of literal equation that describes a relationship between certain quantites
Example: A = l ⋅ w where A is area, l is length, and w is width.
The same variable might be used to represent different quantities acorss different formulas.
For example, in the formula for the area of a rectangle w is used to represent the width of the
rectangle, however in another context w might be used to represent a weight or other value.
Worked examples
Example 1
Solve for x in the following equation:
y = 5( 1 + x )
Approach
We need to rearrange the equation to isolate x. We can use the properties of equality and inverse
operations to solve literal equations for a variable, just as we would for linear equations.
Solution
y = 5( 1 + x )
Divide both sides of the equation by 5
Subtract 1 from both sides of the equation
Symmetric property of equality
Chapter 1 Equations and Inequalities in One Variable 19
Reflection
When rearranging an equation, we reverse the operations acting on the variable we want to isolate
in the reverse order of operations. Whatever is done to one side of the equation, must be done to
the other to keep the equation balanced.
Example 2
Given the formula for Ohm’s law:
V = IR
where V is voltage, I is current and R is resistance.
Write the formula for current.
Approach
The formula for current is Ohm's law with I isolated. We use inverse operations and properties of
equality to get the solution.
Solution
V = IR
Divide both sides of the equation by R
Symmetric property of equality
What do you remember?
1 What benefits come from rewriting literal equations?
2 Solve for x in the following equations:
a y = x + z b c y = −8( 7 + x )
d e f kx = m + nx
3 Solve for m in the following equations:
a b y = 9mx − 5 c x = −2k( n + m ) d x = 3k( n + m )
4 Solve for
5 Solve for R : V = IR − E
20
Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
6 The formula for the force of an object is F = ma, where F is the force, m is the mass of the
object, and a is the object’s acceleration. Write an equation that could be used to solve for
the mass of the object.
7 The formula for the perimeter of a rectangle is P = 2l + 2w, where P is the perimeter, l is
the length, and w is the width. Write an equation that could be used to find the length of the
rectangle.
8 Betsy knows the formula to convert Celsius into Fahrenheit is She wants to
convert a temperature from Fahrenheit to Celcius. Find an equation that she could use to do
this conversion.
9 The displacement of an object is given by Write an equation that could be
used to find v 0 .
10 The bend allowance for sheet metal is given by the formula Write an
equation that could be used to solve for the thickness of the sheet, T.
11 The formula to the distance traveled in a time period is d = rt, where d is the distance
traveled, r is the rate of travel, and t is the time spent traveling.
a
b
Write an equation that could be used to solve for the rate of travel.
Floyd is riding his bike 5 miles to Jue’s house. Jue is expecting Floyd in 15 minutes.
Find the average speed Floyd must ride in order to get to Jue’s on time. Give your
answer in mi/h.
Let’s extend our thinking
12 Determine the similarities and differences of solving 3x + y = z and 3x + 8 = 2. Explain your
thinking.
13 Rufino has submitted the following work to write the equation for the mass of an object, m,
given the kinetic energy, KE, and velocity, v:
Determine whether Rufino is correct by explaining each step of his working if the step is
correct, otherwise fix his error.
Chapter 1 Equations and Inequalities in One Variable 21
1.05 Absolute value
equations
Concept summary
The absolute value of a number is the distance of that number from 0 on a number line. Distance is
expressed as a positive value, and so the absolute value of a number is always positive. Absolute
value is sometimes called magnitude.
To denote the absolute value of a quantity, we use a single vertical bar on either side. The absolute
value of x can be thought of as:
• If x ≥ 0, then | x | = x
• If x < 0, then | x | = -x
Absolute value equation
An equation that contains a variable expression inside absolute value bars.
Example: | x | = 4
Extraneous solution
A solution of an equation that emerges from the process of solving the problem, but is not a
valid solution to the problem.
An absolute value equation can be solved by applying arithmetic operations as usual, but
sometimes one or more solutions to an absolute value equation will be extraneous solutions
depending on the context.
Worked examples
Example 1
Solve each absolute value equation.
a |x | = 6
Solution
The solutions to this equation will be any values of x that have an absolute value of 6.
The solutions are x = 6 and x = −6.
22
Mathspace Florida B.E.S.T - Algebra 1
Reflection
Notice that this equation has different solutions from the equation | x − 6 | = 0 which has only a
single solution of x = 6. This shows that adding or subtracting terms in or out of the absolute value
bars changes the equation, similar to parentheses.
b |2x | = 4
Solution
The solutions to this equation will be any values of x that make 2x have an absolute value of 4.
The solutions are x = 2 and x = −2.
Reflection
Notice that this equation has the same solutions as the equation 2|x | = 4. This demonstrates that
multiplying or dividing a constant term in or out of the absolute value bars does not change the
equation.
c |3x | + 4 = 2x + 9
Approach
This equation is harder to solve because it involves x-terms both inside and outside absolute value
bars. One way to work around this is to split up the problem into two cases: when 3x ≥ 0, and when
3x < 0.
Solution
When 3x ≥ 0, we know that | 3x | = 3x, so we can write and solve the equation:
|3x | + 4 = 2x + 9
3x + 4 = 2x + 9 Take the case where 3x ≥ 0
x + 4 = 9
x = 5
Subtract 2x from both sides
Subtract 4 from both sides
When 3x < 0, we know that | 3x | = −3x, we can write and solve the equation:
|3x | + 4 = 2x + 9
−3x + 4 = 2x + 9 Take the case where 3x < 0
4 = 5x + 9 Add 3x to both sides
−5 = 5x
Subtract 9 from both sides
−1 = x Divide both sides by 5
Using two cases, we have found two solutions: x = 5 and x = −1.
Chapter 1 Equations and Inequalities in One Variable 23
We can substitute each solution into the starting equation to check that they are valid.
|3( 5 ) | + 4 = 2( 5 ) + 9
15 + 4 = 10 + 9
19 = 19
and
|3( −1) | + 4 = 2( −1) + 9
3 + 4 = −2 + 9
7 = 7
Reflection
If substituting either solution into the starting equation resulted in an untrue equation, then that
solution would be extraneous.
Example 2
Clay wants to find the point in a 100 yard dash where his distance in yards from the middle of the
race is equal to 40 less than twice the number of yards he has run so far.
Find the number of yards that Clay can have run to make this equality true.
Approach
The varying factor in this context is the number of yards that Clay has run. We can represent this
with the variable y.
Clay’s distance from the middle of the race can be described as the difference between the
number of yards he has run, y, and 50 yards. We can write this as the absolute value expression
| y − 50|.
Since y represents how far Clay has run in the race, we can express “40 less than twice the yards
he has run so far” as the expression 2y − 40.
Putting these two expressions together, we get the equation that Clay wants to solve:
Solution
|y − 50 | = 2y − 40
To solve this equation, we can use the two cases method.
Taking the case where y − 50 ≥ 0, we have:
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Mathspace Florida B.E.S.T - Algebra 1
Taking the case where y − 50 < 0, we have:
Solving using the two cases method gives us the possible solutions y = −10 and y = 30. We can
see that if we substitute y = −10 into our starting equation we will get
|−10 − 50 | = 2( −10 ) − 40
which is not true. This tells us that y = −10 is an extraneous solution.
We can also see that if we substitute y = 30 into our starting equation we will get
which is true.
|30 − 50 | = 2( 30 ) − 40
So Clay must run exactly 30 yards so that his distance from the middle of the race is equal to
40 less than twice the number of yards he has run.
Reflection
Notice also that the solution y = −10 is made non-valid by the context, since Clay cannot run a
negative number of yards.
What do you remember?
1 Solve each of the following equations:
a |x | = 8 b 9 | x | = 4 c −3 | x | = −12
d |x + 3 | = 7 e |1 − x | = 3 f
g |2x | − 5 = 4 h |−3.9x | + 3 = 22.5 i |3x + 6 | = 9
j −3 | x + 5 | = −9 k 7 − | 4x | = 5 l |4x − 8 | + 1 = 13
m |2x + 9 | = x + 6 n |3x | + 2 = 2x + 7 o −2 | x + 3 | = −14
p q −6 | 4x − 1| = −12 r
2 Solve 3 | x + 2 | − 1 = 3x + 11 and state how many solutions it has.
Chapter 1 Equations and Inequalities in One Variable 25
Let’s practice
3 For each of the following, solve the absolute value equation and then check each answer to
determine which solutions are extraneous and which are valid:
a |x + 6 | = 2x b |3x + 2 | = 4x + 5
c |x − 1| = 5x + 10 d −2 | x | = 6 − 2x
4 Write 5x − 2 = 3 or 5x − 2 = −3 as an equivalent absolute value equation.
5 Write an absolute value equation that represents each of the following:
a All real numbers x that are 8 units away from 0.
b All real numbers x that are 5 units away from −2.
6 The minimum and maximum lengths of AA batteries, b in millimeters, can be represented
with the absolute value equation | b − 49.9 | = 0.6.
Find the minimum and maximum lengths in millimeters of a AA battery.
7 Baby clothing is rated using a TOG rating for what temperatures it is designed for to avoid
the baby being too cold or overheating. A sleep sack with rating 2.5 TOG is designed for
65°F, plus or minus 4°F.
Write an absolute value equation that represents the minimum and maximum temperatures
for that sleep sack.
8 Compare and contrast the process for solving each equation and state their solutions.
• Equation A: | 3x | + 4 = 5
• Equation B: | 3x + 4 | = 5
9 Compare and contrast the process for solving each equation and state their solutions.
Assume a, b > 0.
• Equation A: | ax | = b
• Equation B: a | x | = b
10 Determine the value( s ) of b for which the equation | x | = b has only one solution.
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
11 Describe why the absolute value of a number cannot be negative. Give an example of an
absolute value equation with no solution.
12 Clem and Finn are waiting in line at a petting zoo. Clem is 6 spots in line away from Finn.
Consider an absolute value equation that represents Clem’s possible position, compared to
Finn’s position.
a
b
c
Describe the positions for Finn where there are two possible positions for Clem.
Describe the positions for Finn where there is only one possible position for Clem.
Describe the positions for Finn where there is no possible position for Clem.
13 Roxanne performed the following steps to solve the equation 4 | x | = −2:
1 4 | x | = −2
2 4x = ±2
3
Show that neither of these solutions is valid and explain where her error was.
14 Kaydence is running to meet her friend at the local market. She knows that her average
speed is 7.5 mph ( 0.125 miles per minute ). She tells her friend that she will be there
in 30 minutes, plus or minus 6 minutes depending on the route she ends up taking.
Remember that:
Write and solve an equation for the minimum and maximum distances for her run.
Chapter 1 Equations and Inequalities in One Variable 27
1.06 Multi-step
inequalities
Concept summary
Some mathematical relations compare two non-equivalent expressions - these are known as
inequalities.
We can solve inequalities by using various properties to isolate the variable, in a similar way to
solving equations:
Asymmetric property of inequality
If a > b, then b < a
Transitive property of inequality
If a > b and b > c, then a > c
Addition property of inequality
If a > b, then a + c > b + c
Subtraction property of inequality
If a > b, then a − c > b − c
Multiplication property of inequality
If a > b and c > 0, then a ⋅ c > b ⋅ c
If a > b and c < 0, then a ⋅ c < b ⋅ c
Division property of inequality
If a > b and c > 0, then a ÷ c > b ÷ c.
If a > b and c < 0, then a ÷ c < b ÷ c.
When multiplying or dividing an inequality by a negative value the inequality symbol is reversed.
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Mathspace Florida B.E.S.T - Algebra 1
Linear inequality
An inequality that contains a variable term with an exponent of 1, and no variable terms with
exponents other than 1
Example: 3x − 5 ≥ 4
Solving an inequality using the above properties of inequalities results in a solution set.
Solution set
The set of all values that make the inequality or equation true
Example: 3 ≤ x
We can represent solutions to inequalities algebraically, by using numbers, letters, and/or symbols,
or graphically, by using a coordinate plane or number line.
An algebraic solution to an inequality can be represented as an inequality, such as 3 ≤ x, or in other
ways, including interval notation and set-builder notation.
Interval
A set of numbers that lie between two values
Interval notation
A way to represent a solution set or interval as a pair of numbers using a combination of
square brackets and parentheses.
Example: Inequality solution 3 ≤ x
Interval notation [3, ∞ )
We use square brackets if the endpoint is included and parentheses if the endpoint is not included.
We always use parenthese for infinity. We can join two sets together using the union symbol ∪. We
may see x ∈ which says “x is in”.
Set-builder notation
A shorthand used to write sets. The set {x ∈ |x > 0} is read as “the set of all real values of x
such that x is greater than zero”. Sometimes a colon : is used instead of a vertical line | .
Example: Inequality solution 3 ≤ x Set-builder notation {x ∈ |x ≥ 3}
• Inequality notation: −3 ≤ x < 4
• Interval notation: [−3, 4 )
• Set-builder notation: {x ∈ | − 3 ≤ x < 4}
−5 −4 −3 −2 −1 0 1 2 3 4 5
Chapter 1 Equations and Inequalities in One Variable 29
• Inequality notation: x ≥ −3
• Interval notation: [−3, ∞ )
• Set-builder notation: {x ∈ | x ≥ −3
−5 −4 −3 −2 −1 0 1 2 3 4 5
Two inequalities that have the same set of solutions are called equivalent inequalities.
Worked examples
Example 1
Consider the inequality
a Solve the inequality
Approach
We want to isolate x on one side of the inequality and a number on the other.
Solution
Reflection
Multiply both sides of the inequality by 2
Add 8 to both sides.
Divide both sides by −3. Reverse the direction of the
inequality symbol since we are dividing by a negative
number.
Solving an inequality is similar to solving an equation. However, we need to reverse the direction of
the inequality when multiplying or dividing by a negative number.
b Plot the inequality on a number line.
Solution
Plot the solution set of the inequality x ≥ −6. Note that since we include −6 the point should be filled.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10
Reflection
What if the solution was x > −6?
End points included in the interval are filled points.
End points not included in the interval are unfilled points.
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Mathspace Florida B.E.S.T - Algebra 1
Example 2
Oprah charges $37.72 to style hair, as well as $6 per foil. Pauline would like a style and foils,
but has no more than $95.86 to spend.
a Write an inequality that represents the number of foils Pauline could get.
Approach
Pauline would like a style which is $37.72 and some foils at $6 each. If the number of foils Pauline
can get is N, then we can write an expression that represents the cost to style hair as 6N + $37.72.
Pauline has no more than $95.86 to spend. Remember that “no more than” means “less than or
equal to.”
Solution
6N + 37.72 ≤ 95.86
b Write the solution set to the inequality.
Approach
Solve the inequality and then write the solution set.
Solution
37.72 + 6N ≤ 95.86
6N ≤ 58.14 Subtract 37.72 from both sides of the inequality
N ≤ 9.69 Divide both sides by 6
Inequality notation: N ≤ 9.69
Reflection
We could also write this in interval notation as ( −∞, 9.69].
c Determine the solution set in the context of the question.
Approach
We need to work out what the solution means in context.
Solution
Pauline cannot get part of a foil, so she can get a maximum of 9 foils and a minimum of 0 foils.
In context, the solution set contains the following numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 because
Pauline can get any number between 0 and 9 foils.
Chapter 1 Equations and Inequalities in One Variable 31
Reflection
Sometimes the context restricts the set of solutions.
What do you remember?
1 Explain the differences between the inequalities: 0 ≤ x < 4 and 3 ≤ x ≤ 7.
2 Plot the following inequalities on a number line:
a x > −9 b x ≤ 15
3 Describe the range of values that satisfy each inequality.
a x ≥ 29 b x < −29
4 Which of the following inequalities represents the solution for x in 10 ≤ 6 − 4x?
A x ≤ −4 B x ≥ −1 C x ≤ 1 D x ≥ 4
5 Which of the following number lines represent the solution for 4x − 7 < 5?
A
−5 −4 −3 −2 −1 0 1 2 3 4 5
B
−5 −4 −3 −2 −1 0 1 2 3 4 5
C
−5 −4 −3 −2 −1 0 1 2 3 4 5
D
−5 −4 −3 −2 −1 0 1 2 3 4 5
6 Which of the following inequalities represents the solution for r in “5 more than 2r is less
than 39”?
A r > 17 B r < 17 C r > 22 D r < 22
7 To ride the ferris wheel in Fantasy World, a child must be at least 47 inches tall and less than
64 inches tall. Which of the following number lines represent these conditions?
A
44 46 48 50 52 54 56 58 60 62 64 66
B
44 46 48 50 52 54 56 58 60 62 64 66
C
44 46 48 50 52 54 56 58 60 62 64 66
D
44 46 48 50 52 54 56 58 60 62 64 66
Let’s practice
8 Consider the inequality: 5( x + 3 ) ≤ 35.
a Solve for x.
b State whether the following values satisfy the inequality:
i x = −4 ii x = 8 iii x = 4 iv x = 0
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Mathspace Florida B.E.S.T - Algebra 1
9 For the following inequalities:
i Solve for x. ii Plot the solutions on a number line.
a 3x − 7 < 8 b 4 < 6x − 2 c −6x − 7 ≤ 5 d 2( x − 3 ) < −16
10 Consider the inequality: 4 − 2x < 3x − 2.
a Solve for x.
b
State whether the following values satisfy the inequality:
i ii x = 1 iii iv x = 2
11 Solve the following inequalities:
a 5x − 40 ≥ 50 b −8 − m > 3 c −8( x + 8 ) ≥ −40
d e f
12 Consider the situation: “3 less than 3 groups of p is no more than 24”.
a Solve the inequality. b Find the largest value p can take.
13 Skye has a budget for school stationary of $43, but has already spent $21.14 on books and
folders. Let p represent the amount that Skye can spend on other stationery. Solve for p.
14 When breeding certain types of fish it is recommended that the number of female fish is
more than double the number of male fish.
a
b
Write the inequality for the recommended relationship. Let f be the number of female fish
and m be the number of male fish.
State whether the following combinations align with the recommendation:
i f = 10, m = 8 ii f = 23, m = 9
iii f = 13, m = 10 iv f = 7, m = 4
15 James is saving up to buy a laptop that is selling for $550. He has $410 in his bank account
and expects a nice sum of money for his birthday next month.
a
b
Write the inequality that models the situation in which James can afford the laptop. Let x
represents the amount he is to receive for his birthday.
Plot the solution to the inequality on a numberline.
16 To get a grade of C, Uther must obtain a total score of at least 300 over his four exams. So
far he has taken the first three exams and achieved scores of 68, 60, and 86.
a
b
Solve for x, the score on his last exam to get a C or better.
Describe the solution regarding Uther’s score.
Chapter 1 Equations and Inequalities in One Variable 33
Let’s extend our thinking
17 Ryan wants to save up enough money so that he can buy a new sports equipment set, which
costs $40.00. Ryan has $22.10 that he saved from his birthday. In order to make more
money, he plans to wash neighbors’ windows for $2 per window.
a
b
Let w be the number of windows that Ryan washes. Solve for w, correct to two decimal
places.
State whether each statement is correct. Explain your thinking.
i
ii
iii
iv
Ryan must wash more than 9 windows to be able to afford the equipment.
Ryan must wash at least 8 windows to be able to afford the equipment.
If Ryan washed 8 windows, and 95% of another window, he could afford the equipment.
The number of windows Ryan must wash to be able to afford the equipment must be
greater than or equal to 9.
18 Rochelle tried to solve the following inequality but made a mistake in her work:
Step 0: −4 − 2x > 10
Step 1: −2x > 14
Step 2: x > −7
Determine which step is incorrect and explain the error.
19 Percy tried to plot the solution to the inequality 4x + 28 ≥ −8 on a number line, however, his
answer is incorrect.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10
Identify the errors and explain how to rectify them.
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Mathspace Florida B.E.S.T - Algebra 1
1.07 Compound
inequalities
Concept summary
A compound inequality is a conjuction of two or more inequalities. The set of solutions for a
compound inequality are the values which make all of the inequalities true.
• We use “and” to indicate that a value must
satisfy both inequalities in order to be in the
solution set. For example:
−5 −4 −3 −2 −1 0 1 2 3 4 5
x < 3 and x ≥ −2
We can also write this compound inequality
more simply as −2 ≤ x < 3
• We use “or” to indicate that a value need only
satisfy at least one inequality in order to be in
the solution set. For example:
−5 −4 −3 −2 −1 0 1 2 3 4 5
x > 3 or x ≤ −2
Worked examples
Example 1
Write a compound inequality to represent the solution set shown on the number line.
−5 −4 −3 −2 −1 0 1 2 3 4 5
Approach
We want to describe the compound inequality then write the solution to the compound inequality
algebraically.
Description: x is less than −1 or x is greater than or equal to 2
Solution
Compound inequality: x < −1 or x ≥ 2
Chapter 1 Equations and Inequalities in One Variable 35
Example 2
Find the solution set of the compound inequality x ≥ −5 and x < 3 using set-builder notation and
plot the solution on a number line.
Approach
1. Plot x ≥ −5 on a number line.
2. Plot x < 3 on a number line.
3. Find the solution of the compound inequality by comparing the number lines.
4. Represent the solution graphically.
5. Represent the solution using set-builder notation.
Solution
Plotting x ≥ −5:
The minimum value, −5, is included in the interval and
is represented by a filled circle.
Plotting x < 3:
The maximum value, 3, is not included in the interval
and is represented by a unfilled circle.
Finding the solution to the compound inequality:
We want the solution to both x ≥ −5 and x < 3. The solution to the compound inequality will be the
values greater than or equal to −5 and less than 3.
Solution represented graphically on a number line:
−6 −4 −2 0 2 4 6
−6 −4 −2 0 2 4 6
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10
Solution represented algebraically using set-builder notation: {x ∈ | − 5 ≤ x < 3}
Reflection
We can check our solution( s ) are correct by taking a number in our solution set and seeing if
it works in both inequalities.
For example, we can see that 0 falls within the region shown on the number line,
and −5 ≤ 0 < 3 as required.
Example 3
The formula for converting temperatures from Celsius to Fahrenheit is:
During a recent year, the average temperatures in Tampa, Florida ranged from 59° to 95° Fahrenheit.
Write a compound inequality to solve for the corresponding range of values of C, the temperature
in Florida in degrees Celcius.
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Mathspace Florida B.E.S.T - Algebra 1
Approach
The degrees in Fahrenheit must be between 59 and 95 inclusive, so the minimum is 59 and
the maximum is 95. The middle part of our compound inequality is going to be the formula for
converting temperatures.
Once we have the compound inequality we can solve it.
Solution
Subtract 32 from each section of the inequality
Multiply each section of the inequality by 5
Divide each section of the inequality by 9
The average temperatures in Tampa ranged from 15° to 35° Celcius.
Reflection
When solving a compound inequality, whatever is done to one part of the inequality must be done
to all parts.
What do you remember?
1 For each pair of inequalities, plot the solution to the compound inequality that is Inequality 1
and Inequality 2 on a number line.
a Inequality 1: x ≥ 2 b Inequality 1: x > 3
Inequality 2: x ≤ 7 Inequality 2: x ≤ 7
2 For each pair of inequalities, plot the solution to the compound inequality that is Inequality 1
or Inequality 2 on a number line.
a Inequality 1: x ≤ 3 b Inequality 1: x < 8
Inequality 2: x > 8 Inequality 2: x > 3
3 For the following compound inequalities:
i
ii
iii
Rewrite the compound inequality as a pair of inequalities joined by either and or or.
Plot the solution to the compound inequality on a number line.
Write the solution in interval notation.
a −5 < x < 8 b −7 < x ≤ 5
4 Which of the following sets of inequalities represents the solution for x in 1 ≥ x − 4 > −6?
A x > −2 and x ≥ 5 B x > −2 or x ≥ 5
C x > −2 and x ≤ 5 D x > −2 or x ≤ 5
Chapter 1 Equations and Inequalities in One Variable 37
5 Freshwater temperature can affect the development of various fish species including salmon.
The optimum temperature for salmon ranges from 33°F to 68°F.
Which of the following sets of inequalities represent the temperatures where salmon will
thrive?
A T > 33 or T < 68 B T > 33 and T < 68
C T < 33 and T > 68 D T < 33 or T > 68
6 Which of the following number lines represents the solution for m + 2 < −5 or m − 2 ≥ −5?
A
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
B
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
C
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
D
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
7 Which of the following number lines represents the solution for 9 − 2p > −6 and 12 < 3p?
A
3 4 5 6 7 8 9
B
3 4 5 6 7 8 9
C
3 4 5 6 7 8 9
D
3 4 5 6 7 8 9
Let’s practice
8 For each pair of inequalities, state the solution to the compound inequality that is Inequality
1 and Inequality 2.
a Inequality 1: −3x + 2 ≤ −13 b Inequality 1: 4x + 3 ≥ −17
Inequality 2: −4x − 2 ≥ 26 Inequality 2: 3x − 4 < 5
9 For each pair of inequalities, state the solution to the compound inequality that is Inequality
1 or Inequality 2.
a Inequality 1: 5x − 7 < −22 b Inequality 1: −6x + 8 < 26
Inequality 2: −6x + 8 < −28 Inequality 2: 5x − 4 ≤ 26
10 For the following compound inequalities:
i Solve for x. ii Write the solution in interval notation.
a 16 < 3x + 7 ≤ 37 b −52 < −5x + 3 ≤ −22
c d 8 < 2x − 5( x − 7 ) < 41
e
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Mathspace Florida B.E.S.T - Algebra 1
11 For each of the following scenarios, state whether the compound inequality would use and
or or:
a
Francesco solves a compound inequality and expresses the solution in interval notation
as ( −∞, ∞ ).
b Zenaida solves a compound inequality and find that there are no possible values for x.
12 The velocity, in feet per second, of a tennis ball t seconds after it is projected directly upward
is v = 85 − 32t.
Solve for the range of time t at which the ball will be faster than 37 ft/sec but slower than
69 ft/sec.
13 The formula for converting temperatures from Fahrenheit to Celsius is
During a recent year, the temperatures in Moscow ranged from −20°C to 30°C.
Solve for the corresponding range of values of F.
14 To get a final grade of B, a student must have an average mark on five tests that is greater
than or equal to 80 and less than 90. Maria’s grades on the first four tests were 96, 87, 78
and 94.
Solve for the range of grades x that she could score on the fifth test that would result in her
getting a final grade of B.
Let’s extend our thinking
15 Dora solved a compound inequality and expressed their solution as the interval ( 5, 13].
a
b
Rewrite this solution as a pair of inequalities joined by either and or or.
Write a compound inequality which Dora could have solved, using at least two steps, to
produce this solution.
16 Carlton tried to solve the following compound inequality, but made a mistake in his work:
Step 0: −8 ≤ 10 − 3x < 4
Step 1: −18 ≤
−3x < −6
Step 2: 6 ≤ 3x < 18
Step 3: 2 ≤ x < 6
Determine which step is incorrect and explain the error.
17 Skyler tried to plot the solution to the compound inequality 4x − 1 ≤ 11 and
on a number line, however their answer is incorrect.
−5 −4 −3 −2 −1 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Identify the error( s ) and explain how to rectify them.
Chapter 1 Equations and Inequalities in One Variable 39
1.08 Honors: Absolute
value inequalities
Concept summary
The absolute value of a number is a measure of the size of a number, and is equal to its
distance from zero (0), which is always a non-negative value. Absolute value is sometimes
called “magnitude”.
An absolute value inequality is an inequality containing the absolute value of one more
variable expressions.
Inequality
A mathematical relation that compares two non-equivalent expressions
Solutions to absolute value inequalities usually involve multiple inequalities joined by one of the
keywords “and” or “or”. Solutions with two overlapping regions joined by “and” can be rewritten as
a single compound inequality:
| x | ≥ 1 has solutions of x ≤ −1 or x ≥ 1 or in interval notation (−∞, −1] ∪ [−1, ∞).
| x | < 3 has solutions of x > −3 and x < 3 which is equivalent to −3 < x < 3 or in interval
notation (−3, 3).
In general, for an algebraic expression p(x) and k > 0, we have:
• | p(x)| < k can be written as −k < p(x) < k
• | p(x)| ≤ k can be written as −k ≤ p(x) ≤ k
• | p(x)| > k can be written as p(x) < −k or p(x) > k
• | p(x)| ≥ k can be written as p(x) ≤ −k or p(x) ≥ k
Solutions to absolute value inequalities can also be represented graphically using number lines:
−4 −3 −2 −1 0 1 2 3 4
| x | ≥ 1
−4 −3 −2 −1 0 1 2 3 4
| x | < 3
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Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
Consider the inequality | x | > 2:
a Represent the inequality | x | > 2 on a number line.
Approach
This inequality represents values of x which are “more than 2 units away from 0”. To plot this,
we will need to use two regions. Also note that this inequality doesn’t include the endpoints,
so we will use unfilled points to show this.
Solution
−5 −4 −3 −2 −1 0 1 2 3 4 5
Reflection
We can give a quick check of the answer by thinking about whether this is an “and”-type inequality
or an “or”-type inequality.
In this case, the inequality | x | > 2 has a solution of x < −2 or x > 2. Our solution on the numberline
has two distinct parts, which matches what we expect for an “or”-type inequality.
b Rewrite the solution to | x | > 2 in interval notation.
Approach
We can use either the number line or x < −2 or x > 2 to help write this in interval notation.
Solution
The x < −2 part can be written as (−∞, −2).
The x > 2 part can be written as (−2, −∞).
Since this is an “or”, not “and”, we will find the union of these two sets to give:
(−∞, −2) ∪ (−2, −∞)
Example 2
Consider the inequality |4x − 5| ≤ 3.
a Solve the inequality for x. Express your solution using interval notation.
Chapter 1 Equations and Inequalities in One Variable 41
Approach
In order to solve this inequality, it will be easier to first remove the absolute value by rewriting the
inequality as a compound inequality. We can then solve as normal.
Solution
Rewrite as a compound inequality
Add 5 to each part
Divide throughout by 4
So the solutions are “all values of x between
notation as the interval
and 2 inclusive”. We can express this using interval
b Represent the solution set on a number line.
Approach
The solution set for this inequality consists of a single interval, so it will only need one region on
the numberline. The endpoints are also included this time, which we represent using filled points.
Solution
−4 −3 −2 −1 0 1 2 3 4
Reflection
We can give a quick check of the answer by thinking about whether this is an “and”-type inequality
or an “or”-type inequality.
In this case, we found the solution to the inequality |4x − 5| ≤ 3 to be the compound inequality
≤ x ≤ 2. A compound inequality is a way of writing an “and”-type inequality, which matches the
fact that the numberline has one region between two endpoints.
Example 3
Write an absolute value inequality to represent the set of “all real numbers x which are at least
5 units away from 12”.
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Mathspace Florida B.E.S.T - Algebra 1
Approach
We want to break down the wording of the set into its key parts. These key parts are:
• at least
• away from 12
• 5 units away
Each of these key parts tells us information about the form of the inequality.
Solution
Let’s interpret these key parts:
• “At least” is another way of saying “greater than or equal to”, which tells us the inequality
symbol that we want to use.
• “Away from 12” means that we want to compare the distance between x and 12. We can
represent this as |x − 12|.
• “5 units away from” means that the absolute value will be compared to 5.
Putting this all together, we get “the distance between x and 12 is greater than or equal to 5”,
which we can represent as the absolute value inequality
| x − 12 | ≥ 5
What do you remember?
1 Rewrite the following inequalities without using absolute values:
a | x | < 2 b | x | ≥ 5
c | x | > 13 d | x | ≤ k, for a positive value of k
2 Represent the following inequalities on a number line:
a | x | > 6 b | x | ≠ 3 c | x | < 7
d | x | ≤ 9 e | x | ≥ 5 f | x | > −5
3 Rewrite the following as absolute value inequalities:
a −7 ≤ x ≤ 7 b x < −8 or x > 8 c −0.6 ≤ 2x + 3 ≤ 0.6
4 Solve the following inequalities:
a | x − 7| ≥ 2 b | x − 4| ≤ 9
c | 2x + 3 | ≤ 5 d 2 + | x | ≥ 9
e 4 | x | − 5 < 19 f |11 − 2x | > 5
g 3 | x − 7 | − 12 ≥ 0 h 2 |4.5 − x | > x
i | 3x + 4 | ≥ x + 8 j
Chapter 1 Equations and Inequalities in One Variable 43
5 For each of the following inequalities:
i Express the solution using interval notation.
ii Represent the solution on the number line.
a | x | ≤ 4 b | x | > 1 c | x − 5 | ≥ 2 d | 2x | < 10
e 3 | x − 1| ≥ 9 f g |1 + 4x | < 5 h | 5x + 4| + 3 > 2
Let’s practice
6 Write an absolute value inequality that represents each of the following:
a All real numbers x that are less than 8 units away from 0.
b All real numbers x that are more than 8 units away from 0.
c All real numbers x that are at least 2 units away from 8.
d All real numbers x that are at most 5 units away from −2.
7 In a certain company, the measured thickness, m, of a helicopter blade must not differ from
the standard, s, by more than 0.17 millimeters. The manufacturing engineer expresses this
as the inequality | m − s | ≤ 0.17.
Find the range of values that m can take if s is 17.92 millimeters.
8 Cell phone cases have dimension requirements to ensure the phone will fit properly in the
case. The manufacturing engineer has written a specification that the new length, n, of the
case can differ from the previous length, p, by at most 0.04 centimeters.
Find the range of values for the new length of a cell phone case if the previous length was
18.9 centimeters.
9 If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be
heads. A coin is deemed to be unfair if h, the number of outcomes that result in heads,
satisfies
a
b
Solve the inequality.
State the largest and smallest integer values for the number of heads outcomes that
could occur without the coin being labeled as unfair.
10 For each of the following, write two distinct absolute value inequalities that have the
given solution:
a −2 < x < 2 b x ≤ −9 or x ≥ 9
c 1 ≤ x ≤ 7 d x < −4 or x > 0
e
g
f
−5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5
h
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 0 1 2 3 4 5 6 7 8 9 10
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
11 Use the graph of f (x) = | 2x − 4 | and g (x) = 8
to solve the following equation
and inequalities:
a | 2x − 4 | = 8
b | 2x − 4 | ≤ 8
c | 2x − 4 | > 8
14
12
10
8
6
4
2
−8 −6 −4 −2
−2
−4
y
2 4 6 8 10
x
12 Toya claims that −5 < x < −1.4 is the solution to 3 | x + 2 | > 2x + 1. Explain why her answer
is incorrect.
13 Determine the solution(s) to the inequality | 2x − 3 | ≤ 1 − x. Explain your answer.
14 The speedometer in Najah’s car is displaying 35 mph. They know that the speedometer
is accurate to within 10% of the actual speed.
a
b
Write an absolute value inequality to represent the situation.
The road Najah is driving down has a speed limit of 40 mph. Is it possible for Najah to be
driving over the speed limit?
15 Floyd measures the distance between two cities on a map, which has a scale factor of
1 : 50 000. He determines that the cities are approximately 16 inches apart, with a possible
error of measurement of up to half an inch.
a
b
If x represents the actual distance between the two cities, write an absolute value
inequality to represent the situation.
The actual distance between the two cities is 12.9 miles (to one decimal place).
Determine if Floyd’s measurement was accurate.
16 If x represents temperature in Miami, Florida, describe a possible interpretation of
the inequality | x − 84 | ≤ 7.
Chapter 1 Equations and Inequalities in One Variable 45
2
Equations and
Inequalities in
Two Variables
Chapter outline
2.01 Slope-intercept form 48
2.02 Standard form 60
2.03 Point-slope and connecting forms 68
2.04 Parallel and perpendicular lines 78
2.05 Linear inequalities in two variables 84
2.01 Slope-intercept form
Concept summary
The key features of a linear relationship help us to draw its graph, given an equation or table of
values, or to write the equation given a graph.
Graph
A diagram showing the relationship between two things
on a coordinate plane
y
x
Table of values
Numeric information arranged in columns and rows
The slope-intercept form of a line is:
y = mx + b
m The slope of the line
b The y-intercept of the line
A benefit of slope-intercept form is that we can easily identify two key features from the equation.
Rate of change
The ratio of change in one quantity to the corresponding change in another quantity
Slope
The ratio of change in the vertical direction (y-direction) to
change in the horizontal direction (x-direction).
To interpret the slope, it can be helpful to look at the units.
y
x
48
Mathspace Florida B.E.S.T - Algebra 1
y-intercept
A point where a line or graph intersects the y-axis.
The value of x is 0 at this point, so it often represents
the initial value or flat fee.
y
x
x-intercept
A point where a line or graph intersects the x-axis.
The value of y is 0 at this point.
y
x
Domain constraint
A limitation or restriction of the possible x-values, usually written as an equation, inequality,
or in set-builder notation. The constraint often comes from a context or scenario.
Example: Inequality notation: −2 ≤ x < 3 Set-builder notation: {x ∈ ¡ | − 2 ≤ x < 3}
Worked examples
Example 1
Draw the graph of the line y = −2x + 5 using the slope and y-intercept.
Approach
We will:
1. Identify the slope and y-intercept from the equation
2. Plot the y-intercept
3. Use the rise and run to plot another point
4. Draw the graph of the line
Chapter 2 Equations and Inequalities in Two Variables 49
Solution
The y-intercept is (0, 5).
The slope is m = −2, so we can have:
rise = −2 and run = 1
This means that going down 2 units and right by
1 unit gives another point on the line.
7
6
5
4
3
2
y
y = −2x + 5
1
−4 −3 −2 −1 1 2 3 4
−1
x
Reflection
Sometimes we will see this equation written as y = 5 − 2x, but this does not change the graph.
Example 2
Find the equation of a line that has the same slope as
y = −7x − 9.
and the same y-intercept as
Approach
We will identify the slope of
y = mx + b.
and the y-intercept of y = −7x − 9 and then substitute into
Solution
The slope of
The y-intercept of y = −7x − 9 gives us b = −9.
Substituting in these values, we get:
Reflection
A common error would be to say that 2 is the slope of
because it is the first number
after the equals sign. Remember that the slope is the coefficient of the x-term.
50
Mathspace Florida B.E.S.T - Algebra 1
Example 3
A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of
water. The table of values shows the linear relationship of the amount of water remaining in the
tub, y, after x minutes.
Time in minutes (x) 0 1 2 3
Water remaining in gallons (y) 30 28 26 24
a Determine the linear equation in slope-intercept form that represents this situation.
Approach
We can pick two points to calculate the rate of change for the slope. Then we can recognize that
the y-intercept is given in the table of values.
Solution
Find the slope using the values (0, 30) and (1, 28):
Notice that the initial value, or y-intercept is given in the table as (0, 30).
The equation that represents this situation is y = −2x + 30.
Reflection
If we had not noticed that the y-intercept was given, then we could have substituted in any pair of
values for x and y, and solved for b.
b Draw the graph of this linear relationship with the domain constraint of {x Î ¡ | 0 ≤ x ≤ 15}.
Solution
From the table, the point at x = 0 is (0, 30).
For the point at x = 15, we need to substitute to find the y value.
y = −2x + 30
y = −2 × 15 + 30
y = 0
Chapter 2 Equations and Inequalities in Two Variables 51
The point at x = 15 is (15, 0)
y
30
25
20
15
10
5
x
5
10 15
What do you remember?
1 For each of the following linear equations:
i State the value of m, the slope. ii State the value of b, the y-intercept.
a y = 6x + 4 b y = −3x − 8
2 For each of the following graphs:
i Determine the slope of the line. ii Determine the y-intercept of the line.
iii
Determine the equation of the line.
a
5
y
b
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
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Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
3 Consider the following linear equations:
• Line 1: y = 2 + 6x
• Line 2: y = 5x + 6
• Line 3: y = 6x
State the three lines that have the same slope.
• Line 4:
• Line 5: y = 5 − 6x
• Line 6: y = 6x − 2
4 For each of the following lines, state whether the following lines have a y-intercept or not.
If the line does have a y-intercept, state the coordinates of it.
a y = 2 b 5x = 4y c y = −4x + 1 d x = 1
5 Suppose the Line L 1 has the equation
a Simplify L 1 to the form y = mx + b.
b
c
State the slope of the line.
Find the y-value of the y-intercept of the line.
6 Determine whether each line has the same slope as y = 6x + 3.
a y = 6x b y = 3x + 3 c 12x − 2y = 3 d 18x + 3y = 0
7 Find the equation of the lines matching the following descriptions:
a A line that has the same slope as y = 2 − 4x and the same y-intercept as y = −7x − 3
b A line whose slope is and goes through the point (0, 5)
c A line that is horizontal and goes through the point (9, −2)
d A line that is vertical and goes through the point (11, −9)
8 Graph the following lines on a coordinate plane:
a The line that has a y-intercept of −4 and
whose slope is
b The line y = 2x + 4
c The line y = −2x + 4
6
4
y
d
e
The line
The line x = −3
−6 −4 −2
2
2 4 6
x
9 Select the linear equation that could
represent the following graph:
A y = 2x - 3 B y = −2x - 3
−2
−4
C
D
−6
Chapter 2 Equations and Inequalities in Two Variables 53
10 Select the linear equation that could represent the following table of values:
x −2 −1 0 1 2 3
y −1 1 3 5 7 9
A y = 2x + 3 B y = −2x − 3 C D
11 A linear function is defined by the equation y = 4x − 3.
a
Select the graph that represents the linear function.
A
5
y
B
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
C
5
y
D
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
b
c
Identify the domain of the function.
Identify the range of the function.
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Mathspace Florida B.E.S.T - Algebra 1
12 Consider the following line:
a
Complete the table of values.
5
4
y
b
c
x -1 0 1 2
y
State the values of the slope, m and the
y-intercept, b.
Write the linear equation expressing the
relationship between x and y.
−5 −4 −3 −2
3
2
1
−1
−1
−2
−3
1 2 3 4 5
x
−4
−5
13 Consider the following line:
a
State the values of the slope, m and the
y-intercept b.
5
4
y
b
Write the linear equation expressing the
relationship between x and y.
3
2
1
x
−5 −4 −3 −2
−1
−1
1 2 3 4 5
−2
−3
−4
−5
14 For the following lines passing through the given two points:
i Find the slope of the line.
ii Find the equation of the line in the form y = mx + b.
a (0, 5) and (2, 9) b (−3, 9) and (4, 2)
c (−6, 8) and (6, 0) d (−7, 3) and (−7, −2)
Chapter 2 Equations and Inequalities in Two Variables 55
15 A 12-inch candle decreases in length by 0.4 in/s after it was lit.
a
Select the graph of the linear function that could describe the relationship between the
height of the candle in inches, y, and the number of minutes after it was lit, x.
A
y
B
y
10
10
5
5
x
x
10 20
10 20
C
y
D
y
10
10
5
5
x
x
10 20
10 20
b
c
Identify the slope of the function.
Identify the y-intercept of the function.
16 A tank containing 220 gallons of water is being drained at the rate of 0.1 gallons per minute.
Select the equation of the linear function that could describe the relationship between the
gallons of water, y, remaining in the tank and the time in minutes, x.
A y = 220x + 0.1 B y = 0.1x + 220 C y = −200x + 0.1 D y = −0.1x + 220
17 Consider the following table of values for a linear function:
x -2 -1 0 1 2
y -6 -3 0 3 6
a
Select the graph of the linear function that could represent the table of values.
56
Mathspace Florida B.E.S.T - Algebra 1
A
4
y
B
4
y
3
3
2
2
1
x
1
x
−4 −3 −2
−1
−1
1 2 3 4
−4 −3 −2
−1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
C
4
y
D
4
y
3
3
2
2
1
x
1
x
−4 −3 −2
−1
−1
1 2 3 4
−4 −3 −2
−1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
b
c
Identify the slope of the function.
Identify the y-intercept of the function.
18 A race car starts the race with 60 gallons of fuel. From there, it uses fuel at a rate of 1 gallon
per minute.
a
Complete the table of values.
Number of minutes passed (x) 0 5 10 15 20 60
Amount of fuel left in tank (y)
b
c
Write the equation relating the number of minutes passed (x) and the amount of fuel left
in the tank (y).
Graph the function.
19 A cleaner charges an initial fee of $110 plus $20 per hour. Write an equation to represent
the total amount charged by the cleaner, y, as a function of the number of hours worked, x.
Chapter 2 Equations and Inequalities in Two Variables 57
Let’s extend our thinking
20 Neil is running a 30-mile ultramarathon.
a
b
Write an equation to represent the
distance Neil has left to run, y, in terms
of the number of hours since the start, x.
The domain has been restricted to
x Î [0, 6]. Explain why this restriction
has been put in place.
40
35
30
25
20
15
10
Distance remaining
5
1
Time (hours)
2 3 4 5 6 7 8
21 A train leaves Station A and moves along a straight railroad towards Station B, which is
5280 ft ahead. The average speed of the train during the travel is 88 ft/min.
a
A
Select the graph of the linear function that describes the relationship between the
remaining distance of travel in feet, y, and the time of travel in minutes, x.
y
B
y
5000
5000
4000
4000
3000
3000
2000
2000
C
1000
y
10 20 30 40 50
x
D
1000
y
10 20 30 40 50
x
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
x
x
10 20 30 40 50
10 20 30 40 50
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Mathspace Florida B.E.S.T - Algebra 1
b
Identify the domain restriction of the function.
22 The table shows the water level of a well that is being emptied at a constant rate with a
pump where the time is measured in minutes and the water level in feet.
Time (x) 3 7 10
Water level (y) 56.7 48.3 42
a
b
c
d
e
Write an equation in the form y = mx + b to represent the water level (y), as a function of
the minutes passed (x).
Interpret the slope of this linear function.
Interpret the y-intercept of this linear function.
Find the water level after 18 minutes.
State an appropriate domain constraint on this linear function. Justify your answer.
23 A carpenter charges an initial fee of $250 plus $35 per hour for labor.
a
b
c
d
Write an equation to represent the total amount charged, y, by the carpenter as a
function of the number of hours worked, x.
Interpret the slope of this linear function.
Interpret the y-intercept of this linear function.
Peta has budgeted $2500 to have a ramp built at the community center to increase
accessibilty. The carpenter works 8 hour days and estimated that the project would take
one and a half days. He also priced the materials at $1800.
Determine if Peta has budgeted enough for the ramp. Justify your answer.
24 A diver starts at the surface of the water and begins to descend below the surface at a
constant rate. The table shows the depth below the surface of the diver over the first
5 minutes.
Number of minutes passed (x) (min) 0 1 2 3 4
Depth of diver (y) (ft) 0 1.63 3.26 4.89 6.52
a
b
c
d
e
f
State the rate of change with units of ft/min.
Write an equation for the relationship between the number of minutes passed (x) and
the depth below the surface (y) of the diver.
Determine the depth of the diver after 6 minutes.
Find how long the diver takes to reach 11.41 ft beneath the surface.
The diver has two air tanks, each of which last for 45 minutes. One is used for going
down and one is used for returning to the surface. Determine an appropriate domain
restriction for the function from part (b).
Assuming the diver comes back up at the same rate they go down, state the equation
for the linear function which could describe the return to surface including the domain
restriction. Graph the full scenario.
Chapter 2 Equations and Inequalities in Two Variables 59
2.02 Standard form
Concept summary
The standard form of a linear relationship is a way of writing the equation with all of the variables
on one side:
Ax + By = C
A is a non-negative integer
B, C are integers
A, B are not both 0
To draw the graph from standard form, we can find and plot the x and y-intercepts or convert to
slope-intercept form.
The standard form is helpful when looking at scenarios that have a mixture of two different items.
When we identify the intercepts in a mixture scenario, it can be interpreted as the amount of that
item when none of the other item is included.
Worked examples
Example 1
Draw the graph of the line 5x − 3y = −15 on the coordinate plane.
Approach
We can find both the x and y-intercept and then graph the line using those two points.
Solution
Find the x-intercept by setting y = 0 and solving:
5x − 3y = −15 State the given equation
5x − 3(0) = −15 Set y = 0
5x = −15 Simplify
x = −3 Divide both sides by 5
Find the y-intercept by setting x = 0 and solving:
5x − 3y = −15 State the given equation
5(0) − 3y = −15 Set x = 0
−3y = −15 Simplify
y = 5 Divide both sides by −3
y
7
6
5
4
3
5x − 3y = −15
2
1
−5 −4 −3 −2 −1 1 2 3
−1
x
60
Mathspace Florida B.E.S.T - Algebra 1
Reflection
We could have also converted to slope-intercept and graphed using the slope and y-intercept.
We can decide unless the question specifies how to do it. If the x and y-intercept happened to
be same point, (0, 0), then we need to find another point by substituting another x-value into
the equation and solving for y.
Example 2
Darius wants to buy a mix of garlic and chipotle powders for seasoning tacos. Garlic powder costs
$4/lb. Chipotle powder costs $7/lb. Darius spends exactly $14 on spices.
Let x represent the amount of garlic powder Darius buys and let y represent the amount of chipotle
powder he buys. Write an equation to represent this scenario.
Approach
The cost of each spice will be the cost per pound multiplied by the amount purchased.
The total cost will be the sum of the two spice costs and is equal to $14.
Solution
Cost of just the garlic: 4 × x
Cost of just the chipotle: 7 × y
Total cost: 4x + 7y = 14
What do you remember?
1 State the standard form of a linear equation.
2 State which of the following linear equations are in standard form:
a 2x − 5y = 0 b x = 10
c
d
Chapter 2 Equations and Inequalities in Two Variables 61
Let’s practice
3 Consider the following line:
a
b
State the y-intercept.
Find the slope, m, of the line.
4
3
y
c
d
Find the equation of the line in the form
y = mx + b.
Rewrite the equation of the line in
standard form.
4 Consider the following table of values:
2
1
−4 −3 −2 −1
−1
−2
1 2 3 4
x
x 1 2 3 4 5
−3
y −1 −9 −17 −25 −33
−4
a
b
Write the relationship between x and y in
slope-intercept form.
Rewrite the relationship in standard form.
5 For each of the following equations:
i Calculate the x-value of the x-intercept of the line.
ii Calculate y-value of the y-intercept of the line.
iii Draw the graph of the equation of the line on the coordinate plane.
a 5x − 3y = −15 b 2x + y = 10
6 For each of the following equations:
i Rewrite the equation in slope-intercept form.
ii Draw the graph of the equation of the line on the coordinate plane.
a 4x − 3y = −12 b 2x + y = 10
7 Draw the graph of each of the following linear equations:
a 2x = 6 b 6y = −3 c 3x − 2y = −12 d 5x + 4y = 40
8 Select the linear equation that could represent the following table of values.
x 2 4 6 8 10
y 4 5 6 7 8
A 2x − y = 6 B 2x − y = −6 C x − 2y = 6 D x − 2y = −6
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Mathspace Florida B.E.S.T - Algebra 1
9 Select the linear equation that could represent
the following graph.
A 5x + y = 20 B 5x − y = 20
C x + 5y = 20 D x − 5y = 20
5
y
−5
5
x
−5
10 Consider the following table of values:
x 0 3 6 9 12
y −5 −4.25 −3.5 −2.75 −2
a
Select the graph that could represent the table of values.
A
y
B
y
5
10
5
x
x
−5
5
−10
−5 5 10
−5
−5
−10
C
y
D
y
10
10
5
5
x
x
−10
−5 5 10
−10
−5 5 10
−5
−5
−10
−10
b Identify the slope of the line. c Identify the y-intercept of the line.
Chapter 2 Equations and Inequalities in Two Variables 63
11 A linear function is defined by the equation 2x − 5y = −15.
a
Select the graph that represents the equation.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
b Identify the domain of the function. c Identify the range of the function.
12 On a particular day, the air temperature on a mountain decreases by 5.5°F for every 1000 ft
increase in elavation. The temparature at the base of the mountain is 55°F. Select the linear
equation that could describe the relationship between the height x in ft above the base of
the mountain and the temperature y in °F.
A 1000x − 5.5y = 5500 B 1000x + 5.5y = 5500
C 5.5x − 1000y = 302.5 D 5.5x + 2y = 302.5
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Mathspace Florida B.E.S.T - Algebra 1
13 A white oak tree stands 20 ft tall in 2009. It has been determined that the tree grows by
a rate of 1.25 ft per year.
a Select the graph that could represent the relationship between the height y in feet and
the number of years x since 2009.
A
y
B
y
25
25
20
20
15
15
10
10
5
5
x
x
1
2 3 4 5
1
2 3 4 5
C
y
D
y
25
25
20
20
15
15
10
10
1 2 3 4 5
1 2 3 4 5
b Identify the slope of the line. c Identify the y-intercept of the line.
14 Marvin went to the grocery store to buy x lb of granulated sugar for $0.60/lb and y lb of
powdered sugar for $0.90/lb. The total cost of all the sugar is $12.
a Select the graph of the linear function that relates the number of pounds of granulated
sugar x and the number of pounds of powdered sugar y.
A
21
5
y
x
B
21
5
y
x
18
18
15
15
12
12
9
9
6
6
3
x
3
x
3
6 9 12 15 18 21
3
6 9 12 15 18 21
Chapter 2 Equations and Inequalities in Two Variables 65
C
21
y
D
21
y
18
18
15
15
12
12
9
9
6
6
3
x
3
x
3
6 9 12 15 18 21
3
6 9 12 15 18 21
b
Identify the domain restriction of the function.
15 Salvina needs to buy a mix of red and green apples. She is going to spend $15 in total.
Green apples cost $1.50 lb and red apples cost $3 lb.
a
b
c
d
Write a linear equation in standard form
to represent the scenario. Use the
variables x for green and y for red.
Calculate and interpret the x-intercept of
the linear equation from part (a).
Determine the number of green
apples she could buy if she bought
2 red apples.
State the domain of the linear function.
16 The line kx + 3y = 15 passes through the point (13, −8).
a Find the value of k.
b
c
Solve for the x-value of the x-intercept of the line.
Solve for the y-value of the y-intercept of the line.
Let’s extend our thinking
17 Write an expression for the slope, m, of the line represented by the equation Ax + By = C.
State any restrictions. Explain how this expression might be helpful.
18 Describe a scenario where using the standard form of a linear equation is more useful than
the slope-intercept form.
19 Compare and contrast the standard form and slope-intercept form of a linear equation.
Include how they are similar, how they are different, and the benefits of each form.
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Mathspace Florida B.E.S.T - Algebra 1
20 Effie is a entomologist and is currently studying mosquitos and spiders. She knows that
mosquitos have six legs and spiders have eight legs. In her lab, she has a mix of mosquitos
and spiders. Between all the bugs, there is a total of 240 legs.
a
b
c
d
Write an equation to represent this scenario.
Draw a graph of the linear equation.
State and describe the domain of the equation.
Explain whether or not every point on the line represents a possible solution.
21 Nakisha is selling the baby outfits her daughter has outgrown and is buying second-hand
toddler outfits. She is able to sell each baby outfit for $3 and she buys each toddler outfit for
$5. In the end, she wants to make a profit of exactly $30.
Profit = revenue − cost
Revenue is the money earned
Cost is the money spent
a
b
c
d
Write an equation to represent the scenario, using x for the number of baby outfits sold,
y for the number of toddler outfits bought, and the profit of exactly $30.
Calculate and interpret the x-intercept.
Nakisha has 25 baby outfits available to sell and needs 15 toddler outfits.
Determine if she is able to make her desired profit of $30.
Using the additional information from part (c), state and interpret the domain restriction.
Chapter 2 Equations and Inequalities in Two Variables 67
2.03 Point-slope and
connecting forms
Concept summary
When we are given the coordinates of a point on the line and the slope of that line, then pointslope
form can be used to state the equation of the line.
Coordinates can be given as an ordered pair, in a table of values, read from a graph, or described
in a scenario. The slope can be stated as a value, calculated from two points, read from a graph, or
given as a rate of change in a scenario.
Point-slope form of a linear relationship:
y − y 1 = m(x − x 1 )
m The slope of the line
x 1 The x-coordinate of the given point
y 1 The y-coordinate of the given point
Coordinate
A number used to locate a point on a number line. One of the numbers in an ordered pair,
or triple, that locates a point on a coordinate plane or in coordinate space, respectively.
Ordered pair
A point on a graph written as (x, y). Also called coordinates, or a coordinate pair.
We can also find the equation in point-slope form when given the coordinates of two points on
the line, by first finding the slope of the line. We now have three forms of expressing the same
equation, and each provides us with useful information about the line formed. Point-slope form is
useful when we know, or want to know:
• The slope of the line
• A point on the line
Slope-intercept form is useful when we know, or want to know:
• The slope of the line
• The y-intercept of the line
Standard form is useful when we know, or want to know:
• Both intercepts of the line
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Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
For each of the following equations, determine if they are in point-slope form, standard form,
or slope-intercept form. If they are not in standard form, convert them to standard form.
a 2x + 8y = 10
Solution
This is in standard form as all of the variables are on one side and the constant is on the other side.
b y = 4x − 10
Solution
This is in slope-intercept form.
y = 4x − 10
−4x + y = −10
4x − y = 10
4x − y = 10 is in standard form.
State the given equation
Subtract 4x from both sides
Divide both sides by −1
c
Solution
This is in point-slope form.
State the given equation
5y − 15 = −2(x + 7) Multiply both sides by 5
5y − 15 = −2x − 14
2x + 5y = 1
2x + 5y = 1 is in standard form.
Reflection
Distribute the multiplication
Add 2x and 15 to both sides
There are more steps to convert from point-slope form to standard form, than from point-slope
form to slope-intercept form.
Chapter 2 Equations and Inequalities in Two Variables 69
d x = 8
Solution
Yes, this is in standard form.
Reflection
This is a vertical line.
Example 2
A line passes through the two points (−3, 7) and (2, −3). Write the equation of the line in pointslope
form.
Approach
We will first find the slope of the line using the two points. Then we will pick one of the points to
substitute into the point-slope equation.
Solution
Find the slope of the line:
Formula for slope of a line
Substitute in the points
Simplify the subtraction
= −2 Simplify the fraction
Find the equation of the line in point-slope form:
Reflection
y − y 1 = m(x − x 1 )
y − 7 = −2(x − (−3))
y − 7 = −2(x + 3)
Formula for point-slope form
Substitute known values into the formula
Simplify
We can confirm we have correctly written this in point-slope form: y − y 1 = m (x − x 1 ), as we can
see the coordinates (−3, 7) and a slope of −2 are represented correctly. This is easier to see in the
previous line: y − 7 = −2(x − (−3))
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Mathspace Florida B.E.S.T - Algebra 1
Example 3
A carpenter charges for a day’s work using the given equation, where y is the cost and x is the
number of hours worked:
y − 125 = 50(x − 2)
a Draw the graph of the linear equation from point-slope form.
Approach
We need two points to plot a line. We can read from the equation that one point on the line is
(2, 125). To get another point, we can use the slope from the given point or substitute in an x-value
in the domain and solve for y.
Solution
The given point is (2, 125) Since the slope is 50, we can go up 50 units and right 1 unit from our
given point to plot another point on the line.
225
y
200
175
150
125
100
75
50
25
1 2 3 4
x
b The carpenter works a maximum of 10 hours per day. State the domain constraints for this
scenario.
Solution
Domain: 0 £ x £ 10
Chapter 2 Equations and Inequalities in Two Variables 71
What do you remember?
1 State the point-slope form of a linear equation.
2 For each of the following equations, state the form they are written in:
a y = −4x + 5 b y − 1 = 5(x − 4) c 5x − 3y = 8
3 Write the equation of the following lines in point-slope form:
a A line passing through the point (7, 1) has a slope of 4.
b A line passing through the point (4, 0) has a slope of −5.
c A line passing through the point (1, 6) has a slope of 0.
4 For each of the following lines:
i Find the slope, m, of the line.
ii Write the equation of the line in point-slope form.
a A line passes through the two points (7, 6) and (9, 12).
b A line passes through the two points (4, −8) and (2, 0).
5 For each of the following table of values:
i Find the slope, m, of the line represented in the table.
ii Write the equation of the line in point-slope form.
a
x 1 2 3 4
b
x 1 2 3 4
y −3 2 7 12
y 9 7 5 3
Let’s practice
6 Rewrite the following linear equations into slope-intercept form:
a 2x + 3y = 12 b y − 2 = 3(x − 7)
7 Rewrite the following linear equations into standard form:
a b y + 7 = −3(x + 2)
c d y − y 1 = m(x − x 1 )
8 Select the linear equation that could represent the following table of values.
x -3 -2 -1 0
y 2 -2
A B C D
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Mathspace Florida B.E.S.T - Algebra 1
9 Select the linear equation that could
represent the following graph.
A y + 3 = 2(x + 3)
B y − 7 = 2(x − 2)
C y − 7 = 2(x − 5)
D y − 4 = 2(x − 5)
8
6
4
2
y
x
−8 −6 −4 −2
−2
2 4 6 8
−4
−6
−8
10 State the equation of the given linear
function in point-slope form.
5
4
3
2
y
1
x
−5 −4 −3 −2 −1
−1
1 2 3
−2
−3
11 Sketch the graph of the lines on the coordinate plane given the following equations:
a y + 8 = 3(x + 2) b y + 1 = −2(x + 1)
c
d
12 Consider the following table of values:
x 0 5 10 15 20
y 7 4 1 −2 −5
Chapter 2 Equations and Inequalities in Two Variables 73
a
Select the graph that could represent the table of values.
A
y
B
y
10
10
5
5
x
x
−10
−5 5 10
−10
−5 5 10
−5
−5
−10
−10
C
y
D
y
10
10
5
5
x
x
−10
−5 5 10
−10
−5 5 10
−5
−5
−10
−10
b
c
Identify the slope of the line.
Identify the y-intercept of the line.
13 A linear function is defined by the equation
a
Select the graph that could represent the equation.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
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Mathspace Florida B.E.S.T - Algebra 1
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
b
c
Identify the domain of the function.
Identify the range of the function.
14 The graph of a linear function passes thorugh the point (1, 6) and has a slope of
a
Select the graph that could represent the function.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
b
c
Identify the domain of the function.
Identify the y-intercept of the function.
Chapter 2 Equations and Inequalities in Two Variables 75
15 A race car uses fuel at the rate of 0.9 gallons per minute. After running for 12 minutes,
the car has 48 gallons of fuel left in the tank. Select the linear equation that could describe
the relationship between the number of minutes x the car is running and the number of
gallons of fuel y left in the tank.
A y + 12 = 0.9(x + 48) B y + 48 = 0.9(x + 12)
C y − 12 = 0.9(x − 48) D y − 48 = 0.9(x − 12)
16 A plumber charges a fixed amount for a call out fee plus $30 per hour of work. He charged a
total of $80 for 2 hours of work. He cannot work for more than 8 hours in any given day.
a
A
Select the graph of the linear function that relates the number of hours of work x and
the total charge y for the day.
y
B
y
80
80
60
60
40
40
20
20
x
x
C
y
1
2 3 4
D
y
1
2 3 4
80
80
60
60
40
40
20
20
x
x
b
1 2 3 4
Identify the domain of the function.
1
2 3 4
17 Sally receives an order to make key chains for an upcoming festival. She is told to make
at least 50 and at most 100 key chains, and will be paid $8 for each key chain she makes.
Sally makes 80 key chains and is paid $665 altogether.
a
b
c
Let x represent the number of key chains Sally makes and y represent the amount she
is paid. Write an equation in point-slope form relating x and y.
Convert the equation from part (a) to slope-interept form.
Determine the flat fee that Sally was paid in addition to the $8 per key chain that
she received.
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Mathspace Florida B.E.S.T - Algebra 1
d
State the domain of this linear relationship
18 Gary is flying in a plane which is moving directly away from his home at a constant speed.
At 2 hours into the flight, he checked that he was 1042 miles from his home. He checks
again at 7 hours into the flight to find that he is now 3542 miles from his home.
a
Find the speed of the plane, in miles per hour.
b Write an equation in point-slope form relating the number of hours flying, x,
to the distance Gary is from his home, y, in miles.
c
If the destination of Gary’s flight is 5542 miles from his home, find the duration of
the flight.
Let’s extend our thinking
19 Create a scenario where you would choose to use each of the following forms and explain
why you would use that form over the others:
a Point-slope form, (y − y 1 ) = m(x − x 1 )
b
c
Slope-intercept form, y = mx + b
Standard form, Ax + By = C
20 Rewrite the point-slope form of a linear equation in slope-intercept form. Draw connections
between the two forms.
21 A cup is placed under a faucet which is leaking at a rate of 2 oz per hour. After 4 hours,
the cup contains 15 oz of water.
a
b
c
d
Write an equation in point-slope form relating the volume in the cup, y, to the amount of
time that has passed in hours, x.
Now, rewrite the equation into slope-intercept form.
State and interpret the y-intercept of the linear equation.
Suppose that the cup has a maximum capacity of 18 oz. Determine how long it can be
under the faucet before it starts to over flow. Now, state the domain and range.
22 Paige is planning an event and the
brochure for the caterer at the venue
shows the following graph for the meal
costs.
a
b
c
State and interpret the domain of the
linear function.
Paige concludes that the unit cost per
person is $16.25. Determine if this is
correct. If so, explain how to calculate
this value. If not, explain the error and
state the correct value.
State the equation of the linear
function in point-slope form.
1900 Cost ($)
1800
1700 (100, 1625)
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500 (10, 500)
400
300
200
100
People
10 20 30 40 50 60 70 80 90 100 110
Chapter 2 Equations and Inequalities in Two Variables 77
2.04 Parallel and
perpendicular lines
Concept summary
Knowing that two lines are parallel or perpendicular to each other gives us information about the
relationship between their slopes and graphs.
Parallel lines
Two or more lines that have the same slope and never intersect.
Distinct vertical lines are parallel to each other.
y
x
Perpendicular lines
Two lines that intersect at right angles. Their slopes are opposite
reciprocals. Horizontal and vertical lines are also perpendicular to
each other.
y
x
Opposite reciprocals
Two numbers whose product is −1.
Example:
To determine if two lines are parallel or perpendicular, we will need to find the slopes of the two
lines. We often do this by rearranging to y = mx + b and comparing the values of m.
If m 1 = m 2 , then the two lines are parallel.
If m 1 × m 2 = −1, then the two lines are perpendicular.
If the two lines are parallel, then m 1 = m 2 .
If the two lines are perpendicular, then m 1 × m 2 = −1.
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Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
For each of the following pairs of lines, determine whether they are parallel, perpendicular, or neither.
a y = −3x + 7 and y = 3x − 4
Approach
We need to identify the slopes of both lines to see if they are the same, opposite reciprocals,
or neither.
Solution
For y = −3x + 7, the slope is m = −3.
For y = 3x − 4, the slope is m = 3.
The slopes are not the same. The slopes are not opposite reciprocals.
The lines y = −3x + 7 and y = 3x − 4 are neither parallel nor perpendicular.
b 3x − 5y = 15 and 6x − 10y = 60
Approach
We need to identify the slopes of both lines to see if they are the same, opposite reciprocals, or
neither. First, we want to rearrange both lines into the form y = mx + b to identify their slopes.
Solution
For 3x − 5y = 15,
3x − 5y = 15
−5y = −3x + 15
Given equation
Subtract 3x from both sides
Divide both sides by −5
Simplify
The slope for
For 6x − 10y = 60,
6x − 10y = 60
−10y = −6x + 60
Given equation
Subtract 6x from both sides
Divide both sides by −10
Simplify
Chapter 2 Equations and Inequalities in Two Variables 79
The slope for
The slopes are the same, so lines 3x − 5y = 15 and 6x − 10y = 60 are parallel.
Reflection
Notice that in standard form, Ax + By = C, the values of A and B for 3x − 5y = 15 and
6x − 10y = 60 were in the same ratio, while C was different.
c y = −2x + 3 and x − 2y = 10
Approach
We need to identify the slopes of both lines to see if they are the same, opposite reciprocals,
or neither. We will need to rearrange x − 2y = 10 to slope-intercept form.
Solution
For y = −2x + 3, the slope is m = −2.
For x − 2y = 10,
x − 2y = 10
−2y = −x + 10
Given equation
Subtract x from both sides
Divide both sides by −2
Simplify
The slope of
The slopes are opposite reciprocals, so lines y = −2x + 3 and x − 2y = 10 are perpendicular.
Example 2
Find the equation of the line, L 1 that is parallel to the line
(0, 4). Give your answer in slope-intercept form.
and goes through the point
Approach
The slope of parallel lines are the same, so we can to identify the slope from the line
The point given happens to be the y-intercept. We can substitute the slope in for m and the
y-intecept in for b in y = mx + b.
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Mathspace Florida B.E.S.T - Algebra 1
Solution
The slope of
The y-intercept is b = 4.
The equation of the line
Reflection
If we had been given a point that was not the y-intercept, then we would have needed to substitute
the point in for x and y and solved for b. We could also have used point-slope form and rearranged
to slope-intercept form.
What do you remember?
1 Determine whether the following lines are parallel, perpendicular, or neither:
a
10
y
b
10
y
8
8
6
6
4
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−4
−6
−6
−8
−8
−10
−10
c
10
y
8
6
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−6
−8
−10
2 State the relationship between the slopes of:
a Parallel lines b Perpendicular lines
Chapter 2 Equations and Inequalities in Two Variables 81
3 Find the slope, m, of a line that is:
a Parallel to a line which has a slope of 4
b Parallel to
c Perpendicular to a line which has a slope of −9
d Perpendicular to y = −5x + 6
e Perpendicular to y = 10 − x
4 Determine if the following lines are parallel to or not:
a b 4x − 6y = 37 c 3x + 2y = −13 d
5 Determine if the following lines are perpendicular to y = 4x − 3 or not:
a b 4x + 16y = 96 c x + 4y = −3 d y = −4x − 3
Let’s practice
6 For each of the following pair of lines, determine whether they are parallel, perpendicular, or
neither.
a y = 3x − 7 and y = 3x + 4 b x − y = 6 and x − y = −6.
c d 2x + 3y = 5 and −3x − 2y = 12
7 Find the equation the line that is:
a Parallel to the line and passes through the point (0, −2)
b Parallel to x = 1 and passes through (−4, 6)
c Parallel to the x-axis and passes through (−8, 5)
d Parallel to the y-axis and passes through (−7, 5)
8 Find the equation the line that is:
a Perpendicular to x = 4 and passes through (−7, 1)
b Perpendicular to y = 2x − 1 and passes through the point (2, 3)
c Perpendicular to the x-axis and passes through the point (10, 3)
d Perpendicular to the y-axis and passes through (1, −7)
9 Consider a line that passes through points A(4, 3) and B(8, −10).
a
b
Find the slope, m, of the line.
Find the equation of a line that passes through (−1, 2) and is parallel to the line passing
through A(4, 3) and B(8, −10). Express the equation in slope-intercept form.
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Mathspace Florida B.E.S.T - Algebra 1
10 Consider a line that passes through A(2, −4) and B(3, 9).
a
b
Find the slope, m, of the line.
Find the equation of a line that has a y-intercept of 5 and is perpendicular to the line that
goes through A(2, −4) and B(3, 9). Express the equation in slope-intercept form.
11 Consider line L 1 with equation 3x − 4y − 4 = 0.
a
Write L 1 in slope-intercept form.
b Find the slope of a line, L 2 , that is perpendicular to L 1 .
c Find the equation of L 2 in standard form, given that it passes through the point A(−6, 4).
12 Consider line L 1 with equation
Select the equation of L 2 that is parallel to L 1 and passes through the point the point (−6, 3).
A B C D
13 Consider line L 1 with equation 2x − 3y = 21.
Select the equation of L 2 that is parallel to L 1 and passes through the point the point (0, −6).
A 2x − 3y = −12 B 3x + 2y = −18 C 2x − 3y = 18 D 3x + 2y = −12
14 Consider line L 1 with equation
Select the equation of L 2 that is perpendicular to L 1 and passes through the point the point
(5, −7).
A y = −2x + 3 B y = −2x − 9 C D
15 Consider line L 1 with equation 32x + 24y = 21. Select the equation of L 2 that is
perpendicular to L 1 and passes through the point the point (−9, 11).
A B C D
Let’s extend our thinking
16 Consider the lines 2x − 7y = 2 and mx − 35y = −7.
Given that the two lines are parallel, determine the value of m.
17 Explain why two lines with the standard forms ax + by = c 1 and bx − ay = c 2
are perpendicular.
18 Ammon and Minh work part-time at the same pizzeria, both earning $360 per week. Both
Ammon and Minh are saving up, so they don’t spend any of the money they earn. Ammon
currently has a bank balance of $212 while Minh has $275. If we sketch linear graphs for Ammon
and Minh’s bank balances over time, can we expect to get two parallel lines? Justify your answer.
19 Suppose that the lines px − 4y = 7 and qx + 5y = −14 are perpendicular.
Write an equation relating the values of p and q.
Chapter 2 Equations and Inequalities in Two Variables 83
2.05 Linear inequalities
in two variables
Concept summary
If a linear inequality involves two variables, we can represent it as a region on a coordinate plane
rather than an interval on a number line.
Linear inequality in two variables
An inequality whose solution is a set of ordered pairs represented by a region of
the coordinate plane on one side of a boundary line.
Example: 2y > 4 − 3x
Boundary line
A line which divides the coordinate plane into two regions.
A boundary line of a linear inequality is solid if it is included
in the solution set, and dashed if it is not.
y
x
Depending on the inequality sign, the boundary line will be solid or dashed, and region shaded will
be above or below the boundary line.
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
y > x
y ≥ x
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Mathspace Florida B.E.S.T - Algebra 1
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
y < x
y ≤ x
Worked examples
Example 1
Write the inequality that describes the region shaded on the given coordinate plane.
4
y
3
2
1
x
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
−4
Approach
First, we want to determine the equation of the boundary line. Then we need to determine what
inequality sign to use.
Chapter 2 Equations and Inequalities in Two Variables 85
Solution
To find the equation of the boundary line in
the form y = mx + b:
• The line crosses the y-axis at (0, −3)
• The rise is 3
• The run is 1
So, b = −3 and m = 3.
Now we have y = 3x − 3 as the boundary line.
−4 −3 −2 −1
4
3
2
1
−1
y
1 2 3 4
x
−2
−3
(0, −3)
−4
To determine the inequality:
• The boundary line is solid, so it is ≤ or ≥.
4
y
• Any point in the shaded region would be above
the boundary line.
This means that for every x-value, the y-value for any
point in the region is greater than the y-value of the
point on the line.
So our linear inequality is y ≥ 3x − 3.
3
2
1
−4 −3 −2 −1
−1
−2
1 2 3 4
x
−3
−4
Reflection
It is a good idea to check our answer by substituting in a point in the region to make sure it satisfies
the inequality. For example, the origin, (0, 0), is in the region, so should satisfy the inequality.
y ≥ 3x − 3
0 ≥ 3(0) − 3
0 ≥ −3
It does satisfy the inequality, so we have selected the correct inequality sign.
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Mathspace Florida B.E.S.T - Algebra 1
Example 2
A pick-up truck has a maximum weight capacity of 3000 pounds. Each box of oranges weighs
8 pounds and each box of grapefruits weighs 12 pounds. Let x represents the number of boxes of
oranges in the truck. Let y represents the number of boxes of grapefruit in the truck.
a Write an inequality to represent the number of boxes of oranges and grapefruit that can be in
the truck.
Solution
The weight of all the orange boxes is the product of the weight of one box, 8, and the number
of boxes, x.
Weight of orange boxes: 8x
The weight of all the grapefruit boxes is the product of the weight of one box, 12, and the number
of boxes, y.
Weight of grapefruit boxes: 12y
The total weight is the sum of the weights of orange boxes and grapefruit boxes.
Total weight: 8x + 12y
The truck can carry at most 3000 pounds, so we get our inequality: 8x + 12y ≤ 3000
b Create a graph of the region containing the points corresponding to all the different numbers
of orange and grapefruit boxes that can be loaded into the truck.
Approach
We will graph the region representing 8x + 12y ≤ 3000.
We need to identify our boundary line, then graph it. Then, we need to decide which side of
the boundary line to shade.
Solution
The boundary line is 8x + 12y = 3000. This is a line in standard form, so we can graph it by finding
the intercepts. Find the x-intercept by setting y = 0 and solving:
8x + 12y = 3000
8x − 12(0) = 3000 Setting y = 0
8x = 3000
Stating the given equation
Simplifying
x = 375 Dividing both sides by 8
Find the y-intercept by setting x = 0 and solving:
8x + 12y = 3000
8(0) + 12y = 3000 Setting x = 0
12y = 3000
Stating the given equation
Simplifying
y = 250 Dividing both sides by 12
Chapter 2 Equations and Inequalities in Two Variables 87
The boundary line will be solid because we have ≤ as the sign.
We will shade below the line as the point (0, 0) satisfies the inequality.
300
275
250
225
200
175
150
125
100
75
50
25
y
8x + 12y ≤ 3000
x
25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400
Reflection
The actual possible solutions are the positive integer coordinates since we are working with whole
fruits and have both x ≥ 0 and y ≥ 0. This region shows where all those possible solutions can be.
What do you remember?
1 State the equation of the boundary line for each inequality.
a y ≥ 4x − 3 b c y ≤ −x + 1 d x > 12
2 For each of the following inequalities, state whether they are represented using a solid or
dashed line on a graph:
a y ≤ 4x − 4 b y < 3x − 5 c y ≥ 2x − 3 d y > 5
3 For each of the following shaded regions, determine whether the following points lie in the
shaded region:
i (2, 4) ii (−3, 2) iii (−1, 6) iv (3, −3)
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Mathspace Florida B.E.S.T - Algebra 1
a
6
5
4
3
2
1
−3 −2 −1 −1
−2
−3
−4
−5
−6
y
1 2 3 4 5 6 7
x
b
6
5
4
3
2
1
−7 −6 −5 −4 −3 −2 −1
−1
1 2 3
−2
−3
−4
−5
−6
y
x
4 For each of the following inequalities
i Determine if the origin (0, 0) satisfies the inequality.
ii Graph the region that satisfies the inequality.
a x > 3 b y ≤ −2 c x ≥ −1
Let’s practice
5 For each of the following inequalities:
i Determine whether the point (2, 3) satisifies the inequality or not.
ii Graph the region that satisfies the inequality.
a y ≤ 3x + 5 b c y ≥ 2x + 4 d y < 3x + 2
6 For each of the following inequalities:
i State the coordinates of the x- and y-intercepts of the boundary line.
ii Graph the region that satisfies the inequality.
a 3x + 2y < 12 b 5x + 2y ≥ 10
7 Select the graph that represents the inequality 2x − 7y > 14.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
Chapter 2 Equations and Inequalities in Two Variables 89
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
8 Select the graph that represents the inequality y ≤ −3x + 4.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
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Mathspace Florida B.E.S.T - Algebra 1
9 Select the graph that represents the inequality y > −x − 3.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
10 Select the graph that represents the inequality y ≥ 3x + 1.
A
y
B
y
5
5
x
x
−5
5
−5
5
−5
−5
Chapter 2 Equations and Inequalities in Two Variables 91
C
y
D
y
5
5
x
x
−5
5
−5
5
−5
−5
11 Select the inequality that describes
the following region:
A
8
6
y
B
C
D
4
2
−8 −6 −4 −2
−2
−4
2 4 6 8
x
−6
−8
12 Select the inequality that describes
the following region:
A
8
6
y
B
C
D
4
2
−8 −6 −4 −2
−2
−4
2 4 6 8
x
−6
−8
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Mathspace Florida B.E.S.T - Algebra 1
13 Select the inequality that describes
the following region:
A
8
6
y
4
B
C
2
−8 −6 −4 −2
−2
2 4 6 8
x
D
−4
−6
−8
14 Select the inequality that describes
the following region:
A
8
6
y
B
C
D
4
2
−8 −6 −4 −2
−2
−4
2 4 6 8
x
−6
−8
15 Write the inequality that describes the following shaded regions:
a
5
y
b
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
Chapter 2 Equations and Inequalities in Two Variables 93
c
5
4
3
2
1
−5 −4 −3 −2 −1 −1
−2
−3
−4
−5
y
1 2 3 4 5
x
d
6
5
4
3
2
1
−4 −3 −2 −1
−1
−2
−3
−4
y
1 2 3 4 5 6
x
16 Applicants for a particular job are asked to sit a numeracy test and verbal reasoning
test. Successful applicants must obtain a minimum combined score of 43 for both tests.
Write an inequality relating the applicant’s score on the numeracy test, x, and the verbal
reasoning test, y.
17 Throughout university, Luigi works as a mentor, getting paid $10 per hour, and as a barista
getting paid $13 per hour. The number of hours he works in each job can vary from week
to week, and he needs to be able to at least cover his weekly expenses of $260. Write an
inequality relating the number of hours he works as a mentor, x, and barista, y.
18 Lisa is being careful with her spending so she can later purchase a car. She has allocated no
more than $450 each month for both travel expenses and eating out. On average, each time
she eats out costs $18, and each travel journey costs $5. Write an inequality relating the
number of times she eats out, x, and the number of travel journeys each month, y and then
solve for y.
19 In the new basketball season, Mario is looking to beat his personal best of scoring
102 points in total for the whole season (this does not include points from fouls).
Write an inequality relating the number of ‘two pointers’, x, and the number of ‘three pointers’,
y, he scores throughout the new season and then solve for y.
Let’s extend our thinking
20 Oprah is thinking about how to use her last arcade tokens. A game of Table Tennis costs
2 tokens and a game of Frog Days costs 5 tokens. She has a total of 30 tokens left.
a
b
c
d
Write an inequality relating the number of times Oprah plays Table Tennis, x and the
number of times she plays Frog Days, y.
Graph the region containing the points corresponding to all the different ways Oprah
could spend her remaining tokens across the two games.
Interpret the value of the x-intercept of the boundary line in the given context.
Interpret the value of the y-intercept of the boundary line in the given context.
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Mathspace Florida B.E.S.T - Algebra 1
21 Fiona has set aside $13.20 in her shopping budget for fruit this week. Currently, cantal are
on sale for $1.65 each, while watermelon are on sale for $2.20 each.
a
b
c
Write an inequality relating the number of cantaloupe that Fiona buys, x, and the number
of watermelon that Fiona buys, y.
Graph the inequality on the number plane, and shade the solution set.
Determine whether Fiona can buy the following combinations of fruits. Justify your
answer.
i 4 cantaloupes and 2 watermelons ii 4.9 cantaloupes and 0.1 watermelons
iii 2 cantaloupes and 7 watermelons iv 6 cantaloupes and 5 watermelons
22 A book seller makes a profit of $9 on every book sold online and $5 on every book sold
in store. The company wants to make a profit of at least $270 a day selling books through
online and in-store sales.
a
b
c
Write an inequality relating the number of books sold online, x, and the number of books
sold in store, y on one day.
Graph the region that satifies the inequality.
Given the context, is the ordered pair (−5, 100) a valid solution to the inequality? Justify
your answer.
23 Sean is thinking about how to use his remaining spending money for snacks. A pack of dried
fruit costs $6 and a bag of mixed nuts costs $4. He has $48 remaining.
a
b
c
Write an inequality relating the number of packs of dried fruit, x and the number of bags
of mixed nuts, y that Sean buys.
Graph the region containing the points corresponding to all the different ways Sean
could spend his remaining spending money on these snacks.
Supposing that Sean does not need to spend all of it, how many different ways can Sean
spend his remaining money on some combination of dried fruit and mixed nuts?
Chapter 2 Equations and Inequalities in Two Variables 95
3
Systems of
Linear Equations
and Inequalities
Chapter outline
3.01 Solving systems of equations by graphing 98
3.02 Solving systems of equations by substitution 106
3.03 Solving systems of equations by elimination 111
3.04 Systems of linear inequalities 116
3.01 Solving systems of
equations by graphing
Concept summary
A system of equations is a set of equations which have the same variables.
A solution to a system of equations is any set of values of all variables in that system which is a
solution to each equation in the system.
A solution can also be thought of graphically as the point(s) of intersection of the graphs of the
equations (the points in common to all graphs):
When two lines are parallel and distinct, they have
no points of intersection. The corresponding system
of equations has no solutions.
y
x
When two lines are identical, they intersect at
every point. The corresponding system of equations
has infinitely many solutions.
y
x
98
Mathspace Florida B.E.S.T - Algebra 1
When two lines are not parallel, they have exactly
one point of intersection. The corresponding system
of equations has one solution.
y
x
A solution to a system of equations in a given context is said to be viable if the solution makes
sense in the context, and non-viable if it does not make sense within the context, even if it would
otherwise be algebraically valid.
Worked examples
Example 1
Consider the system of two equations shown in the graph:
10
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
y
2 4 6 8 10
x
Chapter 3 Systems of Linear Equations and Inequalities 99
a How many solutions does this system of equations have?
Approach
The solution(s) to a system of equations can be represented graphically as their point(s)
of intersection.
Solution
This system has one point of intersection and therefore one solution.
Reflection
This system consists of two linear equations. Is it possible for a pair of linear equations to have
more than one solution? Is it possible for a system of two linear equations to have no solutions?
b Determine the solution to the system of equations as an ordered pair (x, y).
Solution
The point of intersection occurs at (−4, −2).
Example 2
Tyson is saving money in order to purchase a new smart-phone for $800 when the latest model
is released. He currently has $350 saved up, and is able to put away $100 each month.
a Write a system of equations to represent the situation.
Approach
To write a system of equations, we will need to define some variables. Let’s choose y to represent
an amount of money (in dollars), and x to represent the number of months that have passed.
Solution
Using these variables, the amount of money Tyson has saved over time can be represented by
y = 350 + 100x. The price of the smart-phone can be represented by y = 800.
100
Mathspace Florida B.E.S.T - Algebra 1
b Sketch the two lines representing these equations on the coordinate plane.
Approach
All of the values involved in the question are multiples of $50, so we can use this for the scale of
the y-axis. Also, both x and y only make sense for positive values in this context, so we only need
to think about the first quadrant.
Solution
y
900
800
700
600
500
400
300
200
100
−1 1 2 3
4 5 6 7 8
x
c If the new phone is to be released in 5 months time, determine if Tyson will be able to afford
it on release.
Solution
The point of intersection on the graph occurs at (4.5, 800), meaning that in 4.5 months Tyson will
have saved $800. Therefore Tyson will have saved enough money before the phone is released.
Reflection
While the model equation of Tyson’s savings is linear, in reality he probably puts money away once
per month or once per week depending on how often he gets paid.
So although the point of intersection is at x = 4.5 months, Tyson might not actually reach $800 in
savings until the end of the 5th month.
Chapter 3 Systems of Linear Equations and Inequalities 101
What do you remember?
1 Select the solution to the system of equations
on the given graph.
A (−1, −7)
4
y
B (−6, −7)
C (−7, −6)
D (1, −6)
E
No solution
−8
−6
2
−4 −2 2 4
−2
x
−4
−6
−8
2 Select the solution to the system of equations
on the given graph.
A (−3, 0)
B (−3, 5)
C (3, 5)
D
E
Infinitely many solutions
No solution
8
6
4
2
−8 −6 −4 −2 2 4 6 8
−2
−4
−6
−8
y
x
3 Solve each system of equations
graphed below:
a
4
2
−4 −2
−2
−4
−6
−8
−10
−12
−14
−16
−18
y
2 4 6 8 10 12
x
b
6 78 5
2 34 1
x
−5 −4 −3 −2 −1 −1 1 2 3 4 5
−2
−3
−4
−5
−6
−7
−8
y
102
Mathspace Florida B.E.S.T - Algebra 1
c
y
d
y
6 7 9
8
1 2 3 4 5 6 7 8 9
2 34 5
1
−3 −2 −1 −1
−2
−3
−4
−5
−6
−7
−8
−9
x
−9−8 −7−6−5−4−3−2 −1 −1
−2
−3
−4
−5
−6
−7
−8
−9
6 7 9
8
1 2 3 4
2 34 5
1
x
4 For each of the following system of equations:
i Sketch the two lines representing the equations on the coordinate plane.
ii Solve the system of equations graphically.
a b c d
5 Consider the system of equations:
a
b
Sketch the two lines representing these equations on the coordinate plane.
How many solutions does this system of equations have?
6 Consider the system of equations:
Select the solution to this system of equations.
A (0, 0) B (−4, 0) C (−4, 8)
D No solution E Infinitely many solutions
7 Consider the system of equations:
Select the solution to this system of equations.
A (0, −5) B (0, −8) C (1, −5)
D (−8, −5) E No solution
Chapter 3 Systems of Linear Equations and Inequalities 103
Let’s practice
8 Sarah is determining the dimensions of a rectangular frame. The length, l, of the frame is
5 cm more than the width, w. The perimeter of the frame is equal to 50 cm.
Sarah came up with the following equations to solve for the length and width of the frame:
a
Select the correct dimensions of the rectangular frame.
A l = 10 cm, w = 15 cm B l = 15 cm, w = 10 cm
C l = 15 cm, w = 20 cm D l = 20 cm, w = 15 cm
b
Is the solution viable in terms of the context?
9 The sum of two mystery numbers is 4. The difference of the two numbers is −2.
a
b
c
Write a system of equations to represent the situation.
Sketch the two lines representing these equations on the coordinate plane.
Use the graph to determine the two mystery numbers.
10 Rochelle and Mohamad are sister and brother. Rochelle’s age is 11 more than 4 times the
age of Mohamad. The sum of their ages is 21.
a
b
c
Write a system of equations to represent the situation.
Sketch the two lines representing these equations on the coordinate plane.
How old are Rochelle and Mohamad?
11 Lauren and Nicole are working on an assignment together. They have broken down the work
into 9 parts of the same size. Lauren works at 2 times the speed of Nicole but has 9 pieces
of work to do for another subject.
a
b
c
Write a system of equations to represent the situation.
Sketch the two lines representing these equations on the coordinate plane.
How many parts of the assignment did each student do?
12 Katrina has to read a book for class and only has 8 days before the test. So far, Katrina has
read 90 pages. She is planning on reading 15 pages each day between now and the test.
The book is 215 pages long.
a
b
c
Write a system of equations to represent the situation.
Sketch the two lines representing these equations on the coordinate plane.
Does Katrina finish her book before the test?
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
13 Write a scenario to represent the system of equations and its solution. Explain what the
solution to the system means in terms of the scenario.
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
y
2 4 6 8 10 12 14 16 18 20 22 24
x
14 Two equations, y 1 and y 2 represent the growth of two different house plants over time. Use
the graph of y 1 and y 2 to support the claim that the two plants will never reach the same
height on the same day.
y 2
y 1
25
20
15
10
−40 −30 −20 −10 10 20 30
−5
5
−10
−15
−20
−25
y
x
15 Describe a situation where it would be unfeasible to solve a system of equations
by graphing.
Chapter 3 Systems of Linear Equations and Inequalities 105
3.02 Solving systems of
equations by substitution
Concept summary
When at least one equation in a system of equations can be solved quickly for one variable,
the system can be solved efficiently by using substitution.
Substitution Method
A method of solving a system of equations by replacing a variable in one equation with an
equivalent expression from another equation
If after solving a system of equations the result is always true, independent of the variables,
then the system has infinitely many solutions. If the result is always false, independent of
the variables, then the system has no solutions.
Worked examples
Example 1
Solve the following system of equations using the substitution method:
Approach
We first want to number our equations to make it easier to work with.
1 y = x + 11
2 y = 3x + 19
Since both equations already have y isolated, we can start by substituting equation 1 into
equation 2 to eliminate y from the equation. We can then solve for x, and substitute this value
back into one of the equations to solve for y.
106
Mathspace Florida B.E.S.T - Algebra 1
Solution
x + 11 = 3x + 19 Substitute equation 1 into equation 2
11 = 2x + 19 Subtract x from both sides
−8 = 2x
Subtract 19 from both sides
−4 = x Divide both sides by 2
y = −4 + 11 Substitute back into equation 1
y = 7
Evaluate the addition
So the solution to the system of equations is x = −4, y = 7.
Example 2
A mother is currently 7 times as old as her son. In 3 years time, she will be 5 times as old as him.
a Write a system of equations for this scenario, where y represents the mother’s current age
and x represents the current age of her son.
Solution
1 y = 7x
2 y + 3 = 5(x + 3)
b Solve the system of equations for their ages.
Approach
Since 1 is already in the form y =
, we can substitute 1 into 2 without rearrangement.
Solution
7x + 3 = 5(x + 3) Substitute equation 1 into equation 2
7x + 3 = 5x + 15
Distribute the multiplication
2x + 3 = 15
Subtract 5x from both sides
2x = 12
Subtract 3 from both sides
x = 6 Divide both sides by 2
y = 7(6) Substitute back into equation 1
y = 42
Evaluate the multiplication
So the mother’s age is 42 and her son’s age is 6.
Chapter 3 Systems of Linear Equations and Inequalities 107
c Does the solution make sense in terms of the context? Explain your answer.
Solution
Yes. 6 years earlier, when her son was born, the mother would have been 36 years old.
It makes sense that the mother is older than the son, and both values are valid ages.
What do you remember?
1 Consider the system of equations:
Select the solution to this system of equations.
A (−1, −5) B (1, 5) C (−5, −1) D (5, 1)
2 Consider the system of equations:
Select the solution to this system of equations.
A (−8, −2) B (−2, 2) C (−2, −8) D (2, −2)
3 Solve the following systems of equations by substitution:
a b c
d e f
4 The total cost of 5 rulers and 3 books is $15.30. If the cost of a ruler is a and a book is
$2.70 more expensive, find a.
5 A rectangular garden bed has a perimeter of 13.8 m. The length of the garden bed is 2.5 m
longer than the width.
a Find the width of the garden bed. b Find the length of the garden bed.
6 Judy and Jorge both walk from their houses to the bus stop every morning. Judy walks
1.1 mi further, and together they walk 3.1 mi. Find the distance that Jorge walks.
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Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
7 At a bookstore, Stella purchased a novel and a magazine for a total cost of $59.
The price of the novel, n, is $10 less than twice the price of the magazine, m.
This situation can be represented by the following system of equations:
a
Select the correct price of the novel and the magazine.
A m = $23, n = $36 B m = $26, n = $33
C m = $33, n = $26 D m = $36, n = $23
b
Is the solution viable in terms of the context?
8 Luke bought some fresh produce from the farmers’ market. He picked up 4 oranges and
5 plums. The total cost of Luke’s fruit was $13.33. Valentina went to the same shop and
bought 1 orange and 1 plum. The total cost of Valentina’s fruit was $2.89.
a
b
Write a system of equations where f represents the price of an orange and g represents
the price of a plum.
Solve for the prices of oranges and plums at the farmers’ market.
9 A man is five times as old as his son. Four years ago the man was thirteen times as old as
his son.
a
Write a system of equations where x represents the age of the man and y represents
the age of his son.
b Solve for x and y.
c
Does the solution make sense in terms of the context? Explain your answer.
10 The number of new jobs created in Miami varies greatly each year. The number of jobs
created in 2013 was 440 000 less than double the number of jobs created in 2004.
This is equivalent to an increase of 10 000 jobs created from 2004 to 2013.
a
Write a system of equations where x represents the number of jobs in 2004 and
y represents the number of jobs in 2013.
b Solve for x and y.
c
Is the solution viable in terms of the context? Explain your answer.
11 Valentina has $2000 to invest, and wants to split it up between two accounts: Account A
earns 8% annual interest, while Account B earns 9% annual interest. Her target is to earn
$177 total interest from the two accounts in one year.
a
Write a system of equations x represents the amount invested in Account A and
y represents the amount invested in Account B.
b Solve for x and y.
c
Would you make the same investment as Valentina? Explain your answer.
Chapter 3 Systems of Linear Equations and Inequalities 109
Let’s extend our thinking
12 The function f (x) = 0.47x + 8.9 represents the US annual bottled water consumption
(in billions of gallons) and the function g(x) = −0.17x + 14.2 represents the US annual soda
consumption (in billions of gallons). For both functions, x is the number of years since 2009.
a
b
Determine the year in which the bottled water and soda consumption in the U.S is
the same.
Is the solution viable in terms of the context? Explain your answer.
13 Create a scenario for each system of equations where the solution is viable in terms of
the scenario. Explain your answer.
a b c
110
Mathspace Florida B.E.S.T - Algebra 1
3.03 Solving systems of
equations by elimination
Concept summary
If at least one pair of variables in a system of equations can be quickly made to have the same
coefficient, or already have the same coefficient, the system can be solved efficiently by using
elimination.
Elimination Method
A method of solving a system of equations by adding or subtracting the equations until only
one variable remains
Worked examples
Example 1
Solve the following system of equations by using the elimination method:
Approach
We first want to number our equations to make it easier to work with.
1 9x + y = 62
2 5x + y = 38
Since the coefficents of y are the same with the same sign, we can start by subtracting 2 from
1 to eliminate y from the equation. We can then solve for x, and substitute this value back into
one of the equations to solve for y.
Solution
(9x + y) − (5x + y) = 62 − 38 Subtract equation 2 from equation 1
4x = 24
Combine like terms
x = 6 Divide both sides by 4
Chapter 3 Systems of Linear Equations and Inequalities 111
5(6) + y = 38 Substitute back into equation 2
30 + y = 38 Evaluate the multiplication
y = 8
Subtract 30 from both sides
So the solution to the system of equations is x = 6, y = 8.
Reflection
We could also have solved this by subtracting 1 from 2 . We could also have substituted x = 6
back into equation 1 .
(5x + y) − (9x + y) = 38 − 62 Subtract equation 1 from equation 2
−4x = −24
x = 6
Combine like terms
Divide both sides by −4
9(6) + y = 62 Substitute back into equation 1
54 + y = 62 Evaluate the multiplication
y = 8
Subtract from both sides
It doesn’t matter which way we choose, but one method will usually be preferable. In this case we
didn’t have to deal with negative numbers when using the first approach.
Example 2
When comparing test results, Verna noticed that the sum of her Chemistry and English test scores
was 128, and that their difference was 16. She scored higher on her Chemistry test.
a Write a system of equations for this scenario, where x represents Verna’s Chemistry test score
and y represents her English test score.
Approach
Since we know that Verna’s Chemistry test score is higher than her English test score, we should
express the difference of the two scores as x − y to get a positive result.
Solution
1 x + y = 128
2 x − y = 16
112
Mathspace Florida B.E.S.T - Algebra 1
b Solve the system of equations to find her test scores.
Approach
Since the equations have the same coefficient of y with opposite sign, we can add the two
equations to eliminate y and solve for x first.
Solution
(x + y) + (x − y) = 128 + 16 Add equation 1 and equation 2 together
2x = 144
Combine like terms
x = 72 Divide both sides by 2
72 + y = 128 Substitute back into equation 1
y = 56
Subtract 72 from both sides
So Verna scored 72 on her Chemistry test and 56 on her English test.
Reflection
In this case, the two equations also had the same coefficient of x with the same sign. So we could
have instead approached this by subtracting the two equations and solving for y first.
c Does the solution make sense in terms of the context? Explain your answer.
Solution
Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain.
What do you remember?
1 Consider the following system of equations:
a Describe how to use multiplication to eliminate the variable y.
b Now, combine the two equations using the method you explained.
2 Consider the following system of equations:
a What value can we multiply each equation by in order to eliminate the variable y?
b Solve for x and y.
Chapter 3 Systems of Linear Equations and Inequalities 113
3 Consider the system of equations:
Select the solution to this system of equations.
A (1, 2) B (−3, 4) C (2, 1) D (4, −3)
4 Consider the system of equations:
Select the solution to this system of equations.
A (−3, 1) B (5, 9) C (1, −3) D (9, 5)
5 Solve each system of equations using the elimination method:
a b c d
e f g h
Let’s practice
6 Solve each system of equations using the elimination method:
a b c d
7 The train in an amusement park has 15 cabs that can hold a total of 72 people. Some cabs
hold 4 people while some hold 6 people.
Let x be the number of four-passenger cabs and y be the number of six-passenger cabs.
This situation can be represented by the following system of equations:
a
Select the correct number of four-passenger cabs and six-passenger cabs.
A x = −2, y = 17 B x = 3, y = 12 C x = 12, y = 3 D x = 17, y = −2
b
Is the solution viable in terms of the context?
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8 When comparing her test, results Judy noticed that the sum of her Geography test score
and Math test score was 137, and that their difference was 29. Judy scored higher on her
Geography test than her math test.
a
b
c
Write a system of equations where x represents Judy’s Geography score and y
represents her Math score.
Solve for Judy’s Geography score using the elimination method.
Now, solve for Judy’s Math score.
9 12 pens and 5 rulers cost $70 while 3 pens and 25 rulers cost $65.
a
b
c
Write a system of equations where x represents the price of a pen and y represents the
cost of a ruler.
Solve for the price of a pen using the elimination method.
Now, solve for the price of a ruler.
10 Fred spent $55.25 to purchase 9 flowers. He bought rhododendrons which cost $6.45 each
and chrysanthemums which cost $5.75 each.
a
b
If R is the number of rhododendrons and C is the number of chrysanthemums that Fred
bought, contruct two equations describing the total number of flowers bought and the
total amount spent in dollars.
Solve for the number of rhododendrons and chrysanthemums that Fred purchased.
Let’s extend our thinking
11 Consider the following system of equations:
Explain how to rewrite the system of equations so that it has integer coefficients.
12 Ricardo solved the following system of equations used to model wildlife populations in a
wildlife sanctuary, where r represents the number of rhinos and h represents the number of
hippopotami.
a
b
Explain why the solution to the system of equations is not viable.
Explain how you might interpret the solution to the system of equations to make sense in
terms of the context.
Chapter 3 Systems of Linear Equations and Inequalities 115
3.04 Systems of linear
inequalities
Concept summary
A system of inequalities is a set of inequalities which have the same variables.
The solution to a system of inequalities is the set containing any ordered pair that makes all of the
inequalities in the system true.
A solution can also be represented graphically as the region of the plane of the plane that satisfies
all inequalities in the system.
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
The solution to a system of inequalities in a given context is viable if the solution makes sense in
the context, and is non-viable if it does not make sense.
Worked examples
Example 1
Consider the following system of inequalities:
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Mathspace Florida B.E.S.T - Algebra 1
Sketch a graph of the solution set to the system of inequalities.
Solution
To sketch the system of inequalities we can first construct the boundary lines for each inequality,
namely y = 3 and y = 4x + 5. When given a strict inequality we will draw a dashed line.
When given a nonstrict inequality we will draw a solid line.
9
8
7
6
5
4
3
2
1
−4 −3 −2 −1 −1
y
1 2 3 4
x
−2
−3
−4
−5
−6
To determine which side of each inequality will be shaded, we can choose some test points that
satisfy each inequality. The test points will indicate which side of the inequality will be shaded.
For y ≤ 3 we will plot (−2, 2) and (4, 0). For y < 4x + 5 we will plot (−2, 5) and (−3, −3).
9
8
7
6
5
4
3
2
1
−4 −3 −2 −1 −1
y
1 2 3 4
x
−2
−3
−4
−5
−6
Chapter 3 Systems of Linear Equations and Inequalities 117
The region that will be shaded is the region which
satisfies both inequalities.
Using the test points we can see that the shading
will occur below the boundary line for y ≤ 3 and to
the left of y < 4x + 5.
9
8
7
6
5
4
3
2
1
−4 −3 −2 −1 −1
y
1 2 3 4
x
−2
−3
−4
−5
−6
Reflection
Since each inequality in the system was already written in terms of y, it would have been possible
to determine the direction of the shading without first plotting the test points.
With some systems of inequalities written in terms of x or in general form, however, it can be less
intuitive to know which direction to shade.
Example 2
Applicants for a particular university are asked to sit a quantitative reasoning test and verbal
reasoning test. Successful applicants must obtain a minimum score of 14 on a quantitative
reasoning test and a minimum combined score of 29 for both tests.
a Write a system of inequalities for this scenario, where x represents the quantitative reasoning
test score and y represents the verbal reasoning test score.
Approach
Since we know that the minimum accepted score for quantitative reasoning is 14, we can
represent this with an inequality showing 14 as the lowest possible solution. A minimum combined
score of 29 means the total of the two scores must sum to 29 or more.
Solution
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Mathspace Florida B.E.S.T - Algebra 1
b. Sketch a graph of the system of inequalities.
Solution
30
y
25
20
15
10
5
5 10 15 20 25 30
x
c. Does the solution (15, 22. ) make sense in terms of the context? Explain your answer.
Solution
No. This would mean that the score for the quantitative reasoning test was 15 and the verbal
reasoning test 22. . By viewing the graph we can see that this point technically satisfies both
inequalities, but a test score is typically a positive integer value or a simple fraction such as
or
and not a non-terminating decimal.
Chapter 3 Systems of Linear Equations and Inequalities 119
What do you remember?
1 Select the graph of the solution set of:
A
10
y
B
10
y
8
8
6
6
4
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−4
−6
−6
−8
−8
−10
−10
C
10
y
D
10
y
8
8
6
6
4
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−4
−6
−6
−8
−8
−10
−10
2 Select the graph of the solution set of:
A
10
y
B
10
y
8
8
6
6
4
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−4
−6
−6
−8
−8
−10
−10
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Mathspace Florida B.E.S.T - Algebra 1
C
10
y
D
10
y
8
8
6
6
4
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−4
−6
−6
−8
−8
−10
−10
3 Select the graph of the solution set of:
A
10
y
B
10
y
8
8
6
6
4
4
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−10 −8 −6 −4 −2 −2
2
2 4 6 8 10
x
−4
−4
−6
−6
−8
−8
−10
−10
C
30
y
D
30
y
24
24
18
18
12
12
−10 −8 −6 −4 −2 −6
6
2 4 6 8 10
x
−10 −8 −6 −4 −2 −6
6
2 4 6 8 10
x
−12
−12
−18
−18
−24
−24
−30
−30
Chapter 3 Systems of Linear Equations and Inequalities 121
4 Select the graph of the solution set of:
A
10
8
6
4
2
−10 −8 −6 −4 −2 −2
y
2 4 6 8 10
x
B
10
8
6
4
2
−10 −8 −6 −4 −2 −2
y
2 4 6 8 10
x
−4
−6
−8
−10
−4
−6
−8
−10
C
10
8
6
4
2
−10 −8 −6 −4 −2 −2
y
2 4 6 8 10
x
D
10
8
6
4
2
−10 −8 −6 −4 −2 −2
y
2 4 6 8 10
x
−4
−6
−8
−10
−4
−6
−8
−10
5 Graph each of the following system of inequalities on the coordinate plane:
a b c d
6 Write the system of inequalities that is represented on each graph:
a
y
12
10
8
6
4
2
−10 −8 −6 −4 −2 −2
2 4 6 8 10
−4
−6
−8
b
x
5
4
3
2
1
−5 −4 −3 −2 −1
−1
−2
−3
−4
−5
y
1 2 3 4 5
x
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Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
7 Consider the graph of
a
b
If the system was changed to
graph the new system of inequalities on
a coordinate plane.
Is (−6, −8) a solution to the system in
part (a)?
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
−12
y
2 4 6 8 10
x
8 Consider the graph of
a
b
If the system was changed to
graph the new system of inequalities on
the coordinate plane.
Is (0, 6) a solution to the system in part (a)?
9 Consider the following graphs:
Graph 1 Graph 2
y
12
10
8
6
4
−6 −4 −2
−2
−4
−6
−8
−10
2 4 6 8 10 12
x
12
10
8
6
4
−6 −4 −2
−2
−4
−6
−8
−10
y
12
10
8
6
2
−12 −10 −8 −6 −4 −2
−2
4
−4
−6
−8
y
2 4 6 8
2 4 6 8 10 12
x
x
Chapter 3 Systems of Linear Equations and Inequalities 123
Graph 1 shows the solution set to the system:
a
b
If Graph 2 shows the solution set when one of the inequalities in the system is changed,
state the new system of inequalities.
Is (3, 0) a solution to either system? Explain how you know.
10 Sheila sells popcorn to raise money during summer break. The popcorn comes in two
flavors, classic butter which costs $2 each and cheese which costs $3 each. Sheila needs to
sell at least $50 worth of popcorn with at least 8 of the cheese popcorn.
Let b be the number of butter popcorn and c be the number of cheddar popcorn.
This situation can be represented by the following system of equations:
Select all the viable solutions.
A b = 6, c = 13 B b = 10, c = 10 C b = 11, c = 9
D b = 13, c = 8 E b = 18, c = 6
11 Yreka Bakery makes two types of cookies: plain and iced. They have enough oven space
to bake 15 dozen cookies each day. Each dozen iced cookies requires 0.8 pounds of icing.
Yreka Bakery can make no more than 32 pounds of icing per day.
If x is the number of dozens of iced cookies and y is the number of dozens of plain cookies,
write a system of inequalities to represent the scenario.
12 Throughout university, Jimmy works as a barista, getting paid $10 per hour, and as a mentor
getting paid $13 per hour. The number of hours he works in each job can vary from week to
week, but he never works more than 27 hours in total each week, and he needs to be able
to at least cover his weekly expenses of $260.
If b represents the number of hours worked as a barista and m represents the number of
hours worked as a mentor, write a system of inequalities to represent the scenario.
Let’s extend our thinking
13 David’s Pizza makes two types of pizzas: Vegan and Meat Lover’s. Based on recent sales
they know they will sell at least twice as many Meat Lover’s Pizzas as they do Vegan Pizzas.
David’s Pizza can use 42 pounds of vegan cheese each day, and each Vegan Pizza requires
0.6 pounds of vegan cheese.
a
b
c
If v is the number of Vegan Pizzas and m is the number of Meat Lover’s Pizzas, write a
system of inequalities to represent the scenario.
Is (28, 70) a viable solution in terms of the scenario? Explain.
Is (25.25, 71) a viable solution in terms of the scenario? Explain.
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Mathspace Florida B.E.S.T - Algebra 1
14 In a team crossfit competition, each team is to be made up of no more than 11 people. For a
particular challenge, each team member must complete the course and the team’s total time
in completing the course must be under 26 minutes. Women take on average 2.6 minutes
and men take on average 2.5 minutes to complete the course.
a
b
c
If w is the number of women and m be the number of men on the team, write a system
of inequalities to represent the scenario.
Graph the system of inequalities.
Give an example of one viable and one non-viable solution to the number of men and
women on the team. Explain your answer for both.
15 Throughout university, Tom works as a mentor, getting paid $15 per hour, and as a fitness
instructor getting paid $18 per hour. The number of hours he works in each job can vary
from week to week, but he never works more than 29 hours in total each week, and he
needs to be able to at least cover his weekly expenses of $270.
a
b
c
d
If x represents the number of hours Tom works as a mentor and y represents the number
of hours he works as a fitness instructor, construct a system of inequalities to represent
the scenario.
Graph the system of inequalities.
If Tom works 6 hours as a mentor in one week, what is the minimum number of hours he
can work as a fitness instructor so that he can cover his expenses?
Is it possible for him to work the same number of hours in both jobs and still be able to
cover his expenses?
Chapter 3 Systems of Linear Equations and Inequalities 125
4 Functions
and Linear
relationships
Chapter outline
4.01 Evaluating functions 128
4.02 Domain and range 132
4.03 Average rate of change 140
4.04 Characteristics of functions from a graph 146
4.05 Linear relationships 153
4.06 Transforming linear functions 162
4.07 Characteristics of linear functions 172
4.08 Comparing linear and nonlinear functions 183
4.09 Linear absolute value functions 189
4.10 Transforming absolute value functions 194
4.01 Evaluating functions
Concept summary
A mathematical relation is a mapping from a set of input values, called the domain, to a set of
output values, called the range. A relation can also be described as a set of input-output pairs.
Input
The independent variable of a relation; usually the x-value
Output
The dependent variable of a relation; usually the y-value
Relations in general have no further restrictions than mapping domain elements to range elements.
By adding the restriction that each input value maps to exactly one output value, we define
a particularly useful type of relation, called a function.
Function
A relation for which each element of the domain corresponds to exactly one element of
the range
Functions are usually written using a particular notation called function notation: for a function f
when x is a member of the domain, the symbol f ( x ) denotes the corresponding member of
the range.
To evaluate a function at a point is to calculate the output value at a particular input value.
Worked examples
Example 1
Consider the equation
where x is the independent variable.
x − 3y = 15
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Mathspace Florida B.E.S.T - Algebra 1
a Rewrite the equation using function notation.
Approach
Since x is the independent variable, we want to rearrange the equation to isolate y, and then
replace y with function notation. We can choose a symbol to represent the function, such as f.
Solution
Rearranging the equation:
−3y = −x + 15
Subtract x from both sides
Divide both sides by −3
We can now rewrite the equation using function notation as
b Evaluate the function when x = 9.
Solution
Substituting x = 9 we have
What do you remember?
1 Consider the function y = 2x − 3.
a State the independent variable. b State the dependent variable.
2 For a function f, what does f ( 3 ) represent?
3 Suppose that ( 7, −6 ) is an ordered pair that satisfies the function g. Write this situation using
function notation.
4 Suppose that the value of the function f is 22 when x = 7. Write this situation using
function notation.
5 Write each of the following equations using function notation:
a y = 2x b y = −x + 7 c y = x 2 + x + 1 d y = 12
Chapter 4 Functions and Linear relationships 129
Let’s practice
6 Assume that y is the dependent variable. Write each of the following equations using
function notation:
a 3x + 4y = 0 b y − 7 = 3( x + 2 ) c 4x + 2y = 18 d xy = 1
7 For each of the following functions, evaluate the function outputs when:
i x = 2 ii x = −3 iii x = a + 1
a f ( x ) = 4x + 1 b g( x ) = ( x + 1)( x + 2)
c h( x ) = x 2 − 3x − 2 d
8 For each of the following functions, identify the effect of the output of the function as:
i the domain increases. ii the domain decreases.
a b f ( x ) = −5
c
x 0 1 2 3 4
d
x −2 −1 0 1 2
y 7 5 3 1 −1
y −2 1 2 1 −2
9 Consider the functions f ( x ) = 7x + 2 and g( x ) = 15 − 2x. Find the value of the following:
a f ( 1) + g( 2 ) b f ( −2 ) − g( 7 ) c f ( 0 ) ⋅ g( 0 ) d
10 Consider the function f ( x ) = x 2 + 2x + k.
a Simplify the expression for f ( 2 ). b If f ( 2 ) = 15, find the value of k.
11 Han is painting a mural which requires gallons of paint per square foot. Han also needs
3 additional gallons of paint to go over the outline of the mural once they are finished.
a
b
c
State the dependent and independent variables in this scenario.
Express the relationship between the independent and dependent variables of the
scenario using function notation, letting x represent the square footage of the mural.
Find the number of gallons of paint that Han will need if the mural has a size of
24 square feet.
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
12 The model P ( t ) = 3500 × 2 t describes the population of fish in a lake after t years,
where t = 0 is the initial time when the population of fish was first recorded.
Find and interpret the value of P ( 2 ).
13 Sharon works at a pizza shop and makes $15 per hour. Her wages can be represented by
the function W ( t ) = 15t, where t represents the number of hours that she works.
a
b
c
Explain what the expression W ( 19 ) represents.
Explain what the equation W ( n ) = 165 represents.
Find her wages after working 26 hours.
14 Mindy can model her tomato plant’s height in inches over time using the function h( t ) = 2 t ,
where t is the number of weeks after Mindy planted it.
a
b
Explain what the equation h( 4 ) = 16 represents.
Find the height of Mindy’s tomato plant after 6 weeks.
15 Consider the function f ( x ) = ax + b. If f ( 4 ) = 7 and f ( −1) = −3, find the values of a and b.
16 Describe a scenario that can be represented by the function
Chapter 4 Functions and Linear relationships 131
4.02 Domain and range
Concept summary
Two defining parts of any function are its domain and range.
The set of all possible input values ( x-values ) for
a function or relation is called the domain.
In the example shown, the domain is the interval of
values −3 < x ≤ 1. Notice that −3 is not included in
the domain, which is indicated by the unfilled point.
−3
Domain
−2
−1
2
1
−1
y
1 2
x
−2
−3
−4
−5
The set of all possible output values ( y-values ) for a function or relation is called the range.
In the example shown, the range is the interval of
values −4 ≤ y ≤ 0. Notice that both endpoints are
included in the range, since the function reaches
a height of y = 0 at the origin.
2
1
y
x
−3
−2
−1
−1
1 2
−2
Range
−3
−4
−5
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Mathspace Florida B.E.S.T - Algebra 1
A domain which is made up of disconnected values is said to be a discrete domain.
A function with a discrete domain.
It is only defined for distinct x-values.
y
x
A domain made up of a single connected interval of values is said to be a continuous domain.
A function with a continuous domain.
It is defined for every x-value in an interval.
y
x
It is possible for the domain of a function to be neither discrete nor continuous.
The domain and range of a function are commonly expressed using inequality notation or
set-builder notation.
Note that if two functions have different domains, then they must be different functions,
even if they take the same values on the shared parts of their domains.
Chapter 4 Functions and Linear relationships 133
Worked examples
Example 1
Consider the function shown in the graph.
8
y
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
2 4 6 8
x
a State whether the function has a discrete or continuous domain.
Solution
The function is defined at every value of x across an interval, so it has a continuous domain.
b Determine the domain of the function using set-builder notation.
Solution
We can see that the function is defined for every x-value between −6 and 8, including −6 but not
including 8.
So the domain of the function can be written as
Domain: {x | − 6 ≤ x < 8}
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Mathspace Florida B.E.S.T - Algebra 1
c Determine the range of the function using set-builder notation.
Solution
We can see that the function reaches every y-value between −6 and 4, including 4 but not
including −6.
So the range of the function can be written as
Range: {y | − 6 < y ≤ 4}
What do you remember?
1 State the definition for the:
a Domain b Range
2 State when the domain of a function is:
a Continuous b Discrete
3 For each of the following graphs:
i
ii
a
State the domain of the relation using inequality notation.
State the range of the relation using inequality notation.
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
y
2 4 6 8
x
b
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
y
2 4 6 8
x
c
8
y
d
8
y
6
6
4
4
2
x
2
x
−8 −6 −4 −2
−2
2 4 6 8
−8 −6 −4 −2
−2
2 4 6 8
−4
−4
−6
−6
−8
−8
Chapter 4 Functions and Linear relationships 135
4 For each of the following graphs:
i State the domain of the relation using set builder notation.
ii State the range of the relation using set builder notation.
a
8
y
b
8
y
6
6
4
4
2
x
2
x
−8 −6 −4 −2
−2
2 4 6 8
−8 −6 −4 −2
−2
2 4 6 8
−4
−4
−6
−6
−8
−8
c
8
y
d
8
y
6
6
4
4
2
x
2
x
−8 −6 −4 −2
−2
2 4 6 8
−8 −6 −4 −2
−2
2 4 6 8
−4
−4
−6
−6
−8
−8
5 For each of the following graphs, identify whether the domain of the function is discrete
or continuous:
a
−4 −3 −2
4
3
2
1
−1
−1
−2
−3
−4
y
1 2 3 4
x
b
−4 −3 −2
4
3
2
1
−1
−1
−2
−3
−4
y
1 2 3 4
x
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Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
6 State the domain and range of the collection
of points shown on the graph.
8
6
4
y
2
−8 −6 −4 −2
−2
−4
−6
−8
2 4 6 8
x
7 Determine for each graph whether or not it has a domain consisting of all the real numbers.
If it does not, state the domain.
a
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
y
2 4 6 8
x
b
14
12
10
8
6
4
2
−8 −6 −4 −2
−2
y
2 4 6 8
x
8 Determine for each graph whether or not it has a range consisting of all the real numbers. If it
does not, state the range.
a
10
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
y
2 4 6 8
x
b
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
y
2 4 6 8
x
Chapter 4 Functions and Linear relationships 137
9 For each of the following:
i Identify an independent variable and a dependent variable.
ii State the domain and range for the scenario.
a
b
c
Uma earns between $200 and $450 selling between 35 and 50 burritos, inclusive.
Jermaine starts reading his book from page 112. Over the course of 17 days,
he reads up to and including page 237.
Brigid Kosgei completes a 26.2 mile marathon in 134 minutes.
10 State the domain and range for each of the following graphs:
a
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
−10
y
x
2 4 6 8 10
b
−4
8
6
4
2
−2
−2
−4
−6
−8
y
2 4 6 8 10
x
Let’s extend our thinking
11 The graph of a function is shown.
a
State the range of the function.
125
y
b
A ball is thrown from an apartment
window in a high-rise building. The height
of the ball above ground over time can
be modelled by the function shown in
the graph, where the ball is thrown at
x = 0. State the range of the function for
this context.
100
75
50
25
x
−6
−4
−2
2 4 6 8 10
−25
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Mathspace Florida B.E.S.T - Algebra 1
12 A comet is travelling through the solar system. It passes by the Earth before curving around
the Sun and heading back out into deep space. The distance of the comet from Earth, y, is
tracked by satellite telescopes and expressed as a function of time, x, since the comet was
first spotted.
a
b
State the domain of this function.
The satellites detect that, at its closest point, the comet was a distance of 0.14 AU
( astronomical units ) from Earth. State the range of the function.
13 A function describing the strictly increasing relation between temperature and a person’s
average resting heart rate has a domain of 50 ≤ T ≤ 105 ( Fahrenheit ) and a range of
40 ≤ f ( T ) ≤ 90 ( beats per minute ), based on experimental results.
a Determine the average resting heart rate at a temperature of 50 °F .
b
c
State the temperature that is expected if a person has an average resting heart rate of
90 beats per minute.
Explain whether or not the function output f ( 10 ) be a reliable estimate for the
corresponding real life scenario.
Chapter 4 Functions and Linear relationships 139
4.03 Average rate
of change
Concept summary
The average rate of change of a function over an interval is the change in value of the dependent
variable per unit change in the independent variable.
The average rate of change of a function can be calculated by dividing the change in function
values between the start and the end of the interval by the length of the interval.
For the function shown, the function values increase
from 3 to 7 over the interval 1 ≤ x ≤ 5.
This means that over the interval 1 ≤ x ≤ 5,
the function has an average rate of change of
We can see this graphically as well: the dashed lines
show an increase of 1 unit in the y-values per unit
increase in the x-values over this interval.
8
7
6
5
4
3
2
y
Worked examples
1
1 2 3
4 5 6 7 8
x
Example 1
A flock of birds migrate to a new island.
The population of birds on that island over the
next six years is shown on the graph.
60
50
40
y
30
20
10
1 2 3
4 5 6
x
140
Mathspace Florida B.E.S.T - Algebra 1
Determine the average rate of change of the bird population over the six year period.
Approach
To find the average rate of change, we want to divide the difference between the initial and final
populations by the length of the time period.
Solution
Looking at the graph we can see that the initial population is 15 birds, and after 6 years the
population is 60 birds.
So we can calculate the average rate of change as
Therefore, the average rate of change is 7.5 birds per year.
Example 2
Consider the function f ( x ) = 2x 2 − 1.
Calculate the average rate of change over the interval −4 ≤ x ≤ 0.
Approach
The average rate of change can be thought of as:
To determine the change in f (x), we want to identify the values of f (x) when x = −4 and when x = 0.
Solution
We know that
So we have that
f (x) = 2x 2 − 1
f (−4) = 2(−4) 2 − 1
= 2 ⋅ 16 − 1
= 31
and
f (0) = 2(0) 2 − 1
= −1
Chapter 4 Functions and Linear relationships 141
Going back to the idea that the average rate of change can be thought of as
we have:
What do you remember?
1 State the definition for the average rate of change over an interval.
2 Consider the interval from the point ( x 1 , y 1 ) to ( x 2 , y 2 ).
a State the change in the dependent variable over the interval.
b State the change in the independent variable over the interval.
c State the average rate of change over this interval.
3 For each of the following equations:
i Find the value of y when x = 2.
ii Find the value of y when x = 7.
iii Find the average rate of change of the equation over the interval 2 ≤ x ≤ 7.
a y = 2x + 3 b y = 4x − 10 c y = −x + 8 d y = −7x + 2
4 Stanton went for a run. The following
graph shows the distance in meters
covered by Stanton at x minutes. Is the
distance’s average rate of change
between 20 and 30 minutes positive,
negative, zero or undefined?
1400
1200
1000
800
Distance
600
400
200
5
10 15 20 25 30 35 40 45
Time
142
Mathspace Florida B.E.S.T - Algebra 1
5 Tori is controlling a drone for a video shoot.
The following table shows the height ( in feet ) of
the drone x minutes after starting its descent.
Is the height’s average rate of change between
0 and 6 minutes positive, negative, zero or
undefined?
Time in minutes Height in feet
0 140
2 135
4 132
6 126
8 120
6 A clock factory’s weekly profit, in dollars, can be modeled by the equation
y = −1875 + 150x − x 2 where x is the selling price of a clock in dollars. Is the profit’s average
rate of change between the $40 selling price and the $110 selling price positive, negative,
zero or undefined?
7 The graph shows Yoichi’s distance from
home over a 10-minute interval.
Is the distance’s average rate of change
between 3 and 8 minutes positive, negative,
zero or undefined?
6
5
4
Distance from home
8 The population of river otters in a particular
area can be modelled by the equation
y = 15 + 5 x , where x is the number of years
from now.
3
2
a Find the value of y when x = 0.
b Find the value of y when x = 2.
c
Determine the population’s average rate
of change between x = 0 and x = 2.
1
1
Time
2 3 4 5 6 7 8 9 10
9 The side profile of a bridge
is overlapped with the coordinate axes
such that an ordered pair ( x, y ) represents a
point on the arch of the bridge that is
x meters horizontally from one end of the
bridge and y meters vertically above the
road. The bridge meets the road at
coordinates ( 0, 0 ) and ( 100, 0 ). Pedestrians
are able to climb the full arch of the bridge.
Find the average steepness of the climb
between x = 10 and x = 60.
54
48
42
36
30
24
18
12
y
(10, 18)
(60, 48)
6
x
20
40 60 80 100
Chapter 4 Functions and Linear relationships 143
Let’s practice
10 Consider the input and output pairs of a function shown in the table.
x 2 7 8 12
y 5 15 16 36
a Find the average rate of change between x = 2 and x = 7.
b Find the average rate of change between x = 7 and x = 8.
c Find the average rate of change between x = 8 and x = 12.
d Determine whether or not the given points display a constant rate of change.
11 The graph shows the price, in dollars,
of a new digital toy for different levels
of supply.
a
b
Find the average rate of change.
Interpret the average rate of change
in the given context.
120
100
80
Price
60
40
20
Supply (thousands of units)
5
10 15 20
12 The graph shows the total number of
tickets, in thousands, purchased for a
premiering movie, with pre-ordered tickets
counted as being purchased on the
0th day.
a
b
Find the average rate of change.
Interpret the average rate of change
in the given context.
25
20
15
10
Tickets (thousands)
5
Days
2
4 6
8 10 12
144
Mathspace Florida B.E.S.T - Algebra 1
13 The graph shows the number
of earthquakes that a particular country
experiences in each year.
a
b
Find the average rate of change.
Interpret the average rate of change
in the given context.
10
8
6
Number of earthquakes
4
2
Year
1
2 3
4 5
14 The path of water projected from a fountain can be modelled by the equation
y = −20x 2 + 80x, where x is the horizontal distance from the nozzle and y is the height.
Find the average rate of change of the water’s height from when it is projected to when it is
two meters horizontally from the nozzle.
15 The value of a particular painting can be modeled by the equation y = 100 ⋅ 2 x , where x is
the number of years from when it was painted. If the painting was made in 2009, find the
average rate of change of its value between 2010 and 2015.
Let’s extend our thinking
16 Sarah and William are finding an expression for the average rate of change of a function
f ( x ) between x = a and x = b.
a
b
Sarah states that, to find the average rate of change, we find the total change in the
function values and divide it by the difference between the x-values. Use Sarah’s method
to find an expression for the average rate of change.
William claims that finding the average rate of change between two points is the same as
finding the slope of the straight line that passes through those two points. Is he correct?
17 Explain why the average rate of change of a linear equation y = mx + b over any interval is
always equal to the slope m.
Chapter 4 Functions and Linear relationships 145
4.04 Characteristics of
functions from a graph
Concept summary
The important characteristics, or key features, of a function or relation include its
• domain
• range
• x-intercepts
• y-intercepts
• maximum value ( the highest output value )
• minimum value ( the lowest output value )
• rate of change over specific intervals
Key features of a function are useful in helping to sketch the function, as well as to interpret
information about the function in a given context. Note that not every function will have each type
of key feature. The rate of change of a function over a specific interval can be broadly categorized
by one of the following three discriptions:
A function is increasing over an interval if, as the input values become higher, the output values
also become higher.
A function is decreasing over an interval if, as the input values become higher, the output values
become lower.
A function is constant over an interval if, as the input values become higher, the output values
remain the same.
4
3
y
Constant
2
1
−4 −3 −2 −1
−1
Increasing −2
−3
−4
1 2 3 4
Decreasing
x
146
Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
Consider the function shown in the following graph:
y
14
12
10
8
6
4
2
−2 −1 1 2
−2
−4
−6
3
4 5 6 7 8
x
a Identify whether the function has a maximum or minimum value, and state this value.
Solution
This function has a minimum value of −4.
b State the range of the function.
Approach
In part (a) we identified that the function has a minimum value of −4. So we know that the function
can’t take values smaller than −4.
Solution
Looking at the function, we can see that it stretches up towards infinity on both sides of
the minimum point. So the function can take any value greater than or equal to −4. That is,
the range of the function is
Range: {y | y ≥ −4}
Chapter 4 Functions and Linear relationships 147
c State the x-intercept( s ) of the function.
Approach
The x-intercept(s) of a function are the points where the function crosses the x-axis. In this case,
by looking at the graph we can see that there are two x-intercepts.
Solution
The x-intercepts of this function are the points (1, 0) and (5, 0).
d Determine the largest interval over which the function is increasing.
Approach
In part (a) we identified that the function has a minimum value. Looking at the graph,
we can see that the function values are decreasing on the left side of the minimum and increasing
on the right side.
Solution
The minimum point occurs at x = 3. The function values are increasing for all x-values to the right of
this minimum. That is to say, the function is increasing for x > 3.
What do you remember?
1 State the definition for the following function characteristics:
a Maximum b Minimum c x-intercept d y-intercept
2 State the definition for when a function over an interval is:
a Increasing b Decreasing c Constant
Let’s practice
3 For the following graphs of the function y = f ( x ):
i Identify whether the function has a maximum or minimum and state its value.
ii State the range of the function.
iii State the interval over which the function is increasing.
iv State the interval over which the function is decreasing.
148
Mathspace Florida B.E.S.T - Algebra 1
a
10
y
b
4
y
8
6
4
2
−4 −2
−2
x
2 4 6 8 10 12
2
−6 −4 −2
−2
2 4 6 8 10
x
−4
−6
−8
−4
−10
−6
−12
c
16
y
d
8
y
12
6
8
4
4
x
2
x
−10 −8
−6 −4 −2 2 4
−4
−8
−6
−4 −2 2 4 6
−2
−8
−4
−12
−6
4 For each of the following graphs, identify the interval( s ) of the domain where the function is:
i Increasing ii Decreasing iii Constant
a
4
y
b
8
y
3
6
2
4
1
x
2
x
−5 −4 −3
−2 −1
−1
1 2 3 4 5
−8 −6 −4 −2
−2
2 4 6 8
−2
−4
−3
−6
−4
−8
Chapter 4 Functions and Linear relationships 149
c
7
y
d
4
y
6
3
5
2
4
3
2
−5 −4 −3
1
−2 −1
−1
1 2 3 4 5
x
−4 −3 −2
1
−1
−1
1 2 3 4
x
−2
−3
−4
5 For the following functions:
i State the coordinates of the x-intercept. ii State the coordinates of the y-intercept.
a
5
4
3
2
1
−5 −4 −3 −2 −1 −1
y
1 2 3 4 5
x
b
10
8
6
4
2
−5 −4 −3 −2 −1 −2
y
1 2 3 4 5
x
−2
−3
−4
−5
−4
−6
−8
−10
6 Consider the functions shown on the following graphs:
i State the coordinates of the x-intercepts.
ii State the coordinates of the y-intercept.
iii As x → ∞, describe the end behavior of the corresponding y-values.
iv As x → −∞, describe the end behavior of the corresponding y-values.
a
12
y
b
10
y
8
8
4
−6 −4 −2
−4
−8
−12
2 4 6 8 10
x
−4 −3 −2
6
4
2
−1
−2
1 2 3 4
x
−16
−4
−20
−6
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Mathspace Florida B.E.S.T - Algebra 1
7 For each graph, state the following:
i Domain ii Range iii x- and y-intercepts iv Rate of change
a
2
y
b
7
y
1
−2 −1 1 2
−1
3 4 5 6
x
6
5
4
−2
3
−3
2
−4
−5
−6
1
−1 1 2 3
−1
4 5 6 7
x
c
5
y
d
4
y
4
3
3
2
−4 −3 −2
2
1
−1
−1
1 2 3 4
x
−3 −2
1
−1
−1
−2
1 2 3 4 5
x
−2
−3
−3
−4
Let’s extend our thinking
8 The graph shows the height ( in meters )
of a soccer ball against the time ( in
seconds ) that has passed after it has been
kicked.
12
10
Height
a
b
c
Find the coordinates of the y-intercept.
Interpret the y-value of the y-intercept in
this context.
Find the coordinates of the x-intercept.
8
6
d
Interpret the x-value of x-intercept in
this context.
4
e
Determine the average rate of change
of the height of the soccer ball.
2
Time
1
2 3 4
Chapter 4 Functions and Linear relationships 151
9 Two construction workers are competing
to see who can lay the most bricks in one
hour. The graph shows the number of
bricks layed and the time, in minutes.
320
280
Bricks layed
a
b
Interpret the rate of change of
the given lines.
Determine who can lay more bricks
in 60 minutes.
240
200
160
Charlie
Neville
c
Explain why the y-intercept of both
lines is 0.
120
80
40
Minutes
10
20 30 40
50 60 70 80
10 Alicia has concluded that the rate of change
is greater in f ( x ) than g( x ) because it is higher
on the given graph. Explain and correct
her error.
5
4
3
y
2
1
x
−5 −4 −3 −2
f(x)
−1
−1
−2
1 2 3 4 5
g(x)
−3
−4
−5
152
Mathspace Florida B.E.S.T - Algebra 1
4.05 Linear relationships
Concept summary
A function that has a constant rate of change is called a linear function. Linear functions can be
written in the form:
f ( x ) = mx + b
m the slope of the line
b the y-value of the y-intercept
The graph of a linear function is a straight line.
In general, any relationship which has a constant rate of change is a linear relationship.
A relationship which does not have a constant rate of change is called a nonlinear relationship.
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
A linear relationship
A nonlinear relationship
Worked examples
Example 1
Emanuel is selling raffle tickets to raise money for charity. The table below shows the cumulative
number of tickets he has sold each hour for the first three hours:
Time ( hours ) 1 2 3
Total ticket sales 14 28 42
Chapter 4 Functions and Linear relationships 153
a State whether Emanuel’s ticket sales represent a linear or nonlinear function.
Approach
A linear function will have a constant rate of change. We can compare the values in the table and
see how much the total ticket sales are increasing by each hour.
Solution
Emanuel sells 14 tickets in the first hour. He then sells 28 − 14 = 14 tickets in the second hour,
and 42 − 28 = 14 tickets in the third hour.
So the rate of change is constant and therefore the ticket sales represent a linear function.
b Determine the rule which relates Emanuel’s ticket sales and time.
Solution
From part (a) we know that Emanuel is able to sell 14 additional raffle tickets each hour.
c If Emanuel’s ticket sales continue in this way, determine the total number of tickets he will have
sold after 6 hours.
Solution
From part (b) we know that Emanuel is selling 14 tickets per hour. So after 6 hours, if the pattern
stays the same, he will have sold 14 ⋅ 6 = 84 raffle tickets.
Example 2
Tiles were stacked in a pattern as shown:
Stack 1 Stack 2 Stack 3 Stack 4
A table of values representing the relationship between the height of the stack and the number of
tiles was partially completed.
Height of Stack 1 2 3 4 5 10 100
Number of tiles 1 3
154
Mathspace Florida B.E.S.T - Algebra 1
a Identify the pattern for a relationship represented between the height of a stack and
the number of tiles.
Solution
The number of tiles from one stack to the next increases by 2. The number of tiles can also be
determined by taking the height of the stack, multiplying by 2, then subtracting 1.
b Complete the table of values representing the relationship between the height of the stack and
the number of tiles.
Solution
We can use the pattern identified in part (a) to help us fill in the table.
Height of Stack 1 2 3 4 5 10 100
Number of tiles 1 3 5 7 9 19 199
c Identify if the relationship between the stack height and number of tiles is linear or not.
Solution
There is a constant rate of change in the table of values and the pattern described a constant
increase, so the relationship is linear.
What do you remember?
1 State whether each of the following graphs represents a linear or nonlinear relationship:
a
−4 −3 −2
4
3
2
1
−1
−1
−2
−3
−4
y
1 2 3 4
x
b
−4 −3 −2
4
3
2
1
−1
−1
−2
−3
−4
y
1 2 3 4
x
Chapter 4 Functions and Linear relationships 155
c
4
y
d
4
y
3
3
2
2
1
x
1
x
−4 −3 −2
−1
−1
1 2 3 4
−4 −3 −2
−1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
e
4
y
f
4
y
3
3
2
2
1
x
1
x
−4 −3 −2
−1
−1
1 2 3 4
−4 −3 −2
−1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
2 Consider the given table of values and
its graph:
Time
Revenue
1 −4
2 3
3 10
4 17
5 24
24
20
16
12
8
4
−4
Revenue
Time Period
1 2 3 4 5
a
b
Determine if the revenue is changing at a constant rate with respect to the time period.
State if the relationship between time and revenue is linear.
156
Mathspace Florida B.E.S.T - Algebra 1
3 The number of goldfish in a pond over a six-month period is recorded in the table and
plotted in a graph:
Number of fish
Number of
months
Number of
fish
200
180
a
b
1 6
2 12
3 24
4 48
5 96
6 192
Determine if the the number of fish
is changing at a constant rate with
respect to the number of months.
State if the relationship between the number of months and the number of fish is linear.
4 The growth of a potted plant over a week is recorded in the table below, with measurements
being taken at the end of each day.
160
140
120
100
80
60
40
20
1
2 3 4 5 6
Months
Day 1 2 3 4 5 6 7
Height ( inches ) 1 3 5 7 9 11 13
Determine if the growth of the potted plant can be represented by a linear function.
5 The following table shows the decay rate of 1000 grams of radioactive element D:
Day 0 1 2 3 4
Mass of D ( grams ) 1000 500 250 125 62.5
Determine if the decay can be modeled by a linear function.
Let’s practice
6 Consider the pattern.
12, 19, 26, 33, …
a Describe the pattern. b Determine if this pattern is linear or not.
c Determine the next value in the pattern.
Chapter 4 Functions and Linear relationships 157
7 Consider the pattern shown with the given figures:
Step 1 Step 2 Step 3 Step 4
a
b
Describe the pattern.
Determine the number of squares there will be in the next step in the pattern.
8 Consider the pattern.
Step 1 Step 2 Step 3 Step 4
a
b
Describe the pattern.
Determine the number of circles there will be in the next step in the pattern.
9 The total volume of water that has dripped from a tap is measured each minute and
displayed in the table.
Time ( minutes ) 1 2 3 4 5 6 7 8
Volume ( mL ) 5 10 15 20 25
a
b
c
State the rule that relates the time and volume.
Determine if the rate of change of the volume is constant with respect to time.
Copy and complete the table.
10 The total distance traveled by a cyclist is measured each hour and displayed in the table.
Time ( hours ) 1 2 3 4 5 6 7 8
Distance ( miles ) 9 18 27 36 45
a
b
c
State the rule that relates the time and distance.
Copy and complete the table.
Determine the distance the cyclist would be able to travel in 15 hours.
158
Mathspace Florida B.E.S.T - Algebra 1
11 Scarlett records the number of push ups she does each day and records them in the table.
Time ( days ) 1 2 3 4 5 6 7 8
Push ups 18 23 28 33 38
a
b
c
State the rule that relates the time and push ups.
Copy and complete the table.
Determine how many push ups Scarlett will be doing on the 20th day.
12 Eileen saves some money each week and deposits it into her savings account.
The weekly balance of this account at the end of each week is displayed in the table.
Time ( weeks ) 2 3 4 5 6 10 20 50
Balance ( dollars ) 425 450 475 500 525
a
b
c
State the rule that relates the time and balance.
Copy and complete the table.
Determine the balance be at the start of the first week.
13 Complete each of the following tables so that they could represent a linear function:
a Input 1 2 3 4 5
Output 8 12 20 24
b Input 3 4 5 6 7
Output −8 −20 −24
c Input 5 10 15 20 25
Output −13 −73
14 The cumulative rainfall over a series
of 4 days is represented on the graph.
a
b
Describe the relationship between time
and volume.
Determine if points on the graph lie on
a straight line.
16
14
12
10
8
6
4
Volume (mL)
2
1
Time (days)
2 3 4 5 6
Chapter 4 Functions and Linear relationships 159
15 The total number of matchbox cars
owned by Sally over a series of years is
represented on the graph.
a
b
Describe the relationship between time
and the total number of matchbox cars.
Determine if the points on the graph lie
on a straight line.
28
26
24
22
20
18
16
14
12
10
8
6
4
2
Number of matchbox cars
Time (years)
1
2 3 4 5 6
16 John is handing out flyers on the street to
passersby. The total number of flyers John
has handed out is recorded each hour and
displayed on the graph.
a
Describe the relationship between
time and the total number of flyers.
180
160
140
120
Total number of flyers
b
c
Determine if points on the graph lie on
a straight line.
If the pattern continues over the next
few hours, determine what the total
number of flyers will be after six hours.
100
80
60
40
20
Time (hours)
1
2 3 4 5 6
160
Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
17 Determine whether the following statements are always, sometimes, or never true.
Explain your answer.
a
A linear function has a constant rate of change.
b A linear function has an end behavior where y → ∞ when x → ∞.
c
A linear function appears as a curve on a graph.
18 State whether the function that would be used to model each scenario is linear or nonlinear.
a
b
c
Water drops from a leaking tap are falling into a bucket at a constant rate. It has to model
the volume of water in the bucket as time passes.
A snowball rolling down a mountain doubles its volume every five seconds. It has to
model the volume of the snowball as time passes.
A raindrop falling from the sky speeds up by 9.8 m/s every second as it is pulled down
by gravity. It has to model the distance travelled by the raindrop as time passes.
19 Explain why a linear function can always be written in the form y = mx + b.
Chapter 4 Functions and Linear relationships 161
4.06 Transforming linear
functions
Concept summary
A transformation of a function is a change in the position, size, or shape of its graph. There are
many ways functions can be transformed:
Vertical compression
A transformation that scales all of the y-values of a function by a constant factor towards
the x-axis
Vertical stretch
A transformation that scales all of the y-values of a function by a constant factor away from
the x-axis
A vertical compression or stretch can be represented algebraically by
g( x ) = af ( x )
where 0 < a < 1 corresponds to a compression and a > 1 corresponds to a stretch.
Reflection
A transformation that produces the mirror image of a figure across a line.
A reflection across the x-axis can be represented algebraically by
g( x ) = −f ( x )
A reflection across the y-axis can be represented algebraically by
Translation
g( x ) = f ( −x )
A transformation in which every point in a figure is moved in the same direction and by the
same distance.
Translations can be categorized as horizontal ( moving left or right, along the x-axis ) or vertical
( moving up or down, along the y-axis ), or a combination of the two.
Vertical translations can be represented algebraically by
g( x ) = f ( x ) + k
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Mathspace Florida B.E.S.T - Algebra 1
where k > 0 translates upwards and k < 0 translates downwards.
Similarly, horizontal translations can be represented by
g( x ) = f ( x − k )
where k > 0 translates to the right and k < 0 translates to the left.
Functions that can be obtained by performing one or more of these transformations on each other
can be collected into groups or families of functions. The function in any family with the simplest
form is known as the parent function, and we frequently consider transformations as coming from
the parent function.
The parent function of the linear function family is the function y = x. Some examples of
transformations are shown below. In each example, the parent function is shown as a dashed line:
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
Vertical compression with scale factor of 0.5:
Reflection across y-axis:
g( x ) = 0.5f ( x ) g( x ) = f ( −x )
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
Vertical translation of 4 units upwards: Horizontal translation of 3 units to the right:
g( x ) = f ( x ) + 4 g( x ) = f ( x − 3 )
Chapter 4 Functions and Linear relationships 163
Worked examples
Example 1
A graph of the function
a Describe the transformation given by
g( x ) = −2f ( x )
is shown below.
10
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
y
2 4 6 8 10
x
Solution
The transformation g(x) is a combination of a vertical stretch of the graph of f (x) by a factor of 2,
and a reflection across the x-axis.
b Draw a graph of g( x ) on the same plane as the graph of f ( x ).
Approach
To perform a vertical stretch, we want to multiply the y-coordinate of each point by 2 without
changing the x-coordinate, so that it is twice as far away from the x-axis. To perform a reflection
across the x-axis, we want to change the sign of the y-coordinate of each point, so that it is on the
opposite side of the x-axis.
Solution
10
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
y
f(x)
x
2 4 6 8 10
g(x)
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Mathspace Florida B.E.S.T - Algebra 1
Example 2
The linear functions f ( x ) and g( x ) are represented on the given graph.
3
y
2
g(x) 1
−4 −3 −2 −1
−1
f(x)
−2
−3
−4
−5
1 2 3 4
x
a Describe the type of transformation( s ) that transforms f ( x ) to g( x ).
Approach
We can look at the slope and intercepts to help describe which transformation has taken place.
Solution
f (x) is increasing, while g(x) is decreasing so there has been a reflection over the x-axis.
g(x) is steeper than f (x), so there has also been a dilation.
b Write an equation for g( x ) in terms of f ( x ).
Approach
The equation will be of the form g(x) = k ⋅ f (x), where k is a real number.
Solution
The slope of f (x) is and the slope of g(x) is −1, so it has be reflected and is twice as steep.
The transformation is g(x) = −2 ⋅ f (x).
Chapter 4 Functions and Linear relationships 165
c Create a table of values for f ( x ) and g( x ) on the same coordinate plane to confirm your answer
to parts ( a ) and ( b ).
Approach
We can use about five corresponding points to check that g(x) = −2 ⋅ f (x). Since f (x) has slope of
we can use even x-values to avoid using fractions or decimals.
Solution
Getting this data from the graph:
3
y
2
g(x) 1
−4 −3 −2 −1
−1
f(x)
−2
−3
−4
−5
1 2 3 4
x
x −4 −2 0 2 4
f (x) −1 0 1 2 3
g(x) 2 0 −2 −4 −6
This table does confirm that g(x) = −2 ⋅ f (x).
What do you remember?
1 Assuming the value of k is positive, state whether the linear function f ( x ) has been translated
up, down, left, or right to produce the graph of g( x ).
a g( x ) = f ( x ) + k b g( x ) = f ( x ) − k c g( x ) = f ( x ) + 3k
2 For the what values of k will the transformation from f ( x ) to g( x ) = kf ( x ) involve a reflection
across the x-axis?
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Mathspace Florida B.E.S.T - Algebra 1
3 For each of the following, state whether f ( x ) has been translated up, down, left, or right:
a g( x ) = f ( x ) + 4 b g( x ) = f ( x ) − 2 c d
4 For each of the following, describe the type transformation from f ( x ) to g( x ):
a g( x ) = 6 f ( x ) b c d g( x ) = − f ( x )
5 For each of the following tables:
i Write an equation for g( x ) in terms of f ( x ) in the form g( x ) = f ( x ) + k or g( x ) = kf ( x )
depending on the transformation.
ii Describe the type of transformation from f ( x ) to g( x ).
a f (x) 2 5 8 11 14
g(x) 4 7 10 13 16
c f (x) −2 −1 0 1 2
g(x) −4 −3 −2 −1 0
b f (x) 1 2 3 4 5
g(x) 4 8 12 16 20
d f (x) 4 6 8 10 12
g(x) −2 −3 −4 −5 −6
6 Consider the following graphs:
i Describe the transformation applied to f ( x ) to produce g( x ).
ii Write an equation for g( x ) in terms of f ( x ) in the form g( x ) = f ( x ) + k or g( x ) = kf ( x )
depending on the transformation.
a
y
b
y
4
4
−4 −3 −2
g(x)
f(x)
3
2
1
−1
−1
−2
−3
−4
1 2 3 4
x
−4 −3 −2
f(x)
3
2
1
−1
−1
−2
−3
g(x)
−4
1 2 3 4
x
c
4
y
d
f(x)
4
y
3
3
2
1
g(x)
f(x)
x
g(x)
2
1
x
−4 −3 −2
−1
−1
1 2 3 4
−4 −3 −2
−1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
Chapter 4 Functions and Linear relationships 167
Let’s practice
7 For each of the following pairs of functions:
i Copy and complete the table.
ii Graph f ( x ) and g( x ) on the same coordinate plane.
a x 0 1 2
f (x) 4 6 8
g(x) = f (x) + 3
b x 0 1 2
f (x) 3 5 7
g(x) = f (x) − 7
c x 0 1 2
f (x) −2 0 2
g(x) = −3 f (x)
d x 0 1 2
f (x) 18 24 30
8 For each graph of f ( x ), graph g( x ) using the given transformation.
a g( x ) = f ( x ) + 5 b g( x ) = 2 f ( x )
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2
−1
−1
−2
1 2 3 4
−4 −3 −2
f(x)
−1
−1
−2
1 2 3 4
−3
f(x)
−4
−3
−4
c g( x ) = −f ( x ) d
4
y
4
y
f(x)
3
2
3
2
1
x
1
x
−4 −3 −2
−1
−1
−2
1 2 3 4
−4 −3 −2
−1
−1
−2
f(x)
1 2 3 4
−3
−3
−4
−4
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Mathspace Florida B.E.S.T - Algebra 1
e g( x ) = f ( x ) − 2 f g( x ) = f ( x ) + 2
f(x)
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2
−1
−1
1 2 3 4
−4 −3 −2
−1
−1
1 2 3 4
−2
−3
f(x) −2
−3
−4
−4
9 For each transformation, describe how the graph of g( x ) is related to the graph of f ( x ).
a g( x ) = f ( x ) − 8 b c g( x ) = −3f ( x ) d
10 Consider the graph of f ( x ) and the table of values for g( x ).
i Describe the transformation applied to f ( x ) to produce g( x ).
ii Write an equation for g( x ) in terms of f ( x ) in the form g( x ) = f ( x ) + k or g( x ) = kf ( x )
depending on the transformation.
a
4
3
2
y
x −1 0 1 2 3
g(x) 6 4 2 0 −2
1
x
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
f(x)
−4
Chapter 4 Functions and Linear relationships 169
b
4
3
y
x −2 −1 0 1 2
g(x) −4 −3 −2 −1 0
2
1
x
−4 −3 −2 −1
−1
1 2 3 4
f(x)
−2
−3
−4
c
f(x)
4
3
2
y
x −4 −2 0 2 4
g(x) 12 8 4 0 −4
1
x
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
−4
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
11 Stavros has a bank account with $100.
Every week, Stavros deposits an
additional $50 into the account. A graph
of Stavros’ bank balance reveals that the
growth of his account is linear. Suppose
that Stavros was instead depositing an
additional $100 every week. Explain
whether or not this could be represented
by a vertical stretch of the current graph.
500
400
300
200
Balance
100
Weeks
1
2 3 4 5 6 7
12 Consider the transformation g( x ) = kf ( x ). Under what conditions will the graphs of f ( x ) and
g( x ) have:
a No points of intersection b Infinitely many points of intersection
13 Oreste draws the graph of g( x ) = −f ( x ),
as shown.
a
b
Describe the error Oreste made.
Describe the type of graphs that f ( x ) could
be for Oreste’s error to still give him the
correct answer.
g(x)
4
3
2
1
y
x
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
f(x)
−4
Chapter 4 Functions and Linear relationships 171
4.07 Characteristics of
linear functions
Concept summary
The characteristics, or key features of a function include its:
• domain and range
• x- and y-intercepts
• maximum or minimum value( s )
• rate of change over specific intervals
Key features of a function are useful in helping to sketch the function, as well as to interpret
information about the function in a given context. In the case of linear functions, they will either be
constant everywhere ( a horizontal line ), increasing everywhere, or decreasing everywhere, due to
their constant rate of change. This also means that they will never have a maximum or minimum
turning point, and will have at most one x-intercept ( except for the horizontal line y = 0 ).
4
y
4
y
3
3
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
An increasing linear function. It has an
A decreasing linear function. It has an
x-intercept at ( −2, 0 ) and a y-intercept at ( 0, 2 ). x-intercept at ( 2, 0 ) and a y-intercept at ( 0, 1).
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Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
Consider the function f ( x ) = 4x − 8.
a Find the value of f ( 0 ).
Approach
To find the value of a function at a particular x-value, we simply substitute that value into the
function.
Solution
We have
f (0) = 4(0) − 8
= −8
Reflection
The y-intercept of a function is the point on the function where x = 0. So we know that this function
has a y-intercept at (0, −8).
b Find the value of x which gives a function value of 0.
Approach
This means that we want to solve the equation f (x) = 0 using the expression we have for f (x).
Solution
We have
f (x) = 0
4x − 8 = 0
4x = 8
x = 2
Reflection
The x-intercepts of a function are the points on the function where y = 0. So we know that this
function has a single x-intercept at (2, 0).
Chapter 4 Functions and Linear relationships 173
c Sketch a graph of the function and label each intercept.
Approach
We found the two intercepts in parts (a) and (b). Since two points determine a line, we can connect
the intercepts to create the graph of the function.
Solution
10
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
y
(2, 0)
x
2 4 6 8 10
(0, −8)
Example 2
Whitney is traveling across the city by taking an Uber ride. The cost of the ride, in dollars,
is given by
C( x ) = 2.4x + 2.8
where x is the distance traveled in miles. The minimum charge for a ride is $10.
a State the range of the function.
Solution
Since x is the distance traveled in miles, we know that x > 0, which corresponds to C (x) > 2.8.
We are also told that the minimum charge is $10.
Putting these together, we have that the range is
Range: {y | y ≥ 10}
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Mathspace Florida B.E.S.T - Algebra 1
b Determine the rate of change of the function, and state what it represents in context.
Approach
The rate of change of a linear function in the form y = mx + b is m, the constant being multiplied by
the variable.
Solution
In this case, the function is C (x) = 2.4x + 2.8, so the rate of change is 2.4.
Since the output of C (x) is a value in dollars, and x is a value in miles, the rate of change is 2.4
dollars per mile, and it represents the additional cost of the trip per each extra mile traveled.
c If Whitney is travelling a distance of 4 miles, determine the cost of her trip.
Approach
We want to find the value of C (4), and compare it to the minimum cost. The larger of the two values
will be the cost of Whitney’s trip.
Solution
We have
C(4) = 2.4(4) + 2.8
= 9.6 + 2.8
= 12.4
Since this is larger than the minimum amount, the cost of her trip will be $12.40.
Example 3
Consider functions f and g shown in the graph.
Compare the following features of each function:
4
3
y
g(x)
−4 −3 −2 −1
2
1
−1
−2
1 2 3 4
x
−3
f(x) −4
Chapter 4 Functions and Linear relationships 175
a Domain
Approach
The domain of a function is the set of all x-values that correspond to a point on the graph.
Solution
As these are linear functions, with no domain restrictions from context, all real values of
x correspond to a point on each line.
So the domain of both functions is all real values of x.
b Range
Approach
The range of a function is the set of all y-values that correspond to a point on the graph.
Solution
We can see that all values of y correspond to a point on the graph of f, so its range will be all real
values of y.
On the other hand, g is a constant function and so it only takes one y-value - in this case, 2.
So the range of g is just y = 2.
c Intercepts
Approach
The x-intercepts of a function are the points where it intersects the x-axis. Similarly, the y-intercept
of a function is the point where it intersects the y-axis.
As these are linear functions, they will have at most one of each type of intercept.
Solution
We can see that function f crosses the x-axis at
the point (2, 0), and crosses the y-axis at the point
(0, −2).
g(x)
4
3
2
1
y
x
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
f(x) −4
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Mathspace Florida B.E.S.T - Algebra 1
Function g crosses the y-axis at the point (0, 2). Since it is a constant function, it is parallel to the
x-axis and so it does not have an x-intercept.
4
y
g(x)
−4 −3 −2 −1
3
2
1
−1
−2
1 2 3 4
x
−3
f(x) −4
d Slope
Approach
The slope of a line is the ratio of its change in the vertical direction to the change in the horizontal
direction.
That is, the slope is a measure of how far up (or down) the y-values change for each unit change in
the x-values.
Solution
For function f we can see that at any point on the graph, if we move 1 unit to the right
(in the x-direction), we also move 1 unit up (in the y-direction). So the slope of f is 1.
Function g is a constant function, which has no change in its y-values. That is, at any point on the
graph, moving 1 unit to the right corresponds to 0 units of change in the vertical direction.
So the slope of g is 0.
What do you remember?
1 For each function, complete the following:
i Find the coordinates of the intercepts. ii Determine the slope of the function.
iii Determine the domain of the function. iv Determine the range of the function.
a f ( x ) = 35x − 15 b f ( x ) = −3x + 7
Chapter 4 Functions and Linear relationships 177
2 For each of the following functions, evaluate the function at two values for x and find the
slope of the function:
a f ( x ) = 6x + 3 b f ( x ) = −2x + 4
c
d
3 Using the following graphs, write the equation in function notation:
a
8
6
4
2
−8 −6 −4 −2
−2
f(x)
−4
−6
−8
y
2 4 6 8
x
b
g(x)
8
6
4
2
−8 −6 −4 −2
−2
−4
−6
−8
y
2 4 6 8
x
Let’s practice
4 Consider the following table of values of a function.
x −2 −1 0 1 2
f ( x ) −5 −1 3 7 11
a
Select the graph that could represent the function:
A
5
4
3
2
1
−5 −4 −3 −2 −1 −1
y
1 2 3 4 5
x
B
5
4
3
2
1
−5 −4 −3 −2 −1 −1
y
1 2 3 4 5
x
−2
−3
−4
−5
−2
−3
−4
−5
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Mathspace Florida B.E.S.T - Algebra 1
C
5
y
D
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
b
c
Find the coordinates of the y-intercept.
Find the slope of the function.
5 Consider the function
a
Select the graph of the function:
A
8
y
B
8
y
6
6
4
4
2
x
2
x
−8 −6 −4 −2
−2
2 4 6 8
−8 −6 −4 −2
−2
2 4 6 8
−4
−4
−6
−6
−8
−8
C
8
y
D
8
y
6
6
4
4
2
x
2
x
−8 −6 −4 −2
−2
2 4 6 8
−8 −6 −4 −2
−2
2 4 6 8
−4
−4
−6
−6
−8
−8
b
c
Find the domain of the function.
Find the range of the function.
Chapter 4 Functions and Linear relationships 179
6 Consider the function whose graph passes through points ( 0, −5 ) and ( −1, 1).
a
Select the graph of the function:
A
5
y
B
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
C
5
y
D
5
y
4
4
3
3
2
2
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−5 −4 −3 −2 −1 −1
1
1 2 3 4 5
x
−2
−2
−3
−3
−4
−4
−5
−5
b
c
Find the slope of the function.
Find the coordinates of the x-intercept.
7 For each of the following equations, assuming that y is a function of x:
i Rewrite the equation using the function notation f ( x ).
ii
iii
iv
Find the value of f ( −2 ). Write answer using function notation.
Find the value of f ( 3 ). Write answer using function notation.
Graph the function.
a y = −3x + 4 b y − 2 = 3( x + 4 ) c 3x + 5y = 14 d 7x − 5y = −4
8 For each of the following functions:
i Find the value of f ( 0 ).
ii
iii
Find the value of x which makes f ( x ) equal to zero.
Graph the function, labelling each intercept.
a f ( x ) = −3x + 12 b f ( x ) = 5x − 15 c f ( x ) = −x − 4 d f ( x ) = 5( x − 2 )
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Mathspace Florida B.E.S.T - Algebra 1
9 For each of the following functions, use the given domain to find the range of the function:
a f ( x ) = 3x + 2 has a domain of x > 7
b f ( x ) = −x − 13 has a domain of x ≤ 2
c f ( x ) = 2x − 5 has a domain of −5 ≤ x ≤ 2
d f ( x ) = −5x + 6 has a domain of 3 ≤ x < 6
10 Two functions f and g are shown in the graph.
Compare the following features of each
function:
a
b
c
d
Domain
Range
Intercepts
Slope
g
−4 −3 −2 −1
4
3
2
1
−1
y
1 2 3 4
x
−2
f
−3
−4
11 Two functions are described by the equations f ( t ) = 3t − 6 and g( t ) = 3t + 9. Compare the
following features of each function:
a Domain b Range c Intercepts d Slope
12 Each time Serena finishes reading a book, her mother adds a dollar to Serena’s reading jar.
At the start of the holidays, Serena had $31 in her reading jar. During the holidays, Serena
reads at a rate of five books a week.
a
b
Write a function which represents the number of dollars in Serena’s reading jar in terms
of how many holiday days have passed.
If Serena’s holidays last for four weeks, determine how many dollars will be in her
reading jar by the end of the holidays.
13 The height of a candle in inches can be represented as a linear function of time in minutes
since the candle was lit. At 2 minutes after being lit, the candle had a height of 11 inches.
At 4 minutes after being lit, the candle had a height of 8 inches.
a
b
c
Determine the rate of change of the candle’s height per minute.
Find the height of the candle before it was lit.
Determine how long the candle can be lit for before it completely melts.
Chapter 4 Functions and Linear relationships 181
14 During a fundraiser marathon, Conner sprains his ankle and needs to walk for the rest of the
race. Connor’s distance in miles remaining in the race can be represented as a function of
how many hours have passed since his injury: f ( t ) = 18 − 2t.
a
b
c
d
Describe what the variable t represents in the function.
Find the remaining distance to the finish line after Connor has been walking for 3 hours.
Determine how many hours of walking it will take for Connor to reach the finish line.
Determine the constraints on the domain and range in terms of the context.
15 Tyrell and Veronica live 500 meters away from school, and they often race each other home.
As Tyrell is younger, Veronica gives him a 20 second head start so that they can arrive home
at approximately the same time. Tyrell takes 32 seconds to jog 100 meters, on average.
Let T ( x ) represent Tyrell’s distance from home x seconds after they started a race, and V ( x )
represent Veronica’s distance from home at the same time.
a
b
c
Determine the range of each function in terms of the context.
Determine the domain of each function in terms of the context.
Compare the x-intercept of each function, and describe what they represent in terms of
the context.
Let’s extend our thinking
16 Use numbers 1 through 5 at most once to write a linear equation and to identify a point on
the function. The linear function has the following key features:
• Rate of change is greater than −1
• y-intercept is greater than 2
Write an equation of a line in the form f ( x ) = mx + b and a point in ( x, y ) form.
17 Give an example of a scenario that could be modeled by the function f ( x ) = 25x − 12.
18 Describe how to find the x- and y-intercepts of a linear equation when it is presented as a
function f of x.
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Mathspace Florida B.E.S.T - Algebra 1
4.08 Comparing linear
and nonlinear functions
Concept summary
We can use key features to compare linear and non-linear functions. Some additional key features
that we might look at are:
End behavior
Describes the trend of a function or graph at its left and right ends; specifically the y-value that
each end obtains or approaches
Positive interval
A connected region of the domain in which all function values lie above the x-axis
Negative interval
A connected region of the domain in which all function values lie below the x-axis
Worked examples
Example 1
Consider the two functions shown in the graphs below.
y
4
3
2
4
3
2
y
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
−4
−4
Chapter 4 Functions and Linear relationships 183
a State the intercepts of each function.
Solution
Both functions have a y-intercept at (0, −1).
Also, both functions have an x-intercept at (1, 0).
The second function has an additional x-intercept at (−1, 0).
b Compare the end behavior of the two functions.
Solution
On the right side, both functions take larger and larger positive values as x gets further from zero.
That is, as x → ∞, y → ∞ for both functions.
On the left side, the first function takes larger and larger negative values as x gets further from
zero. That is, as x → −∞, y → −∞ for the first function.
On the other hand, the second function takes larger and larger positive values as x gets further
from zero on the left side. That is, as x → −∞, y → ∞ for the second function.
Reflection
Note that although both functions tend towards infinity to the right, the way they do so is different.
The first function increases at a constant rate, while the second function increases at an
increasing rate.
c State the interval( s ) over which each function is positive or negative.
Solution
The first function is positive for x > 1 and negative for x < 1.
The second function is positive for both x > 1 and x < −1, and is negative for −1 < x < 1.
d Determine whether each function is linear or non-linear.
Approach
A linear function has a constant rate of change. It also has no turning points, with at most one
x-intercept.
Solution
The first function is linear, while the second function is non-linear.
184
Mathspace Florida B.E.S.T - Algebra 1
What do you remember?
1 Consider the functions shown below:
4
y
4
y
3
3
2
1
x
2
g(x)
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
f(x)
−2
−2
−3
−3
a
b
c
−4
Identify the domain of each function
Identify the range of each function
State whether each function is linear or nonlinear
−4
2 Consider the functions shown below:
4
y
4
y
3
g(x)
3
f(x)
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
−2
−3
−3
a
b
c
−4
Identify the increasing interval( s ) for each function
Identify the decreasing interval( s ) for each function
State whether each function is linear or nonlinear
−4
Chapter 4 Functions and Linear relationships 185
3 Consider the functions shown below:
4
y
4
y
3
3
f(x)
2
2
1
x
1
x
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
1 2 3 4
−2
g(x)
−2
−3
−3
a
b
c
−4
Identify the intercepts of each function
Describe the end behavior of each function
State whether each function is linear or nonlinear
−4
4 A few values for two functions are shown in the following table and plotted on the graph:
x f (x) g(x)
1 −4 −6
2 −2 −3
3 0 0
4 2 3
6
5
4
3
2
1
y
f(x)
x
5 4 0
6 6 −3
−1 1 2 3
−1
−2
−3
−4
−5
−6
4 5 6 7 8 9
g(x)
Assume the domain of each function is all real numbers.
a Compare the intervals where the function is positive and negative for each function
b State whether each function is linear or nonlinear
186
Mathspace Florida B.E.S.T - Algebra 1
Let’s practice
5 A few values for two functions are shown in the following table:
x f ( x ) g( x )
−2 −8 5
−1 −7 2
0 −5 −1
1 −1 −4
2 7 −7
Assume the domain of each function is all real numbers.
a Draw a graph of each function
b Compare the intervals where the function is increasing and decreasing for each function
c State whether each function is linear or nonlinear
6 The perimeters and areas for a sequence of similar triangles are shown in the following table:
Triangle Perimeter Area
1 18 30
2 27 67.5
3 36 120
4 45 187.5
a
b
Draw a graph of the perimeter and area functions
State whether each function is linear or nonlinear
7 An amount of $1000 is invested into two
different accounts for a fixed period of
time, with account C earning compound
interest and account S earning simple
interest. The balance of each account
over time is shown in the following
graph:
2500
2000
y
S(t)
a
b
State the domain of each function
Describe how each function
increases over its domain
1500
C(t)
c
State whether each function is linear
or nonlinear
1000
1
2 3 4 5 6 7 8 9 10
t
Chapter 4 Functions and Linear relationships 187
8 Consider the function shown in the
graph:
4
y
Bonita claims that this graph is a linear
function, since it is a horizontal line at each
point in its domain. Explain why this is
incorrect.
−4 −3 −2 −1
3
2
1
−1
1 2 3 4
x
−2
−3
−4
Let’s extend our thinking
9 Three functions f, g, and h each pass through the origin and have the following end behaviors:
• The values of f tend towards ∞ as x gets far from zero on both sides.
• The values of g tend towards ∞ as x → −∞, while the values tend towards −∞ as x → ∞.
• The values of h tend towards 3 as x gets far from zero on both sides.
Determine which of these could describe a linear function. Use graphs to support your answer.
10 Two species of parrot finch, Red-faced
and Blue-faced, are introduced to a new
location at the same time. The population
of the Red-faced species after t months
can be modeled by the function
R ( t ) = 5t + 100. The population of the
Blue-faced species is modeled by the
function shown in the graph.
a
b
c
Determine which species has the
largest population after the first year
has passed. Explain your answer.
Determine if the two species’ will
ever have the same population at the
same time after the initial introduction.
Explain your answer.
If both species continue to increase
in population in the same way, determine which species will reach a population of 1000
first. Explain your reasoning.
11 Determine whether each of the following statements are always, sometimes, or never true:
a The valus of a linear function tend towards ∞ as x → ∞.
b
c
Graphs of nonlinear functions are curves.
260
240
220
200
180
160
140
120
100
Linear functions cannot have more than one positive interval.
80
Population
4
B(t)
t (months)
8 12 16 20 24
188
Mathspace Florida B.E.S.T - Algebra 1
4.09 Linear absolute
value functions
Concept summary
The absolute value of a number is its distance from zero on a number line. An absolute value
is indicated by vertical lines on either side. For example, the absolute value of −3 is 3, which is
written as |−3| = 3.
An absolute value function is a function that contains a variable expression inside absolute value
bars; a function of the form f ( x ) = a |x − h| + k
x −2 −1 0 1 2 3 4
f ( x ) 3 2 1 0 1 2 3
For example, consider the absolute value function
f ( x ) = |x − 1|. We can complete a table of values for
the function.
4
3
2
1
y
x
Since the absolute value of an expression is nonnegative,
the graph of this absolute value function
does not go below the x-axis, as the entire
expression is inside the absolute value. If the
function has a negative value for k, such as
y = |x − 1| − 2, it can go below the x-axis, but will
still have a minimum value, in this case y = −2.
−4 −3 −2 −1
−1
−2
1 2 3 4
If the function has a negative value of a, the graph
of the function will open downwards, and instead
have a maximum value.
−3
−4
Worked examples
Example 1
Consider the function f ( x ) = |3x − 6|.
a Complete the table of values for this function.
x 0 1 2 3 4 5
f ( x )
Chapter 4 Functions and Linear relationships 189
Solution
x 0 1 2 3 4 5
f (x) 6 3 0 3 6 9
b Sketch a graph of the fuction.
Solution
8
y
7
6
5
4
3
2
1
x
−2 −1 1 2
−1
3
4 5 6 7 8
−2
What do you remember?
1 Determine whether the following graphs show an absolute value function:
a
−2
2
1
−1 −1
y b
1 2
x
−2
2
1
−1 −1
y c
1 2
x
−2
2
1
−1 −1
y d
1 2
x
−2
2
1
−1 −1
y
1 2
x
−2
−2
−2
−2
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Mathspace Florida B.E.S.T - Algebra 1
2 Consider the table of values of a function.
x −3 −2 −1 0 1 2
y 2 1 0 1 2 3
Select the graph that could represent the function:
A
−4
2
−2
−2
−4
y B
x
2
4
2
−2
−2
y C
x
2 4
2
−2
−2
−4
y D
x
2 4
4
2
−4 −2
−2
y
2
x
3 Consider the function y = |x|.
a State the values of x that can be substituted into the function.
b State the values of y that the function could take.
4 Consider the function y = |5x|.
Select the graph that represents the function:
A
−4 −2
12
10
8
6
4
2
y B
2 4
x
4
2
−4 −2
−2
y C 2
2 4
x
−4 −2
−2
−4
−6
−8
−10
−12
y x D 12
10
2 4
8
6
4
2
−4 −2 −2
−4
y
2 4
x
5 Consider the graph of y = f ( x ) as shown.
a For what values of x is f ( x ) positive?
6
y
b For what values of x is f ( x ) negative?
c For what values of x is |f ( x )| = f ( x ).
d For what values of x is |f ( x )| = −f ( x ).
4
2
x
e Graph y = |f ( x )|. −6 −4 −2
−2
2 4 6 8
−4
−6
−8
Chapter 4 Functions and Linear relationships 191
Let’s practice
6 An absolute value function has its vertex at ( −1, 3 ), and goes through the point A ( 2, 0 ).
Select the graph that represents the function:
A
3
2 1
−6 −4 −2−1
2
−2
−3
−4
y B
x
3
2 1
−6−4−2−1
2 4
−2
−3
−4
y C 3
x
2 1
−4−2−1
−2
−3
2 4 6
−4
y D 3
x
y
2
1
−4−2−1
−2
−3
2 4 6
−4
x
7 Consider the function y = |5 − x|.
a Complete the given table.
b Graph the function.
8 Consider the function
a Complete the given table.
b Graph the function.
9 Consider the function y = 2 |x + 3| − 4.
a Complete the given table.
b Graph the function.
10 Consider the function
a Complete the given table.
b Graph the function.
11 Consider the function
a Complete the given table.
b Graph the function.
x 1 2 3 4 5 6 7
y
x 2 3 4 5 6 7 8
y
x −5 −4 −3 −2 −1 0
y
x −4 −3 −2 −1 0 1
y
x −15 −10 −5 0 5 10
y
12 Michael is training for the land speed record and takes his new car for a test drive by driving
straight down a closed highway and back. His distance y in miles from the end of the
highway x minutes after he takes off is given by the function y = |5x − 40|.
a
Select the graph that represents the function:
192
Mathspace Florida B.E.S.T - Algebra 1
A
40
y
B
40
y
35
35
30
30
25
25
20
20
15
15
10
10
5
x
5
x
2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16
C
40
y
D
40
y
35
35
30
30
25
25
20
20
15
15
10
10
5
x
5
x
2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16
b
c
d
How far does he drive in total?
How long does Michael take to reach the end of the highway?
The next day Michael goes for another drive down the same route and his distance from
the end of highway is given by y = |4x − 40|. Is Michael driving faster or slower than the
previous day?
Let’s extend our thinking
13 Compare the function y = 3x − 9 and the function f ( x ) = |3x − 9|.
14 Describe a method of sketching an absolute value function y = |f ( x )|, given a graph of
y = f ( x ).
15 Explain why the function y = |x| doesn’t go below the x-axis.
Chapter 4 Functions and Linear relationships 193
4.10 Transforming
absolute value functions
Concept summary
Absolute value functions can be transformed in the same ways as linear functions. Recall the types
of transformations that we have looked at:
Translation
A transformation in which every point in the graph of the function is shifted the same distance
in the same direction.
Reflection
A transformation that flips a function across a line of reflection, producing a mirror image of
the original function.
Vertical compression
A transformation that pushes all of the y-values of a function towards the x-axis.
Vertical stretch
A transformation that pulls all of the y-values of a function away from the x-axis.
The parent function of the absolute value function family is the function y = |x|. Other linear
absolute value functions can be obtained by transformations of this parent function.
194
Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
A graph of a function f ( x ) is shown below.
10
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
y
2 4 6 8 10
x
a Sketch a graph of the function g( x ) = −f ( x ).
Approach
The function g(x) is formed by changing the sign of the function values of f (x). So the x-intercept
will stay the same, and every point above the x-axis will change to a point the same distance below
the x-axis.
Solution
10
8
6
4
2
−10 −8 −6 −4 −2
−2
−4
−6
−8
−10
y
2 4 6 8 10
x
Chapter 4 Functions and Linear relationships 195
b Describe the transformation from f ( x ) to g( x ).
Solution
The transformation from f (x) to g(x) is a reflection across the x-axis.
Example 2
Consider the following table of values for two absolute value functions f ( x ) and g( x ):
x 0 1 2 3 4 5 6
f ( x ) 6 3 0 3 6 9 12
g( x ) 12 9 6 3 0 3 6
a Describe the transformation from f ( x ) to g( x ).
Solution
Looking at the values in the table for f (x), we can see that it has a vertex at (2, 0) and has a rate of
change of 3 on either side.
Comparing this to the value for g(x) we can see that it has the same rate of change, but has its
vertex at (4, 0) instead.
So we can get from f (x) to g(x) by translating 2 units to the right.
Reflection
Since absolute value functions are symmetric about a vertical axis, it is also possible to describe
this transformation as a reflection across the line x = 3.
b Write a function that describes the relationship between f ( x ) and g( x ).
Solution
We can write “a translation by 2 units to the right” as g(x) = f (x − 2).
Reflection
Since it is also possible to describe this transformation as a reflection across the line x = 3, we can
also express this transformation as g(x) = f (6 − x).
196
Mathspace Florida B.E.S.T - Algebra 1
What do you remember?
1 Assuming the value of k is positive, state whether the graph of f ( x ) has been translated up,
down, left, or right to produce the graph of g( x ).
a g( x ) = f ( x ) + k b g( x ) = f ( x + k )
c g( x ) = f ( x − k ) d g( x ) = f ( x ) − k
2 Consider the following graphs:
i Describe the transformation applied to f ( x ) to produce g( x ).
ii Write an equation for g( x ) in terms of f ( x ) in the form g( x ) = f ( x ) + k or g( x ) = kf ( x )
depending on the transformation.
a
10
f(x)
g(x) 8
6
4
2
−10 −8 −6 −4 −2
−2
y
2 4 6
x
b
10
8
6
f(x)
4
2
−10 −8 −6 −4 −2 2
−4
g(x)
−6
−8
−10
y
2 4 6
x
c
g(x)
10
f(x)
8
6
y
d
g(x)
10
8
f(x)
6
y
4
4
2
x
2
x
−8 −6 −4 −2
−2
2 4 6 8
−8 −6 −4 −2
−2
2 4 6 8 10
Chapter 4 Functions and Linear relationships 197
Let’s practice
3 Consider the following graphs:
i How was f ( x ) transformed to produce the graph of g( x )?
ii State the equation of g( x ) in terms of f ( x ).
a
y
b
y
f(x)
10
g(x)
8
8
f(x)
6
g(x)
6
4
4
2
−8 −6 −4 −2
−2
2 4 6 8
x
2
−8 −6 −4 −2
−2
−4
x
2 4 6 8 10
c
f(x)
10
y
d
g(x)
8
y
8
g(x)
6
4
2
−8 −6 −4 −2
−2
2 4 6 8
x
f(x) 6
4
2
−6 −4 −2
−2
x
2 4 6 8 10
4 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = f ( −x ).
b Describe the transformation from f ( x )
to g( x ).
10
8
6
4
y
−10 −8 −6 −4 −2
2
−2
2 4
x
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Mathspace Florida B.E.S.T - Algebra 1
5 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = f ( x ) + 3.
b Describe the transformation from f ( x )
to g( x ).
10
8
6
4
y
−10 −8 −6 −4 −2
2
−2
2 4 6
x
6 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = f ( x ) − 5.
b Describe the transformation from f ( x )
to g( x ).
10
8
6
4
y
2
−6 −4 −2 2 4 6 8
−2
−4
10 12
x
7 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = f ( x + 3 ).
b Describe the transformation from f ( x )
to g( x ).
10
8
6
y
4
2
−4 −2 2 4 6 8
−2
10
x
Chapter 4 Functions and Linear relationships 199
8 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = f ( x − 5 ).
b Describe the transformation from f ( x )
to g( x ).
10
8
6
4
y
2
−6 −4 −2 2 4 6 8
−2
10
x
9 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = 4f ( x ).
b
Describe the transformation from f ( x ) to
g( x ).
8
6
y
4
2
−10 −8 −6 −4 −2 2 4
−2
6
x
10 Consider the graph of the function y = f ( x ):
y
a
b
Graph y = g( x ) where
Describe the transformation from f ( x ) to
g( x ):
8
6
4
2
−6 −4 −2 2 4
−2
6
x
200
Mathspace Florida B.E.S.T - Algebra 1
11 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = −2f ( x ).
b
Describe the transformation from f ( x ) to
g( x ).
8
6
y
4
2
−8 −6 −4 −2 2 4
−2
6
x
12 Consider the graph of the function y = f ( x ):
a Graph y = g( x ) where g( x ) = f ( 2x ).
b
Describe the transformation from f ( x ) to
g( x ).
10
8
6
4
y
−8 −6 −4 −2 2 4 6
2
−2
8
x
13 For each of the given f ( x ) and g( x ):
i Complete the table.
ii Describe the transformation from f ( x ) to g( x ).
a f ( x ) = |x − 2| b f ( x ) = |2x|
g( x ) = 3f ( x ) g( x ) = f ( −2x )
c f ( x ) = |2x − 1| d f ( x ) = |3x + 4|
g( x ) = f ( x ) + 5 g( x ) = f ( x ) − 3
e f ( x ) = |3x − 4| f f ( x ) = |2x − 6|
g( x ) = f ( x − 1) g( x ) = f ( x + 3 )
x f (x) g(x)
−3
−2
−1
0
1
2
3
4
5
Chapter 4 Functions and Linear relationships 201
14 Consider the following tables:
i Describe the transformation from f ( x ) to g( x ).
ii Write a function that describes the relationship between f ( x ) and g( x ).
a
x f (x) g(x)
b
x f (x) g(x)
−1 3 12
−4 4 5
0 2 8
−3 2 3
1 1 4
−2 0 1
2 0 0
−1 2 3
3 1 4
0 4 5
4 2 8
1 6 7
5 3 12
2 8 9
c
x f (x) g(x)
d
x f (x) g(x)
−3 2 −4
−4 4 1
−2 1 −2
−3 2 −1
−1 0 0
−2 0 −3
0 1 −2
−1 2 −1
1 2 −4
0 4 1
e
x f (x) g(x)
f
x f (x) g(x)
−2 8 0
−4 4 20
−1 6 2
−3 2 16
0 4 4
−2 0 12
1 2 6
−1 2 8
2 0 8
0 4 4
3 2 10
1 6 0
4 4 12
2 8 4
5 6 14
3 10 8
15 Describe how the graph of y = f ( x ) is shifted to get the graph of the following:
a y = f ( x ) + 4 b y = f ( x + 6 )
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
16 Determine whether each of the following statements are true or false. Explain your thinking.
a
b
c
Absolute value functions are always positive.
The vertex of an absolute value function is always a minimum or maximum point.
The vertex of an absolute value can occur on the x-axis.
17 Maddox and Paulina are going to a lake 2 miles from their home to go ice skating.
Their mother told them that if the ice is not thick enough they must turn around and come
straight home. It took them 20 minutes to walk to the lake at a constant rate and when they
arrived Maddox could tell the ice was not thick enough so they immediately turned around
and went home which took them an additional 20 minutes.
a
b
c
Graph Paulina and Maddox’s distance from home over their entire journey
If Paulina and Maddox had walked twice as fast, describe how your graph would change.
Describe the transformations if instead you were graphing Paulina and Maddox’s
distance from the lake.
Chapter 4 Functions and Linear relationships 203
5 Exponential
Functions
Chapter outline
5.01 Operations with numerical radicals 206
5.02 Rational exponents 213
5.03 Exponential relationships 220
5.04 Characteristics of exponential functions 228
5.05 Exponential growth and decay 237
5.06 Percent growth and decay 249
5.07 Simple interest 259
5.08 Compound interest 264
5.01 Operations with
numerical radicals
Concept summary
Radical expressions have many parts as shown in the following diagram:
Index
The number on a radical symbol that indicates which type of root it represents. For instance,
the index on a cube root is 3. The index on a square root is usually not written, but would be 2.
Radical
A mathematical expression that uses a root, such as a square root
or a cube root
Radicand
The value or expression underneath the radical symbol
Perfect square
A number that is the result of multiplying two of the same integer together
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Mathspace Florida B.E.S.T - Algebra 1
Perfect cube
A number that is the result of multiplying three of the same integer together
If a radical expression is written such that there are no factors that can be removed from
the radicand, and with no radicals in the denominator (if the expression is a fraction),
then the expression is said to be in simplified radical form.
For square roots this means there are no remaining factors of the radicand that are perfect
squares, and for cube roots this means there are no remaining factors of the radicand that are
perfect cubes. The same operations that can be applied to rational expressions can also be
applied to radical expressions.
• Multiplication: For radicals with the same index, multiply the coefficients, multiply the
radicands, and write under a single radicand before checking to see if the radicand can be
simplified further
• Division: For radicals with the same index, divide the coefficients, divide the radicands,
and write under a single radicand before checking to see if the radicand can be
simplified further
• Addition and subtraction: Add or subtract like radicals (radicals with the same index and
radicand) by adding the coefficients and keeping the radicand the same, if there are no like
radicals check to see if any of the radicals can be simplified first.
Worked examples
Example 1
Rewrite the expressions in simplified radical form.
a
Approach
To simplify a square root it can be helpful to see if the radicand has any factors that are perfect
squares. In this case, 150 = 25 ⋅ 6 and 25 is a perfect square.
Solution
Knowing this, we can simplify the original expression as follows:
Since 6 does not have any factors that are perfect squares, the original expression is fully
simplified as
Chapter 5 Exponential Functions 207
b
Approach
To simplify a cube root it can be helpful to see if the radicand has any factors that are perfect
squares. In this case, 72 = 8 ⋅ 9 and 8 is a perfect cube.
Solution
Knowing this, we can simplify the original expression as follows:
Since 9 does not have any factors that are perfect cubes, the original expression is fully simplified
as
Example 2
Simplify the radical expressions.
a
Approach
To simplify the product of two square roots we can multiply them together in the usual way.
Then we can look at the result and factor out the square root of any perfect square factors, so that
the solution is in simplified radical form.
Solution
Following this process, we have:
So the fully simplified form is
Combining the radicals
Multiplying the radicands
Rewriting to find a perfect square factor
Taking out the perfect square factor
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Mathspace Florida B.E.S.T - Algebra 1
b
Approach
To simplify the quotient of two radicals it can be helpful to see if the radical in the numerator
can be written as the product of two or more radicals, one of which is equal to the radical in
the denominator.
Solution
In this case, 48 = 24 ⋅ 2, and so we can rewrite the original expression as follows:
So we have that
fully simplifies to
Example 3
Simplify the following expressions:
a
Approach
Since both terms in the expression include
we can combine like terms to simplify.
Solution
b
Approach
Since both terms in the expression include
we can combine like terms to simplify.
Solution
Chapter 5 Exponential Functions 209
c
Approach
Since the two terms in the expression have different radicands, we first want to rewrite each term
in simplified radical form and see if they have any like terms. Since
and
we can then combine like terms.
Solution
What do you remember?
1 How do you know if a radical is in simplified radical form?
2 Simplify each of the following radicals:
a b c d
3 Determine whether each expression is in simplified radical form:
a b c d
4 Rewrite each expression in simplified radical form:
a b c d
5 Simplify the following:
a b c
3 3
2 7
d
6 Simplify the following:
a b c d
7 Simplify the following:
a b c d
Let’s practice
8 Simplify the following:
a b c
d e f
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Mathspace Florida B.E.S.T - Algebra 1
9 Simplify the following:
a b c d
e f g h
10 Rewrite the following expressions in simplified radical form:
a b c d
11 Find the area of the following rectangles:
a
b
12 Solve for the height of a triangle with and whose base measures
13 Aviva and Jillian are traveling from the shore
at Point A to their home at Point C. Jillian
wants to swim home and travels directly from
A to C. Aviva wants to avoid the water and
decides to run home via point B.
a
b
c
How far does Jillian have to swim to
get home?
How far does Aviva have to run to
get home?
How much further did Aviva travel than Jillian?
Let’s extend our thinking
14 Kurt and Ursula are comparing answers to the question:
“Which of the following is larger: or ?”
• Ursula claims that is the largest
• Kurt claims is the largest
Who is correct? Explain how you know.
Chapter 5 Exponential Functions 211
15 Describe a method you could use to compare and without a calculator.
16 Uma solved for the height of a triangle whose area is and whose base measures
ft.
Has Uma made an error? If so, identify and correct the error.
17 Give two different pairs of values for k and m which make the following equation true:
18 Sasha is simplifying the expression and says that the answer is
Is she correct? Explain how you know.
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Mathspace Florida B.E.S.T - Algebra 1
5.02 Rational exponents
Concept summary
Expressions with rational exponents are expressions where the exponent is a rational number (can
be written as an integer fraction). In general, a rational exponent can be rewritten as a radical (or a
radical as a rational exponent) in the following ways:
The laws of exponents can also be applied to expressions with rational exponents, where m, n, p
and q are integers and a and b are nonzero real numbers
Product of powers
Quotient of powers
Power of a power
Power of a product
Identity exponent
a 1 = a
Chapter 5 Exponential Functions 213
Zero exponent
a 0 = a
Negative exponent
Worked examples
Example 1
For each expression, convert between exponential and radical forms.
a
Approach
We can use
be simplified.
to first rewrite the expression, then check to see if the radical can
Solution
b
Approach
We can use
to rewrite the expression.
Solution
Example 2
Rewrite the following as an expression with a single exponent:
a
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Mathspace Florida B.E.S.T - Algebra 1
Approach
We can use the product of powers,
to rewrite the expression.
Solution
Product of powers law
Evaluate the addition
Evaluate the division
b
Approach
We can use the power of a product
to rewrite the expression.
Solution
Power of a product law
Evaluate the multiplication
c
Approach
We can use the power of a power
to rewrite the expression.
Solution
Power of a power law
Evaluate the multiplication
Simplify the fraction
Chapter 5 Exponential Functions 215
d
Approach
We can use the quotient of powers
to rewrite the expression.
Solution
Quotient of powers law
Evaluate the subtraction
Evaluate the division
Identity exponent
Example 3
Evaluate the following expressions:
a
Approach
We can use the definition of negative exponents,
to evaluate the expression.
Solution
Definition of negative exponents
Definition of rational exponents
Evaluate the radical
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Mathspace Florida B.E.S.T - Algebra 1
b
Approach
We can use the quotient of powers,
to simplify the expression.
Solution
Quotient of powers law
Evaluate the subtraction
Evaluate the division
Definition of zero exponent
Reflection
It is possible to simplify this expression more quickly by recognizing that a number divided by itself
always results in 1.
c
Approach
We can use the definition of zero exponents to simplify the expression, then apply the law of
negative exponents followed by multiplication to fully evaluate the expression.
Solution
Definiton of zero exponent
Definition of negative exponent
Evaluate the exponent
Evaluate the division
Evaluate the multiplication
Chapter 5 Exponential Functions 217
What do you remember?
1 Rewrite the following exponents in radical form:
a b c d
2 Rewrite the following radicals in exponential form:
a b c d
3 Evaluate the following:
a b c d
4 Rewrite each expression using a single exponent:
a b c d
e f g h
Let’s practice
5 The volume of a cube, V, with a surface area, A, is given the formula
Find the volume of a cube with a surface area of 54 in 2 .
6 Find the value of x that makes each equation true.
a b c
d e f
7 Simplify the following expressions:
a b c
d e f
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
8 Kaley and Lucia are discussing the laws of exponents.
Kaley thinks that the exponent can only be written as the radical .
Lucia thinks that the exponent can only be written as .
Is either student correct? Explain your reasoning.
9 Brody and Veena are evaluating the radical expression Here are their works:
Brody:
Veena:
Describe the differences in the two students’ work.
10 Show that 3 5 ⋅ 3 6 is equivalent to 3 3 ⋅ 3 8 .
11 Identify and correct the error in the following work:
12 A colony of bacteria doubles each hour for three hours. After three hours, the scientists change
the temperature conditions and the colony triples each hour for three hours. Represent the
bacteria growth rate over the six hours using three different exponential expressions.
13 Identify and correct the error in the following work:
Chapter 5 Exponential Functions 219
5.03 Exponential
relationships
Concept summary
Exponential relationships include any relations where the outputs change by a constant factor for
consistent changes in x, and form a pattern.
In the table, we can see the change in output is increasing by a factor of 3, and can describe this
pattern as “the number triples each time”.
x 0 1 2 3
y 1 3 9 27
This relationship can be shown on a coordinate plane, with the curve passing through the points
from the table.
y
• This graph shows an exponential relationship.
30
• y approaches a minimum value as x decreases
and approaches ∞ as x increases.
25
20
15
10
−4 −3 −2 −1
5
f(x)
1 2 3 4
x
An exponential relationship can be modeled by a function with a variable in the exponent,
known as an exponential function:
f (x) = ab x
a The initial value
b The growth or decay factor
The initial value is the output value when x = 0, and the growth or decay factor is the
constant factor.
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Mathspace Florida B.E.S.T - Algebra 1
Worked examples
Example 1
Consider the following pattern:
Step 1 Step 2 Step 3
a Describe the pattern in words.
Approach
We can see that the first step is made up of 2 squares, the second step is made up of 4 squares,
and the third step 8 squares. If we only considered the first two steps we would not know if this
relationship was linear or exponential, we would just know it has increased by 2. By considering
the increase from step 2 to step 3, we can see that it has increased by 4 squares, so we know
the relationship is not linear.
Solution
The number of squares doubles each step.
Reflection
We could have constructed a table of values showing the step number and the number of squares
in each step to see the pattern in another form.
b Determine the number of squares the next step if the pattern continues.
Approach
Using the pattern we described in part (a), we have to double the number of squares in step 3 to
find the number of squares in the next step.
Solution
2 ⋅ 8 = 16
There are 16 squares in the next step.
Chapter 5 Exponential Functions 221
Example 2
For the following exponential function:
x 1 2 3 4
f (x) 5 25 125 625
a Identify the growth factor.
Approach
We can find the growth factor by dividing a term by the previous term, that is by evaluating.
We can see that when x = 1, f (x) = 5 and when x = 2, f (x) = 25.
Solution
Reflection
We could have chosen other values and arrived at the same result. For example
b Determine the value of f (5).
Approach
Using the growth factor found in part (a), we know that as x increases by 1, f (x) increases by a
factor of 5. This means f (5) = 5 × f (4).
Solution
f (5) = 5 × 128 = 3125
Example 3
A large puddle of water starts evaporating when the sun shines directly on it. The amount of water
in the puddle over time is shown in the table.
Hours since sun came out 0 1 2 3 4 5
Volume in mL 1024 512 256 64
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Mathspace Florida B.E.S.T - Algebra 1
Assuming the relationship is exponential, complete the table and describe the relationship
between time and volume.
Approach
We can find the value of b by dividing the amount of water in the puddle after one hour by the
amount that was present at the start. Using this value for b, we can then find the missing values.
Solution
Using this value for b, we know that the volume after 3 hours will be half of 256 and the time after
5 hours will be half of 64.
Hours since sun
came out
Volume in mL
0 1024
1 512
2 256
3 128
4 64
5 32
Reflection
This exponential relation is an example of one that decreases over time. We can see that it
represents decay instead of growth because in this case b is less than one.
What do you remember?
1 Consider the functions shown in the following tables of values:
i Describe the pattern in words. ii Find the next two terms.
a
x 0 1 2 3
b
x 0 1 2 3
y 5 10 20 40
y 5 10 15 20
c
x 0 1 2 3
d
x 0 1 2 3
y 2.5 7.5 22.5 67.5
y 40000 4000 400 40
Chapter 5 Exponential Functions 223
2 Consider the pattern:
a
b
Describe the pattern in words.
Determine the number of circles the next
step in the pattern contains.
Step 1 Step 2 Step 3
3 Consider the pattern:
a
b
Describe the pattern in words.
Determine the number of hexagons the
next step in the pattern contains.
Step 1 Step 2 Step 3
4 Consider the function y = 5 x .
a Copy and complete the following table of values:
x 0 1 2 3 4 5
y 1 5 3125
b
Determine if y = 5 x represents a linear or an exponential relationship.
Let’s practice
5 Determine if the following functions represent a linear, exponential or another type
of relationship:
a y = 4 x b y = 4x + 7 c y = 4x 2 + 7 d y = 4 ⋅ 7 x
6 Determine if the following representations of functions are linear, exponential or another type
of relationship:
a
4
2
−5 −4 −3 −2 −1
−2
−4
−6
−8
−10
y
1 2 3 4 5
x
b
−4 −3 −2
−1
80
70
60
50
40
30
20
10
y
1 2 3 4
x
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Mathspace Florida B.E.S.T - Algebra 1
c
f (x) 1 2 3 4
d
f (x) 1 2 3 4
x 0.8 1.6 2.4 3.2
x 0.125 0.5 2.0 8.0
7 For the following exponential functions:
i Find the growth or decay factor. ii Determine the value of f (5).
a
x 1 2 3 4
b
x 1 2 3 4
f (x) 0.123 0.246 0.492 0.984
f (x) 10 35 122.5 428.75
c
x 1 2 3 4
d
x 1 2 3 4
f (x) 2 30 450 6750
f (x) 4096 1024 256 64
8 Consider the pattern:
a
b
Find the growth factor.
Determine the number of triangles the
next step in the pattern contains.
Step 1 Step 2 Step 3
9 For each of the following scenarios, state whether the type of function to be used should be
linear, exponential or neither:
a
b
c
d
Water drops from a leaking tap are falling into a bucket at a constant rate.
A radioactive element is decaying such that every 10 seconds there is half as many
atoms left.
A snowball rolling down a mountain doubles its volume every five seconds.
A raindrop falling from the sky increases in speed by 9.8 m/s every second as it is pulled
down by gravity.
10 The total volume of water that has dripped from
a tap is measured each minute and displayed in
the following table:
a
b
c
Describe the relationship between time
and volume.
Complete the values in the table over the next
two minutes.
Find the volume after 8 minutes.
Time (minutes) Volume (mL)
1 3
2 9
3 27
4 81
5
6
Chapter 5 Exponential Functions 225
11 Hannah saves some money each week and puts
it in her piggy bank. The total amount in her piggy
bank each week is displayed in the
following table:
a
b
c
Describe the relationship beween time
and balance.
Complete the values in the table over the next
two weeks.
Find the balance after 8 weeks.
Time (weeks) Balance (dollars)
1 4
2 8
3 16
4 32
5
6
Let’s extend our thinking
12 The cumulative rainfall over a series of
4 days is represented on the graph.
a
b
c
Describe the relationship between time
and volume.
Determine if the points represent
a linear or exponential function.
Explain your reasoning
Assuming the pattern continues over
the next few days, find the cumulative
volume on day seven.
36
32
28
24
20
16
12
8
Volume (mL)
4
1
2 3 4 5
Time (days)
13 Consider the two functions shown:
a
Find the average rate of change for
f (x) and g (x) from x = 0 to x = 3.
20
18
y
b
c
d
e
Find the average rate of change for
f (x) and g (x) from x = 0 to x = 1.
Find the average rate of change for
f (x) and g (x) from x = 2 to x = 3.
Find the average rate of change for
f (x) and g (x) from x = 3 to x = 4.
Describe what is happening to the
rates of change for both f (x) and g (x).
16
14
12
10
8
6
4
f(x)
g(x)
f
Determine which is an exponential
function and which is a linear function.
Explain your reasoning.
2
1
2 3 4 5 6 7 8
x
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Mathspace Florida B.E.S.T - Algebra 1
14 Consider the table of values for
the functions for x ≥ 1:
f (x) = (1.05) x and g (x) = 5x.
Is Ivan correct to conclude that g (x) is
always greater than f (x) for all values
of x ≥ 1? Explain your answer.
x 1 2 3 4 5
f (x) 1.05 1.10 1.16 1.22 1.28
g (x) 5 10 15 20 25
15 Several points have been plotted on
the coordinate plane:
a
Find the three points that form a linear
relationship between x and y.
b For each 1 unit increase in x,
determine how much the linear function
increases by.
c
Find the three points that form an
exponential relationship between
x and y.
d For each 1 unit increase in x,
determine the constant ratio for
the exponential function.
40
35
30
25
20
15
10
5
y
A
B
C
D
E
F
G
x
1
2 3 4 5 6
Chapter 5 Exponential Functions 227
5.04 Characteristics of
exponential functions
Concept summary
To draw the graph of an exponential function we can fill out a table of values for the function and
draw the curve through the points found. We can also identify key features from the equation:
f (x) = ab x
a The initial value gives us the the value of the y-intercept
b We can use the constant factor to identify other points on the curve
The constant factor, b, can be found by finding the common ratio.
We can determine the key features of an exponential function from its graph:
• The graph is increasing
• y approaches a minimum value of 0
• The domain is −∞ < x < ∞
• The range is 0 < y
• The y-intercept is at (0, 3)
• The common ratio is 4
• The horizontal asymptote is y = 0
27
24
21
18
15
12
9
6
3
y
f(x)
1 2
x
• The graph is decreasing
• y approaches a minimum value of 0
• The domain is −∞ < x < ∞
• The range is 0 < y
• The y-intercept is at (0, 10)
• The common ratio is
• The horizontal asymptote is y = 0
25
20
15
10
y
5
f(x)
x
−4
−3
−2
−1
1 2 3 4
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Mathspace Florida B.E.S.T - Algebra 1
Asymptote
A line that a curve or graph approaches as it heads toward positive
or negative infinity
y
x
Worked examples
Example 1
Draw a graph of y = 2.5(4) x by first finding the common ratio and the y-intercept.
Approach
The function has a common ratio of 4, and
a y-intercept at (0, 2.5). We can find other
points on the curve using a table of values.
x −2 −1 0 1 2 3
y 0.15625 0.625 2.5 10 40 160
Solution
−2
−1
45
40
35
30
25
20
15
10
5
y
1 2 3
x
Reflection
When drawing the graphs of exponential functions we want to be sure the y-intercept is clearly
displayed and that the exponential curve is also visible. Be sure to choose a scale for the y-axis
that will show all important characteristics. In this case, we chose to scale by 5s which allows us to
read both the y-intercept at 2.5, a second point at (1, 10), the horizontal asymptote at y = 0 and the
steep slope that all exponential functions have.
Chapter 5 Exponential Functions 229
Example 2
Consider the table of values for the function
x −5 −4 −3 −2 −1 0 1 2 3 4 5 10
y 486 162 54 18 6 2
a Describe the behavior of the function as x increases.
Approach
We want to identify if the values of y are increasing or decreasing as x increases.
Solution
As x increases, the function decreases at a slower and slower rate.
Reflection
We can see that the equation has a constant factor that is less than 1. This is why the function
is decreasing.
b Determine the y-intercept of the function.
Approach
The y-intercept occurs when x = 0. We can read these coordinates from the table.
Solution
(0, 2)
Reflection
We can see that the equation has an initial value of 2. This is the value of the y-intercept,
and the result of substituting x = 0 into the equation.
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Mathspace Florida B.E.S.T - Algebra 1
c State the domain of the function.
Approach
The domain is the complete set of possible values for x. For exponential functions, the graph
extends indefinitely in both horizontal directions.
Solution
All real x.
Reflection
All exponential equations of the form y = ab x have a domain of all real x.
d State the range of the function.
Approach
The range is the complete set of possible values for y. We can see graph extends indefinitely up
towards the left but it approaches an asymptote at y = 0 towards the right.
Solution
y > 0
Reflection
All exponential equations of the form y = ab x have a range of y > 0 for positive values of a.
What do you remember?
1 Consider the exponential functions shown in the following tables of values:
i Determine the common ratio. ii Graph the exponential function.
a
x −1 0 1 2 3
b
x −1 0 1 2 3
y 1 3 9 27 81
y 2 10 50 250
c
x −1 0 1 2 3
d
x −1 0 1 2 3
y
y 4000 400 40 4 0.4
Chapter 5 Exponential Functions 231
2 State whether the following are increasing or decreasing exponential functions:
a b y = 9 × 3 x c d
3 Graph the exponential function with the following properties:
a An initial value of 1 and a common ratio of 4.
b An initial value of 4 and a common ratio of 2.
c An initial value of 64 and a common ratio of
4 Select the exponential function that
represents the following graph.
A
B
C
4 x
5 x
5(4) x
10
5
y
D
−5(4) x
x
−10
−5
5 10
−5
−10
5 Select the exponential function that
represents the following graph.
10
y
A
y = −5 x
B
C
y = −9 x
5
x
D
−10
−5
5 10
−5
−10
6 Consider the table of values:
x 1 2 3 4 5
y 3
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Mathspace Florida B.E.S.T - Algebra 1
Select the exponential function that could represent the table.
A B C D
7 Consider the table of values:
x −4 −3 −2 −1 0 1 2 3 4
y 112 56 28 14 7
Select the exponential function that could represent the table.
A B C D
Let’s practice
8 Consider the graph of the equation y = 4 x :
a
b
State the equation of the horizontal
asymptote.
Explain what the horizontal asymptote
means for an exponential function.
5
4
3
2
y
1
x
−3
−2
−1
1 2 3
−1
9 Do either of the functions have x-intercepts? Explain your answer.
10 Draw the graphs of the functions
Then answer the following questions:
a State whether the following statements are true for all of the functions:
i All of the curves have a maximum value.
ii All of the curves pass through the point (1, 2).
iii All of the curves have the same y-intercept.
iv None of the curves cross the x-axis.
Chapter 5 Exponential Functions 233
b
c
State the y-intercept of each curve.
Describe what happens to the values of y as x gets increasingly larger.
11 Consider the graph of the functions
y = 3 x and
15
y
a
b
c
d
State the coordinates of the point of
intersection of the two curves.
Describe what happens to the values
of y for each function as x gets
increasingly larger.
Describe the rate of change for
each function.
Describe what other features these
functions have in common.
y = 3 x
−5 −4 −3
−2
12
9
6
3
−1
−3
−6
1
⎛ 1 ⎞
y = ⎜ ⎟
⎝ 3 ⎠
2 3 4 5
x
x
12 Consider the table of values for the function .
x −5 −4 −3 −2 −1 0 1 2 3 4 5 10
y 32 16 8 4 2 1
a
b
c
d
Describe the behavior of the function as x increases.
Determine the y-intercept of the function.
State the domain of the function.
State the range of the function.
13 Consider the function y = 4(2 x ).
a Find the y-value of the y-intercept of the curve.
b Can the function values ever be negative?
c As x approaches infinity, determine the value that y approaches.
d Graph y = 4(2 x ).
e List the domain and range for the function.
14 Consider the function
a
b
Find the y-value of the y-intercept of the curve.
Complete the table of values for
x −3 −2 −1 0 1 2 3
y
234
Mathspace Florida B.E.S.T - Algebra 1
c
d
e
Find the horizontal asymptote of the curve.
Graph
List the domain and range of the function.
15 Explain how the answers to part (a) for questions 8 and 9 relate to the functions and the
general form of an exponential function f (x) = a (b) x .
16 Compare the domain and range for f (x) = 4(2) x and from questions 8 and 9.
What do you notice?
17 Consider the given graph of y = 5 x .
a
Describe a transformation of the graph
of y = 5 x that would obtain y = −5 x .
10
8
y
b
Graph y = 5 x and y = −5 x on the same
coordinate plane.
6
4
y = 5 x
c
Compare the domain and range of
y = 5 x and y = −5 x .
−3
−2
−1
2
−2
1
2 3
x
−4
−6
−8
−10
Let’s extend our thinking
18 Consider the graphs of the two exponential
functions R and S:
18
y
a
b
One of the graphs is of y = 4 x and
the other graph is of y = 6 x .
Identify which is the graph of y = 6 x .
Explain your answer.
For x < 0, is the graph of y = 6 x above
or below the graph of y = 4 x ?
Explain your answer.
16
14
12
10
8
6
S
4
R
2
x
−3
−2
−1
1
2 3
Chapter 5 Exponential Functions 235
19 Consider the function
a
State whether the following functions are equivalent to
i ii y = 2 -x iii y = −2 x iv y = −2 -x
b Describe a trasformation that would obtain the graph of from the graph of
y = 2 x .
c Graph the functions y = 2 x and on the same coordinate plane.
20 Consider the equation y = −10 x .
a
Jenny thinks she has found a set of solutions for the equation as shown in the table:
x −2 −1 0 1 2 3
y −1 −10 −100 −1000
She notices that all the y values are negative and concludes that for any value of x, y must
always be negative. Is she correct? Explain your answer.
b Graph y = −10 x .
c Find the values of x for which y = 0.
21 Consider the original graph y = 3 x . The function values of the graph are multiplied by 2 to
form a new graph.
a
For each point on the original graph, find the point on the new graph.
Point on original graph (0, 1) (1, 3) (2, 9)
Point on new graph (−1, ) (0, ) (1, ) (2, )
b State the equation of the new graph.
c Graph the functions y = 3 x and y = 2 × 3 x on the same coordinate plane.
d For negative x-values, is the graph of y = 2 × 3 x above or below y = 3 x ?
e For positive x-values, is the graph of y = 2 × 3 x above or below 3 x ?
236
Mathspace Florida B.E.S.T - Algebra 1
5.05 Exponential growth
and decay
Concept summary
Exponential functions can be classified as exponential growth or exponential decay based on the
value of the constant factor.
Exponential growth
f (x) = ab x
a The initial value of the exponential function
b The constant factor of the exponential function
The process of increasing in size by a constant percent rate of change.
Sometimes called percent growth. This occurs when b > 1
y
8
7
1 2 3 4
−4−3−2−1
6
5
4
3
2
1
x
Exponential decay
The process of reducing in size by a constant percent rate of change.
Sometimes called percent decay. This occurs when 0 < b < 1
y
8
7
1 2 3 4
−4−3−2−1
6
5
4
3
2
1
x
Growth factor
The constant factor of an exponential growth function
Decay factor
The constant factor of an exponential decay function
Chapter 5 Exponential Functions 237
Worked examples
Example 1
Consider the exponential function:
a Classify the function as either exponential growth or exponential decay.
Approach
To classify an exponential function we want to identify the constant factor, b, and determine if b > 1
or 0 < b < 1.
Solution
In this function b = 4. Since 4 > 1, we would classify this function as exponential growth.
b Identify the initial value.
Approach
In the general form of an exponential function, y = ab x , the initial value is represented by the
variable a, which is the factor, or coefficient, that does not have a variable exponent.
Solution
The initial value is
c Identify the growth or decay factor.
Approach
In the general form of an exponential function, y = ab x , the growth or decay factor is represented
by the variable b, which is the factor with a variable exponent.
Solution
The growth factor is 4.
238
Mathspace Florida B.E.S.T - Algebra 1
Example 2
Write an equation that models the function shown in the table.
x 0 1 2 3
y 3 9 27 81
Approach
To create the equation we need to identify the intial value and growth or decay factor for the
function modeled in the table.
For functions in the form y = a(b) x , the y-intercept represents the initial value and the growth or
decay factor can be found using the ratio of two successive outputs.
Solution
The initial value is 3 and the growth factor is so the equation for this function is y = 3(3) x .
Example 3
Write an equation that models the function shown in the graph.
32
y
−4 −3 −2 −1
28
24
20
16
12
8
4
1 2 3 4
x
Chapter 5 Exponential Functions 239
Approach
To create the equation we need to identify the intial value and growth or decay factor for the
function modeled in the graph.
For functions in the form y = a(b) x , the y-intercept represents the initial value and the growth or
decay factor can be found using the ratio of two successive outputs.
Solution
The initial value is 16. To find the growth factor we will take two points, (0, 16) and (1, 4) and create
a ratio of their outputs:
The equation for this function is
Reflection
When determining the growth or decay factor be sure to use the first output as the denominator
and the second output as the numerator.
Example 4
A sample contains 300 grams of carbon-11, which has a half-life of 20 minutes.
a Write a function, A, to represent the amount of the sample remaining after n minutes.
Approach
We will start with the general form of an exponential function, f (x) = a(b) x and input all known values
to find the function.
Solution
The initial value of the function is 300 grams and half-life means that the function has a
decay factor of so we can start with the function Since the decay factor is
happening every 20 minutes we need to adjust the exponent of the function to be
Reflection
By using
in the exponent we can see that the function will halve its value once in 20 minutes
since n = 20 gives us twice in 40 minutes since n = 40 gives us etc.
240
Mathspace Florida B.E.S.T - Algebra 1
b Evaluate the function for n = 30 and interpret the meaning in context.
Approach
In this context n represents the time, in minutes, and the output represents the amount of the
sample remaining. We will evaluate the function and apply these units to interpret the meaning of
the solution.
Solution
remaining after 30 minutes had passed.
which tells us that there were approximately 106.1 grams of carbon-11
What do you remember?
1 Classify each of the following situations as exponential growth, decay, or neither.
a A store owner checked the sales report for the previous month and found that each
week the sales were of the previous week’s sales.
b
c
d
In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are
found to triple each hour.
1 gram of sugar is poured into a cup of water and immediately begins to dissolve.
Each second the amount of sugar remaining is
previous second.
of the amount present in the
Search engines give every web page on the internet a score (called a Page Rank)
which is a rough measure of popularity/importance. One such search engine uses a
logarithmic scale so that the Page Rank is given by: R = log 11 x, where x is the number of
views in the last week.
2 For each of the following exponential functions, where x represents time:
i Classify as either exponential growth or exponential decay.
ii State the initial amount.
iii State the growth or decay factor.
a y = 0.68(6) x b c d
Chapter 5 Exponential Functions 241
3 Consider the points given in the graph.
a
Copy and complete the table of values.
20
18
y
b
c
x 1 2 3 4
y
State the growth factor.
Determine the equation relating
x and y.
16
14
12
10
8
6
4
4 Consider the given table of values.
a
b
State the initial amount.
Classify as either exponential
growth or exponential decay.
c State the growth or decay factor. d Write an equation relating x and y.
5 Consider the table of values.
a
Number of days passed (x) 1 2 3 4 5
Population of shrimp (y) 5 25 125 625 3125
Classify the population of shrimp over time as either exponential growth
or exponential decay.
b Find the equation linking population, y, and time, x, in the form y = a x .
c
Graph the population over time.
6 Some bacteria double every hour which means 1 bacterium can be 2 x bacteria after x hours.
This can be expressed as f (x) = 2 x . Select the graph that represents the function.
A
y
2
B
1
2 3 4 5
x 0 1 2 3 4
y 60000 12000 2400 480 96
y
x
10
10
5
5
x
x
−10
−5
5 10
−10
−5
5 10
−5
−5
−10
−10
242
Mathspace Florida B.E.S.T - Algebra 1
C
y
D
y
10
10
5
5
x
x
−10
−5
5 10
−10
−5
5 10
−5
−5
−10
−10
7 Consider the table of values.
x 0 1 2 3 4 5 6
y 9 3 1
Select the graph that could represent the function.
A
y
B
y
10
10
5
5
x
x
−10
−5
5 10
−10
−5
5 10
−5
−5
−10
−10
C
y
D
y
10
10
5
5
x
x
−10
−5
5 10
−10
−5
5 10
−5
−5
−10
−10
Chapter 5 Exponential Functions 243
8 Select the graph that represents
A
y
B
y
10
10
5
5
x
x
−10
−5
5 10
−10
−5
5 10
−5
−5
−10
−10
C
y
D
y
10
10
5
5
x
x
−10
−5
5 10
−10
−5
5 10
−5
−5
−10
−10
9 The population of a certain city can be expressed by the function, P (t) = 10000(2) t where t is
the number of years since 1995. Select the graph that represents the function.
A
60000
P(t)
B
60000
P(t)
50000
50000
40000
40000
30000
30000
20000
20000
10000
t
10000
t
1
2 3 4 5
1
2 3 4 5
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Mathspace Florida B.E.S.T - Algebra 1
C
60000
P(t)
D
60000
P(t)
50000
50000
40000
40000
30000
30000
20000
20000
10000
t
10000
t
1
2 3 4 5
1
2 3 4 5
Let’s practice
10 The number of layers, y, resulting from
a rectangular piece of paper being folded in
half x times, is shown in the graph.
a
Write the equation linking y and x in
the form y = a x .
10
9
8
7
y
b
c
d
Interpret the meaning of the y-intercept
in this context.
If a rectangular piece of paper is folded
10 times, find the resulting number
of layers.
If a rectangular piece of paper of
thickness 0.02 mm is folded 11 times,
find the total resulting thickness.
6
5
4
3
2
1
1
2 3 4 5
x
11 For each of the following graphs of a
population, P, over time, find the equation of the curve in the form P (x) = Ab x :
a
250
200
P
b
450
400
350
P
150
300
100
50
t
250
200
150
t
1 2 3 4
1 2 3 4
Chapter 5 Exponential Functions 245
12 A futsal ball has less bounce than a full size soccer ball, and is expected to bounce back off
the ground to 25% of the height from which it falls, and after every subsequent bounce.
a
b
Write a function, y, to represent the height of the nth bounce, if it was dropped from an
initial height of 5 meters.
Find the height of the fourth bounce in centimeters. Round your answer to two
decimal places.
13 In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found
to double each hour.
a
Complete the table of values.
Number of hours passed (x) 0 1 2 3 4
Number of bacteria (y) 1
b Find the equation linking the number of bacteria, y, and the number of hours passed, x.
c
d
e
At this rate, how many bacteria will be present in the petri dish after 15 hours?
Graph the number of bacteria over time.
Interpret the meaning of the y-intercept in this context.
14 One person in a city is infected with a virus. During the first day, before they start to show
symptoms and decide to stay home sick, they infect five more people with the virus.
a
Given each new person infects an average of five new people on their first day of being
sick, copy and complete the table showing the number of newly infected people for
that day.
Day 0 1 2 3 4
Number of people infected 1
b If this trend continues, write an expression for the number of people infected on day n.
c
d
There are 345 800 people in the city in total. Given the trend continues, after how many
days will the entire city have been infected?
If everyone wears a mask and socially distances they can restrict the spread so that each
new person only infects an average of two new people on their first day, determine the
number of people that would be infected on day 10
15 During a sudden outbreak, scientists must decide between two anti-bacterial treatments
that are currently being trialled to try to control the outbreak. In the laboratory, they apply
Treatment A and Treatment B to two samples of the bacteria, each containing 300 microbes.
They keep track of the number of microbes in each sample. The table shows the results:
Number of hours (t) 0 2 4 6
Number of microbes using Treatment A 300 310 320 330
Number of microbes using Treatment B 300 900 2700 8100
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Mathspace Florida B.E.S.T - Algebra 1
a
b
c
Identify which treatment causes the number of microbes to increase linearly,
and describe the rate of change.
Determine which treatment will better control the number of microbes.
Justify your choice.
Scientists approximate that they’ll have a more effective treatment in 10 hours.
By that time, what will be the difference between the number of microbes in the
two samples?
Let’s extend our thinking
16 Winston and Rasiah want to organise a lunch time dominos tournament. They want a
knockout tournament, where the winner of each round progresses to the next round until
there are only two players left. The diagram shows the draw for the final, semi-final and
quarter final rounds.
a
Complete the table of values for the total number of players, p, that the competition can
accommodate given a number of rounds, r.
b
Number of rounds (r) 1 2 3 4
Number of players (p)
Winston and Rasiah want to make sure that each round of the tournament has every spot
filled. Find the values of p for which a tournament can be formed.
c Determine the equation relating r and p.
d
Over two weeks they can fit in 8 rounds of play. Determine how many players can they
accept into the tournament.
Chapter 5 Exponential Functions 247
17 An online business that sells phones online has been growing very quickly over the last year.
They have asked you to model their growth over the next three quarters, and further into
the future.
a
Looking at the quarterly data, you start by assuming the number of customers changed
in the same way each quarter over the first year.
Complete the table under this assumption to predict the number of phones sold over the
next three quarters.
Quarter (N) 1 2 3 4 5 6 7
Online Customers (C) 480 2880 17 280 103 680
b
c
d
Find an equation for the number of sales (C) in terms of the number of quarters that have
passed (N).
Predict the number of sales they will have 2.75 years after they started the business.
Considering your answer to part (c), determine if this is a reasonable goal for the company.
18 A teaspoon of sugar, weighing 4 grams, is poured into a cup of hot water and immediately
begins to dissolve. Each second the amount of sugar remaining is
the previous second.
a
Complete the table of values:
of the amount present in
Seconds passed (t) 0 1 2 3 4 5
Undissolved sugar in grams (y)
b Write an equation linking undissolved sugar, y, and time, t.
c
d
e
f
g
Find the difference in undissolved sugar between the first and second seconds.
Find the difference in undissolved sugar between the second and third seconds.
Describe the change in the amount of undissolved sugar over time.
According to this model, will all the sugar eventually dissolve? Explain your answer.
The water is heated more so that when the experiment is performed again,
the sugar dissolves twice as quickly. Determine if more or less sugar will dissolve
between the second and third seconds. Explain your reasoning.
19 Carbon-14 is a form of carbon that is found in all living plants and animals. When the plant
or animal dies, carbon-14 is no longer taken in, and as such, it is useful in estimating the age
of fossils. It has a half-life of 5730 years, which means it takes approximately 5730 years for
half of any amount of carbon-14 to decay.
a
b
c
Write a function, A, to represent the amount of carbon-14 remaining in a sample
containing 15 grams of carbon-14, after n years.
Find the amount of isotope left after 20 000 years. Round your answer to two
decimal places.
Carbon-11 is far more reactive and has a half-life of 20 minutes. Determine how much
carbon-11 would remain from a sample of 15 grams after 3 hours. Round your answer to
two decimal places.
248
Mathspace Florida B.E.S.T - Algebra 1
5.06 Percent growth
and decay
Concept summary
Exponential functions, both growth and decay, can be thought of in terms of their percent change:
f (x) = a(1 ± r) x
a The initial value of the exponential function
r The growth or decay rate of the exponential function, usually expressed as a decimal value
For functions in the form f (x) = a(1 + r) x , r is a growth rate, while for functions in the form
f (x) = a(1 − r) x , r is a decay rate.
Growth rate
The fixed percent by which an exponential function increases
Decay rate
The fixed percent by which an exponential function decreases
Worked examples
Example 1
Classify the exponential function f (x) = 5(1 − 0.03) x as either exponential growth or exponential
decay and identify both the initial value and the rate of growth or decay.
Approach
We know that an exponential function represents growth if the function is in the form f (x) = a(1 + r) x
and decay if it is in the form f (x) = a(1 − r) x .
Solution
This function represents exponential decay because it is in the form f (x) = a(1 − r) x .
The initial value is 5 and the decay rate is 0.03 or 3%.
Reflection
Growth and decay rates are represented as percentages.
Chapter 5 Exponential Functions 249
Example 2
Consider the exponential function modeled by the graph.
18
y
−4 −3 −2 −1
16
14
12
10
8
6
4
2
1 2 3 4
x
a Identify the initial value and growth rate.
Approach
The initial value is represented by the y-intercept. To find the growth rate, first find the constant
factor by evaluating the ratio of two consecutive outputs, then find |constant factor − 1|.
Solution
The initial value of the function is 4. The constant factor is
and the growth rate is
|1.5 − 1| = 0.5 = 50%.
b Write the equation of the function in the form y = a(1 ± r) x .
Solution
y = 4(1 + 0.5) x
250
Mathspace Florida B.E.S.T - Algebra 1
Example 3
Justin purchased a piece of sports memorabilia for $2900, and it is expected to increase in value
by 9% per year.
a Write a function, y, to represent the value of the piece of sports memorabilia after v years.
Approach
Since the memorabilia is predicted to increase in value we will use the growth rate form of
the function, f (x) = a(1 + r) x .
Solution
The initial value of the function is $2900 and the growth rate is 9% so the function is
y = 2900(1 + 0.09) v
Reflection
Remember to convert the rate as a percentage to a decimal by dividing it by 100.
b Evaluate the function for v = 8 and interpret the meaning in context.
Approach
In this context v represents the time, in years, and the output, y represents the value of Justin’s
sports memorabilia. We will evaluate the function and apply these units to interpret the meaning of
the solution.
Solution
A = 2900(1 + 0.09) 8 ≈ 5778.43 which tells us that the memorabilia will be worth approximately
$5778.43 after 8 years have passed.
What do you remember?
1 Classify each of the following situations as exponential growth, decay, or neither.
a
b
Mr. Alvarado bought $3800 worth of stocks in 2019. The value of the stocks has been
decreasing by 1% each year.
A piece of land was purchased for $80 000. The value of the land has slowly been
decreasing by 2% annually.
Chapter 5 Exponential Functions 251
c
d
Claire owns a chain of bakeries that consisted of 100 stores in 2000. There are 7% more
stores in the chain each year.
Michael has $1300 in his checking account. He gets paid $600 every 2 weeks.
2 For each of the following exponential functions, where x represents time:
i Classify as either exponential growth or exponential decay.
ii State the initial amount.
iii State the growth or decay rate.
a f (x) = 0.723(1 − 0.05) x b f (x) = 5.9(1 + 0.25) x
c f (x) = 10 000(1 + 0.13) x d f (x) = 2500(1 − 0.123) x
3 Consider the points shown on an
exponential growth function.
a
b
c
Copy and complete the table
of values.
x 0 1 2 3
y
State the initial value.
State the growth rate.
d Write the function relating x and y.
4 The local rat population is changing
according to the exponential function
f (t) = 840(1 − 0.04) t where t is the number
of years that have passed.
a
b
950
900
850
800
750
700
650
600
550
500
450
Determine the size of the initial population of rats.
Determine the rate at which the population is decreasing.
y
1 2 3 4
x
5 Consider the function where t represents time.
a
b
c
d
Find the initial value of the function.
Express the function in the form f (t) = 2(1 − r) t , where r is a decimal.
Does the function represent growth or decay of an amount over time?
State the rate of change per time period as a percentage.
6 Consider the table of values.
x 0 1 2 3 4
y 200 188 176.72 166.12 156.15
252
Mathspace Florida B.E.S.T - Algebra 1
Select the graph that could represent the function.
A y
B
y
200
200
150
150
100
100
50
50
x
x
1 2 3 4
1 2 3 4
C
y
D
y
200
200
150
150
100
100
50
50
x
x
1 2 3 4
1 2 3 4
7 Consider the table of values.
x 0 1 2 3 4
y 1500 1470 1440.60 41411.79 1383.55
Select the graph that could represent the function.
A y
B
y
2500
2500
2000
2000
1500
1500
1000
1000
500
x
500
x
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Chapter 5 Exponential Functions 253
C
y
D
y
2500
2500
2000
2000
1500
1500
1000
1000
500
x
500
x
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
8 The number of members at the local country club is expected to increase by 19% per year
from the current number of 160. The manager of the club is set to receive a bonus when the
number of members rises above 727. This can be expressed as y = 160(1.19) x .
Select the graph that represents the function.
A
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
y
1 2 3 4 5 6 7 8 9
x
B
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
y
1 2 3 4 5 6 7 8 9
x
C
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
y
1 2 3 4 5 6 7 8 9
x
D
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
y
1 2 3 4 5 6 7 8 9
x
254
Mathspace Florida B.E.S.T - Algebra 1
9 Candice purchased a rare vase worth $2000 and it is expected to increase in value by 7%
per year. This can be expressed as y = 2000(1 + 0.07) x . Select the graph that represents
the function.
A
4500
4000
3500
3000
2500
2000
1500
1000
500
y
1 2 3 4 5 6 7 8 9 10
x
B
4500
4000
3500
3000
2500
2000
1500
1000
500
y
1 2 3 4 5 6
x
C
4500
y
D
4500
y
4000
4000
3500
3500
3000
3000
2500
2500
2000
2000
1500
1500
1000
1000
500
x
500
x
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
Let’s practice
10 For each of the following graphs of a population, P over time, write the exponential growth
or decay function that models it:
a
100
90
80
70
60
50
40
30
20
10
P
1 2 3 4
t
b
360
320
280
240
200
160
120
80
40
P
1 2 3 4
t
Chapter 5 Exponential Functions 255
c
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
P
t
d
320
300
280
260
240
220
200
180
160
140
120
100
P
t
1 2 3 4 5 6 7 8 9
1 2 3 4
11 Marlow purchased a piece of sports memorabilia for $1300, and it is expected to increase in
value by 8% per year.
a
b
Write a function, y, to represent the value of the piece of sports memorabilia after
v years.
Find the value of the piece of sports memorabilia after 10 years.
12 Due to favorable market conditions, a company’s annual costs are expected to decrease
by 5% per year for the next few years. Their annual costs for the current financial year
are $322 000.
a
b
Write a function, y, to represent the company’s annual costs in m years.
Find the company’s annual costs in 5 years.
13 Tina currently follows 174 people on Snapface. The number of people she follows is growing
at a rate of 5% per month. Find how many people she will be following in 6 months.
14 The population of Juanita’s home town is 27 400 and has a growth rate of 6% each year.
Find the estimated population after 7 years.
15 The population of some native bees is declining at a rate of 8% per year. If there are 10 000
bees in a hive now, find the number of bees in 5 years time.
16 A new car purchased for $38 200 depreciates at a rate r each year. The value of the car for
the first two years is shown in the table below:
Years passed (n) 0 1 2
Value of car ($A) 38 200 37 818 37 439.82
a Find the value of r.
b
c
d
Write the rule for A, the value of the car n years after it is purchased.
Find the value of the car after 6 years.
A new motorbike purchased for the same amount depreciates according to the model
V = 38 200(1 − 0.03) n . Determine which vehicle depreciates more rapidly.
256
Mathspace Florida B.E.S.T - Algebra 1
17 A sample contains 60 grams of iodine-131, an isotope of iodine, which has a half-life of
8 days.
a
b
Write a function, A, to represent the amount of iodine-131 remaining in the sample
after n days.
Find the amount of the isotope left after 72 days. Round your answer to two
decimal places.
18 When a heated substance such as water starts to cool, its temperature T at time t minutes
after being left to cool is given by an exponential function.
a
b
The function can be written in the form T = a (1 − r) t . Explain what the variables a AND
r mean within context.
Substance P starts off at a temperature of 145°C and is left to cool in a room whose
temperature is a constant 0°C. By using the table below, find the equation for the
temperature of the substance.
c
d
Minutes passed (t) 0 1 2 3 4
Temperature (T) 145 87 52.2 31.32 18.792
Find the temperature after 10 minutes, correct to two decimal places.
Will the temperature of the substance ever reach 0°C?
e The temperature of another substance Q which also starts off at a temperature of 145°C
is modelled by T = 145(1 − 0.1) t . Which substance will cool more rapidly?
Let’s extend our thinking
19 The population of a town is expected to double in the next 20 years. Determine the rate, r,
the town will it grow each year, assuming growth is exponential. Give your answer as a
percentage correct to two decimal places. Explain your reasoning.
20 A flashmob suddenly appears at the foodcourt of the local mall and starts dancing.
They originally start with 16 performers, and by the end of every minute the number of
performers increases by 50%. The song ends after 4 minutes.
a
b
c
d
Write a function, f (t), to model the number of performers over time.
Determine the domain and range of the function.
Explain what the range means within context.
Determine if this function could continue to model this situation if the song continued.
Explain your reasoning.
Chapter 5 Exponential Functions 257
21 Juanita and Pedro are taking part in a study, and have to eat the same food and in equal
quantity. They were chosen such that Juanita incorporates lots of sugar into their diet while
Pedro consumes limited amounts of sugar in their diet.
After eating, their blood sugar level peaked, and was then continually measured,
with the following functions modelling their blood sugar level at time t hours:
• Juanita: f (t) = 179(1 − 0.7) t
• Pedro: g (t) = 132(1 − 0.4) t
a
State the highest blood sugar level measured for:
i Juanita ii Pedro
b
Find the rate, as a percentage, that the blood sugar level decreased each hour for:
i Juanita ii Pedro
c
Compare the reactions to sugar between Juanita and Pedro.
22 In a memory study, subjects are asked to memorize some content and then asked to recall
what they can remember each day after. The given graphs represent the percentage of
content remembered by two participants Maximilian (P) and Lucy (Q) after t days.
100%
Recall percentage
80%
60%
40%
20%
2
P
Q
4 6 8 10 12 14 16 18 20
Days
a
b
Each day, Maximilian finds that he has forgotten 15% of what he could recount the day
before. Form a model for the percentage P that Maximilian can still recall after t days.
Determine which of the following could be the model for the percentage that Lucy can
still recall after t days.
A Q = (1 − 0.15) t B Q = (1 − 0.17) t C Q = (1 − 0.13) t D Q = (1 − 0.19) t
c
d
Using your answer from part (c), determine the percentage of the previous day’s content
did Lucy forgets.
According to the models, determine if either of them will completely forget the content.
Explain your reasoning.
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5.07 Simple interest
Concept summary
Simple interest is a method for computing interest where the interest is computed from the original
principal alone, no matter how much money has accrued so far.
A = P (1 + rt)
A future value, or final amount
P principal, or present value
t number of years
r rate of interest per year
If we want to determine the total amount of interest, we can use the formula:
I = Prt
I Interest
P principal, or present value
r rate of interest per year
t number of years
Because P and r are constant and t, the time, is variable, this is a linear function. Therefore,
the growth of a sum of money by simple interest is linear growth. We can see this in the fact that
simple interest causes an account balance to grow by a constant amount per interval of time.
Worked examples
Example 1
Taylor’s investment of $5000 earns interest at a simple interest rate of 2.5% per year for 5 years.
a Use the simple interest formula to find the final balance of their investment.
Approach
To apply the simple interest formula we need to identify which variables we have given
information for. In this example we know that the principal is P = 5000, the rate is r = 0.025 and the
time is t = 5 years.
Solution
Substituting all given values into the formula gives us:
A = P (1 + rt)
A = 5000(1 + 0.025(5))
Simple interest formula
Substitute in values for the variables
Chapter 5 Exponential Functions 259
A = 5000(1.125) Simplify
A = 5625
Taylor will have $5625 after 5 years.
Reflection
Evaluate the product
When solving using the simple interest formulas, the rate must be in decimal form. To find the
decimal form of a percentage rate, divide the percentage value by 100.
In this example, we found the rate was 0.025 because
b Taylor wanted to have $6000 at the end of their investment term. Find the initial amount
they would need to invest at the same rate for the same length of time to have an ending value
of $6000.
Approach
In this example we know the ending balance is A = 6000, the rate is still r = 0.025, the time is still
t = 5, and the unknown value is the principal P.
Solution
Substituting all given values into the formula gives us:
A = P (1 + rt)
Simple interest formula
6000 = P (1 + 0.025(5)) Substitute in values for the variables
6000 = 1.125P Simplify
5333.33 = P Divide
Taylor would need to invest $533.33 to have an ending balance of $6000 after investing their
money for 5 years at a simple interest rate of 2.5%.
Example 2
Rei borrows $1200 at a simple interest rate of 8.2%, to help buy a new car. She doesn’t want to
pay more than $300 in interest over the duration of the loan. Calculate the maximum number of
years Rei can take to repay the loan while keeping her total interest at $300 or less.
Approach
We can use either formula to help solve this, but because we know the total maximum interest Rei
wants to pay, we will use the formula I = Prt. In this example we know the maximum total interest is
I = 300, the rate is 0.082 and the unknown value is the number of years t.
Since we don’t want the interest to exceed $300, we will setup an inequality.
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Mathspace Florida B.E.S.T - Algebra 1
Solution
Substituting all given values into the formula gives us:
I ≥ Prt
Simple interest formula
300 ≥ 1200(0.082) t Substitute in values for the variables
300 ≥ 98.4t Simplify
3.048 ≥ t Divide
Rei would need to repay the loan within approximately 3 years in order to pay no more than
$300 in interest.
What do you remember?
1 Calculate the ending balance of the following investments, assuming simple interest is
constantly accruing:
a
b
c
d
e
$1030 at 8% per year for 2 years
$1050 at a semiannual rate of 1.1% for 9 years
$7000 at 1.8% per quarter for 9 years
$8510 at a rate of 7% per year for 25 months
$6730 at 7% per year for 33 days (Note: 1 year = 365 days)
2 Calculate the simple interest charged on the following loans, assuming interest is constantly
accruing:
a
b
c
d
$9300 at 2% per year for 6 years
$7580 at a rate of 6% per year for 13 months
$9040 at 7% per year for 41 weeks (Note: 1 year = 52 weeks)
$7180 at 6% per year for 193 days (Note: 1 year = 365 days)
Let’s practice
3 Amelia made loan repayments totalling to $4320 on a loan of $4000 with simple interest
gained over 4 years.
a
b
Calculate the total interest charged on the loan.
Calculate the annual interest rate.
4 Dave invested $848 at 8% per year simple interest. He wants to know the number of years
it will take the investment to grow to $1458.56.
a
Calculate the total amount in interest that will be earned on the investment.
b Calculate the number of years it will take the investment to grow to $1458.56.
Chapter 5 Exponential Functions 261
5 $3973.00 is to be invested to earn $148.99 at a simple interest rate of 3% per year.
Find the number of years it will take to earn the interest. Round your answer to two
decimal places.
6 $85 620 is to be invested to earn $8990.10 at a simple interest rate of 7% per year.
Find the number of years it will take to earn the interest.
7 Dave invests in government bonds with a simple interest rate of 3% per year.
Calculate the size of Dave’s initial investment if he earned $517.20 interest after 10 years.
8 Find the principal investment that will earn:
a
b
c
d
$865 simple interest, at 3% annually over 6 years
$390 simple interest, at 2.5% annually over 4 years
$1200 simple interest, at 1% per month over 2 years
$3260 simple interest, at 2.5% per week over 6 months
9 Beth takes out a car loan of $3000 at a simple interest rate of 8% per year.
She plans to repay the loan over 4 years through regular monthly payments.
a
b
Calculate the total interest that Beth will be charged over the duration of the loan.
Calculate the value of each loan repayment.
10 Bianca takes out a loan of $800 to pay for an online course. Simple interest is added to the
loan value at 9% per year, charged monthly. Calculate the interest she pays in total if she
repays the loan in 9 months.
11 James’s investment of $3000 earned simple interest of 2% per year for the first 10 years and
6% per year for the next 7 years. Calculate the total amount of interest earned.
12 For each of the following investments find the annual interest rate, r, as a percentage
rounded to one decimal place.
a
b
c
$2220 invested over 8 years earns $728.16 in interest
$7500 invested over 10 quarters earns $1781.25 in interest
$6600 invested over 20 months earns $528.00 in interest
13 Asha was calculating the principal that would earn $520 simple interest over 6 months at a
rate of 4% annually, but her answer of approximately $2167 didn’t make sense.
Find and explain the error in Asha’s work:
I = Prt
520 = P (0.04)(6) Substitute the known values into the equation
520 = 0.24P Multiply
2166.67 = P Divide both sides by 0.24
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Mathspace Florida B.E.S.T - Algebra 1
Let’s extend our thinking
14 Scott and Kirsten are looking to invest $15 600 and $26 700 respectively for 9 years.
Principal
Rate per year
Between $7400 and $17 400 3%
Between $17 400 and $27 400 4%
Greater than $27400 5%
a
b
Use the interest rates in the table to compare their investments.
Scott suggests that instead they should combine their investments to earn more in
interest. Determine the additional interest they would earn by combining their principals.
15 If $2196 is invested at 2% simple interest annually for 8 years, what simple interest rate, r,
would earn the same amount of interest in only 5 years?
16 Jolanda has $5000 to invest at a 1% simple interest rate and Idan has $5 000 000 to invest
at the same rate.
a
b
Calculate the difference in their earnings after 5 years.
Determine the interest rate Jolanda would need to earn the same interest as Idan after
5 years.
17 Explain the similarities between simple interest and linear growth.
Chapter 5 Exponential Functions 263
5.08 Compound interest
Concept summary
Compound interest is a method for computing interest where the interest is computed from the
original principal combined with all interest accrued so far.
A future value, or final amount
P principal, or present value
r rate of interest per year
n number of times interest is compounded per year
t number of years
Because compound interest has the variable t in the exponent while P, r, and n are constant,
it is considered an exponential function. We can see this in how a sum of money earning
compound interest grows by a constant percent rate per unit of time.
We can think of compound interest as a repeated application of simple interest.
Worked examples
Example 1
Frasier’s investment of $200 earns interest at a rate of 4% per year, compounded annually for
5 years.
a Use the compound interest formula to find the final balance of his investment.
Approach
To apply the compound interest formula we need to identify which variables we have given
information for. In this example we know that the principal is P = 200, the rate is r = 0.04 and the
time is t = 5. If the interest is compounded annually, then we also know n = 1 since annually means
once per year.
Solution
Frasier will have $243.33 after 5 years.
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Mathspace Florida B.E.S.T - Algebra 1
b Frasier wanted to have $300 at the end of his investment term. Find the initial amount would he
need to invest at the same rate to have an ending value of $300.
Approach
In this example we know the ending balance is A = 300, the rate is still r = 0.04, the time is still
t = 5 and the interest is still compounded annually so n = 1.
Solution
Substituting all given values into the formula gives us:
Compound interest formula
Substitute in values for the variables
Simplify
Divide
Frasier would need to invest $246.58 to have an ending balance of $300 after investing his money
for 5 years at a rate of 4% compounded annually.
Example 2
A $1610 investment earns interest at 4.5% per year compounded quarterly over 10 years.
Use the compound interest formula to calculate the value of this investment to the nearest cent.
Solution
We are given that the present value is P = 1610 and the annual rate is r = 0.045. The investment
is compounded quarterly which means four times per year so n = 4 for t = 10 years. Substituting
these values into the compound interest formula gives us:
Compound interest formula
Substitute in values for the variables
Simplify
If we evaluate the expression for A, rounding to two decimal places, we find that the value of this
investment is $2518.65
Chapter 5 Exponential Functions 265
Example 3
Decide if each expression represents simple interest or compound interest and explain your decision.
a A = 500(1 + 0.02) 6
Approach
Simple interest will take the form, A = P (1 + rt) and compound interest will take the form
Solution
Compound interest, since the expression is exponential.
Reflection
This is compound interest where P = 500, r = 0.02, n = 1 and t = 6.
b A = 500(1 + 0.05(3))
Approach
Simple interest will take the form, A = P (1 + rt) and compound interest will take the form
Solution
Simple interest, since the expression is linear.
Reflection
This is simple interest where P = 500, r = 0.05 and t = 3.
What do you remember?
1 State whether the following statements are true or false about compound interest:
a Interest is earned on the principal.
b The interest in any time period is calculated using only the original principal.
c Interest is earned on any accumulated interest.
d The amount of interest earned in any time period changes from one period to the next.
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Mathspace Florida B.E.S.T - Algebra 1
2 $4400 is invested for three years at a rate of 5% per year, compounded annually.
a Copy and complete the table to determine the final value of the investment.
Balance + interest Total balance Interest earned
First year − $4400 $220
Second year $4400 + $220 $4620 $231
Third year $4620 + $ $ $
Fourth year $4851 + $ $ −
b
Calculate the total interest earned over the three years.
3 Roberto’s investment of $70 000 earns interest at 5% per year, compounded annually
over 3 years.
a
b
Calculate the value of the investment after 3 years.
Calculate the amount of interest earned.
4 Hannah borrows $5000 at a rate of 4.2% per year, compounded annually.
a
b
After 4 years, Hannah repays the loan all at once. How much money does she pay back
in total?
How much interest was generated on the loan over the four years?
Let’s practice
5 $1000 is invested at 2% per year, compounded annually. The table tracks the growth of the
principal over three years.
Time Period
n (years)
Value at beginning
of time period
Value at end
of time period
Interest earned
in time period
1 $1000 $ $
2 $ $1040.40 $
3 $1040.40 $1061.21 $
a
b
Copy and complete the table.
Calculate the total interest earned over the three years.
6 Calculate the value of the following investments:
a A $7230 investment earns interest at 3% per year, compounded quarterly over 7 years.
b A $5520 investment earns interest at 0.8% per year, compounded monthly over 2 years.
c A $1980 investment earns interest at 0.45% per year, compounded weekly over 2 years.
d A $3000 investment earns interest at 4.2% per year, compounded daily over 14 years.
Chapter 5 Exponential Functions 267
7 Calculate the amount of interest earned by the following investments:
a
b
c
d
Beth’s investment of $4800 earns interest at 3.3% per year, compounded monthly over
13 years.
James’s investment of $4210 earns interest at 0.78% per year, compounded weekly
over 16 years.
Davinda’s investment of $5390 earns interest at 2.1% per year, compounded daily over
11 years.
Una’s investment of $80 000 earns interest at 0.5% per year, compounded semiannually
over 10 years.
8 When Yuri got his first job, he invested $10 000 in an account with a 1.3% annual
growth rate.
a
b
Write a function, A (t), that represents the value of this investment t years after Yuri
started his job.
Calculate how much more the investment will be worth when Yuri has been working for
12 years than when he had been working for 5 years.
9 Jerome has just won $50 000. When he retires in 15 years, he wants to have $94 000 in
his fund which earns 6% interest annually. Determine how much of his winnings he needs to
invest now to achieve this.
10 Victoria has been promised an inher