chapter 1 student notes

Chapter 1 

Functions and Their Graphs 

Course Number 

Section 1.1 Lines in the Plane 

Objective: In this lesson you learned how to find and use the slope of a 

line to write and graph linear equations. 

Instructor 

Date 

Important Vocabulary 

Define each term or concept. 

Slope The number of units a line rises or falls vertically for each unit of horizontal 

change from left to right. 

Parallel Two distinct nonvertical lines are parallel if and only if their slopes are 

equal. 

Perpendicular Two distinct nonvertical lines are perpendicular if and only if their 

slopes are negative reciprocals of each other. That is, m 1 = – 1/m 2 . 

I. The Slope of a Line (Pages 3−4) 

The formula for the slope of a line passing through the points 

(x 1 , y 1 ) and (x 2 , y 2 ) is m = (y 2 − y 1 )/(x 2 − x 1 ) . 

What you should learn 

How to find the slopes of 

lines 

To find the slope of the line through the points (− 2, 5) and 

(4, − 3), . . . subtract 5 from − 3 and divide this result by the 

difference of 4 and − 2. 

A line whose slope is positive rises from left to right. 

A line whose slope is negative falls from left to right. 

A line with zero slope is horizontal . 

A line with undefined slope is vertical . 

II. The Point-Slope Form of the Equation of a Line 

(Pages 5−6) 

The point-slope form of the equation of a line is 

y − y 1 = m(x − x 1 ) . 


How to write linear 

equations given points on 

lines and their slopes 

This form of equation is best used to find the equation of a line 

when . . . you know at least one point that the line passes 

through and the slope of the line. 

Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE 

Copyright © Houghton Mifflin Company. All rights reserved. 1

2 Chapter 1 Functions and Their Graphs 

The two-point form of the equation of a line is 

y − y 1 = (y 2 − y 1 )/(x 2 − x 1 )⋅(x − x 1 ) . 

The two-point form of equation is best used to find the equation 

of a line when . . . you know the coordinates of two points on 

the line. 

Example 1: Find an equation of the line having slope − 2 that 

passes through the point (1, 5). 

y = − 2x + 7 

The approximation method used to estimate a point between two 

given points is called linear interpolation . The 

approximation method used to estimate a point that does not lie 

between two given points is called linear extrapolation . 

A linear function has the form f(x) = mx + b . Its graph 

is a line that has slope m and a y-intercept at 

(0, b) . 

III. Sketching Graphs of Lines (Pages 7−8) 

The slope-intercept form of the equation of a line is 

y = mx + b , where m is the slope and the 

y-intercept is ( 0 , b ). 


How to use slopeintercept 

forms of linear 

equations to sketch lines 

Example 2: Determine the slope and y-intercept of the linear 

equation 2 x − y = 4 . 

The slope is 2 and the y-intercept is (0, − 4). 

The equation of a horizontal line is y = b . The slope of a 

horizontal line is 0 . The y-coordinate of every point on the 

graph of a horizontal line is b . 

The equation of a vertical line is x = a . The slope of a 

vertical line is undefined . The x-coordinate of every point 

on the graph of a vertical line is a . 

The general form of the equation of a line is 

Ax + By + C = 0 . 


Copyright © Houghton Mifflin Company. All rights reserved.

Section 1.1 Lines in the Plane 3 

Every line has an equation that can be written in 

form . 

general 

When a graphing utility is used to sketch a straight line, the 

graph of the line may not visually appear to have the slope 

indicated by its equation because . . . of the viewing window 

used for the graph. 

Example 3: Use a graphing utility to graph the linear equation 

2 x − y = 4 using (a) a standard viewing window, 

and (b) a square window. 

(a) 

(b) 

IV. Parallel and Perpendicular Lines (Pages 9−10) 

The relationship between the slopes of two lines that are parallel 

is . . . that the slopes are equal. 


How to use slope to 

identify parallel and 

perpendicular lines 

The relationship between the slopes of two lines that are 

perpendicular is . . . that the slopes are negative reciprocals of 

each other. 

A line that is parallel to a line whose slope is 2 has slope 2 . 

A line that is perpendicular to a line whose slope is 2 has slope 

− 1/2 . 

What must be done to make the graphs of two perpendicular 

lines appear to intersect at right angles when they are graphed 

using a graphing utility? 

The perpendicular lines must be graphed using a square viewing 

window. 




Example 4: Use a graphing utility to graph the perpendicular 

lines y = 2x 

− 3 and y = −0 .5x 

+ 5 using (a) a 

standard viewing window, and (b) a square 

window. 

(a) 

(b) 

Additional notes 

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Homework Assignment 

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Exercises 



Section 1.2 Functions 5 

Section 1.2 Functions 

Objective: In this lesson you learned how to evaluate functions and find 

their domains. 

Course Number 

Instructor 

Date 



Function A function f from a set A to a set B is a relation that assigns to each element 

x in the set A exactly one element y in the set B. 

Domain The set of inputs of the function f. 

Range The set of all outputs for the given set of inputs of the function f. 

Independent variable A variable in an equation that represents a function that can 

take on any value for which the function is defined. 

Dependent variable A variable in an equation that represents a function whose value 

depends on the value of the independent variable. 

I. Introduction to Functions (Pages 16−18) 

A rule of correspondence that matches quantities from one set 

with items from a different set is a relation . 


How to decide whether a 

relation between two 

variables represents a 

function 

In functions that can be represented by ordered pairs, the first 

coordinate in each ordered pair is the input and the 

second coordinate is the output . 

Some characteristics of a function from Set A to Set B are . . . 

1) Each element of A must be matched with an element 

of B. 

2) Some elements of B may not be matched with any 

element of A. 

3) Two or more elements of A may be matched with the 

same element of B. 

4) An element of A (the domain) cannot be matched with 

two different elements of B. 

To determine whether or not a relation is a function, . . . 

decide whether each input value is matched with exactly one 

output value. 




If any input value of a relation is matched with two or more 

output values, . . . the relation is not a function. 

Some common ways to represent functions are . . . 

1) Verbally in a sentence 

2) Numerically in a table or list of ordered pairs 

3) Graphically by points on a graph in the coordinate plane 

4) Algebraically by an equation in two variables 

Example 1: Decide whether the table represents y as a function 

of x. 

x − 3 − 1 0 2 4 

y 5 − 12 5 3 14 

Yes, this table represents y as a function of x. 

II. Function Notation (Pages 18−20) 

The symbol f(x) is function notation for the value 

of f at x or simply f of x. The symbol f(x) corresponds to the 

y-value for a given x. 

Keep in mind that f is the name of the function, 

whereas f(x) is the output value of the function at the 

input value x. 


How to use function 

notation and evaluate 

functions 

In function notation, the input is the independent 

variable and the output is the dependent variable. 

3 

Example 2: If f ( w) 

= 4w 

− 5w 

− 7w 

+ 13 , describe how to 

find f (−2) 

. 

Replace each occurrence of w in the function by 

− 2 and evaluate the resulting numerical 

expression. 

A piecewise-defined function is . . . a function that is defined 

by two or more equations over a specified domain. 

2 



Section 1.2 Functions 7 

III. The Domain of a Function (Page 20−21) 

The implied domain of a function defined by an algebraic 

expression is . . . the set of all real numbers for which the 

expression is defined. 


How to find the domains 

of functions 

In general, the domain of a function excludes values that . . . 

would cause division by zero or result in the even root of a 

negative number. 

For example, the implied domain of the function f ( x) 

= 5x 

− 8 

is . . . the set of all real numbers greater than or equal to 8/5, or 

[8/5, ∞). 

IV. Applications of Functions (Page 22) 

Example 3: The price P (in dollars) of a child’s handmade 

sweater is given by the function P(s) = 3s + 15, 

where s represents the size (size 1, size 2, etc.) of 

the sweater. Use this function to find the price of a 

child’s size 5 handmade sweater. 

$30 


How to use functions to 

model and solve real-life 

problems 

V. Difference Quotients (Page 23) 

A difference quotient is defined as . . . 

[f(x + h) − f(x)]/h, h ≠ 0. 


How to evaluate 

difference quotients 

Describe a real-life situation which can be represented by a 

function. 

Answers will vary. 






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Section 1.3 Graphs of Functions 9 

Section 1.3 Graphs of Functions 

Objective: In this lesson you learned how to analyze the graphs of 

functions. 

Course Number 

Instructor 

Date 



Graph of a function The collection of ordered pairs (x, f(x)) such that x is in the 

domain of f. 

Greatest integer function The greatest integer function is denoted by [|x|] and is 

defined as the greatest integer less than or equal to x. 

Step function A function whose graph resembles a set of stair steps. 

Even function A function whose graph is symmetric with respect to the y-axis. 

Odd function A whose graph is symmetric with respect to the origin. 

I. The Graph of a Function (Pages 30−31) 

Explain the use of open or closed dots in the graphs of functions. 

The use of dots, open or closed, at the extreme left or right points 

of a graph indicates that the graph does not extend beyond these 

points. If no such dots are shown, assume that the graph extends 

beyond these points. 


How to find the domains 

and ranges of functions 

and how to use the 

Vertical Line Test for 

functions 

To find the domain of a function from its graph, . . . examine 

the graph to see for which x-values the graph is plotted. 

To find the range of a function from its graph, . . . 

graph to see for which y-values the graph is plotted. 

examine the 

The Vertical Line Test for functions states . . . that a set of 

points in a coordinate plane is the graph of y as a function of x if 

and only if no vertical line intersects the graph at more than one 

point. 




Example 1: Decide whether each graph represents y as a 

function of x. 

(a) 

(b) 

(a) Yes, this is a graph of y as a 

function of x 

(b) No, this is not a graph of y as 

a function of x 

II. Increasing and Decreasing Functions (Page 32) 

A function f is increasing on an interval if, for any x 1 and x 2 in 

the interval, . . . x 1 < x 2 implies f(x 1 ) < f(x 2 ). 

A function f is decreasing on an interval if, for any x 1 and x 2 in 

the interval, . . . x 1 < x 2 implies f(x 1 ) > f(x 2 ). 

A function f is constant on an interval if, for any x 1 and x 2 in the 

interval,. . . f(x 1 ) = f(x 2 ). 


How to determine 

intervals on which 

functions are increasing, 

decreasing, or constant 

Given a graph of a function, to find an interval on which the 

function is increasing . . . search for an interval over which 

the graph rises from left to right. 


function is decreasing . . . search for an interval over which 

the graph falls from left to right. 


function is constant . . . search for an interval over which the 

graph is level. 

III. Relative Minimum and Maximum Values 

(Pages 33−34) 

A function value f(a) is called a relative minimum of f if . . . 

there exists an interval (x 1 , x 2 ) that contains a such that 

x 1 < x < x 2 implies f(a) ≤ f(x). 


How to determine 

relative maximum and 

relative minimum values 

of functions 



Section 1.3 Graphs of Functions 11 

A function value f(a) is called a relative maximum of f if . . . 

there exists an interval (x 1 , x 2 ) that contains a such that 

x 1 < x < x 2 implies f(a) ≥ f(x). 

The point at which a function changes from increasing to 

decreasing is a relative maximum . The point at which 

a function changes from decreasing to increasing is a relative 

minimum . 

To approximate the relative minimum or maximum of a function 

using a graphing utility, . . . use the zoom and trace features to 

approximate the lowest or highest point on the graph, OR use a 

built-in program or feature for finding minimum or maximum 

values. 

Example 2: Suppose a function C represents the annual 

number of cases (in millions) of chicken pox 

reported for the year x in the United States from 

1960 through 2000. Interpret the meaning of the 

function’s minimum at (1998, 3). 

The lowest number of cases of chicken pox reported 

during this period was 3 million cases in 1998. 

IV. Graphing Step Functions and Piecewise-Defined 

Functions (Page 35) 

Describe the graph of the greatest integer function. 

The graph of the greatest integer function jumps vertically one 

unit at each integer and is constant (a horizontal line segment) 

between each pair of consecutive integers. 


How to identify and 

graph step functions and 

other piecewise-defined 

functions 

Example 3: Let 

f ( x) 

= x , the greatest integer function. Find 

f (3.74) . 

3 

To sketch the graph of a piecewise-defined function, . . . 

sketch the graph of each equation of the piecewise-defined 

function over the appropriate portion of the domain. 




V. Even and Odd Functions (Pages 36−37) 

A graph is symmetric with respect to the y-axis if, whenever 

(x, y) is on the graph, (− x, y) is also on the graph. A graph 

is symmetric with respect to the x-axis if, whenever (x, y) is on 

the graph, (x, − y) is also on the graph. A graph is 

symmetric with respect to the origin if, whenever (x, y) is on the 

graph, (− x, − y) is also on the graph. A graph that is 

symmetric with respect to the x-axis is . . . 

not the graph of a function (except for the graph of y = 0). 


How to identify even and 

odd functions 

A function f is even if, for each x in the domain of f, 

f(−x) = f(x) . 

A function f is even if, for each x in the domain of f, 

f(−x) = f(x) . 


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Section 1.4 Shifting, Reflecting, and Stretching Graphs 13 

Section 1.4 Shifting, Reflecting, and Stretching Graphs 

Objective: In this lesson you learned how to identify and graph shifts, 

reflections, and nonrigid transformations of functions. 

Course Number 

Instructor 

Date 



Vertical shift A transformation of the graph of y = f(x), represented by h(x) = f(x) ± c, 

in which the graph is shifted upward or downward c units respectively (c is a positive 

real number). 

Horizontal shift A transformation of the graph of y = f(x), represented by 

h(x) = f(x ± c), in which the graph is shifted to the left or to the right c units 

respectively (c is a positive real number). 

Rigid transformations Transformations in which the basic shape of the graph is 

unchanged. 

Nonrigid transformations Transformations that cause a distortion (a change in the 

original shape of the graph). 

I. Summary of Graphs of Parent Functions (Page 42) 

Sketch an example of each of the six most commonly used 

functions in algebra. 


How to recognize graphs 

of parent functions 

Constant Function 

y 

Identity Function 

y 

x 

x 

Absolute Value Function 

y 

Square Root Function 

y 

x 

x 




Quadratic Function 

y 

Cubic Function 

y 

x 

x 

II. Vertical and Horizontal Shifts (Pages 43−44) 

Let c be a positive real number. Complete the following 

representations of shifts in the graph of y = f (x) 

: 

1) Vertical shift c units upward: h(x) = f(x) + c 

2) Vertical shift c units downward: h(x) = f(x) − c 

3) Horizontal shift c units to the right: h(x) = f(x − c) 

4) Horizontal shift c units to the left: h(x) = f(x + c) 


How to use vertical and 

horizontal shifts and 

reflections to graph 

functions 

Example 1: Let f ( x) 

= x . Write the equation for the 

function resulting from a vertical shift of 3 units 

downward and a horizontal shift of 2 units to the 

right of the graph of f (x) 

. 

f(x) = | x − 2 | − 3 

III. Reflecting Graphs (Pages 45−46) 

A reflection in the x-axis is a type of transformation of the graph 

of y = f(x) represented by h(x) = − f(x) . A reflection in 

the y-axis is a type of transformation of the graph of y = f(x) 

represented by h(x) = f(− x) . 


= x . Describe the graph of g( 

x) 

= − x 

in terms of f . 

The graph of g is a reflection of the graph of f in 

the x-axis. 



Section 1.4 Shifting, Reflecting, and Stretching Graphs 15 

IV. Nonrigid Transformations (Page 47) 

Name three types of rigid transformations: 

1) Horizontal shifts 

2) Vertical shifts 

3) Reflections 


How to use nonrigid 

transformations to graph 

functions 

Rigid transformations change only the position of the 

graph in the coordinate plane. 

Name four types of nonrigid transformations: 

1) Vertical stretch 

2) Vertical shrink 

3) Horizontal shrink 

4) Horizontal stretch 

A nonrigid transformation y = cf (x) 

of the graph of y = f (x) 

is 

a vertical stretch if c > 1 or a vertical shrink if 

0 < c < 1. A nonrigid transformation y= f( cx) 

of the graph of 

y = f (x) is a horizontal shrink if c > 1 or a 

horizontal stretch if 0 < c < 1. 


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Section 1.5 Combinations of Functions 17 

Section 1.5 Combinations of Functions 

Objective: In this lesson you learned how to find arithmetic 

combinations and compositions of functions. 

Course Number 

Instructor 

Date 

I. Arithmetic Combinations of Functions (Pages 51−53) 

Just as two real numbers can be combined by the operations of 

addition, subtraction, multiplication, and division to form other 

real numbers, two functions f and g can be combined to create 

new functions such as the sum, difference, product, and 

quotient 

of f and g to create new functions. 


How to add, subtract, 

multiply, and divide 

functions 

The domain of an arithmetic combination of functions f and g 

consists of . . . all real numbers that are common to the 

domains of f and g. In the case of the quotient f(x)/g(x), there is 

the further restriction that g(x) ≠ 0. 

Let f and g be two functions with overlapping domains. 

Complete the following arithmetic combinations of f and g for all 

x common to both domains: 

1) Sum: ( f + g)( 

x) 

= f(x) + g(x) 

2) Difference: ( f − g)( 

x) 

= f(x) − g(x) 

3) Product: ( fg )( x) 

= f(x) · g(x) 

⎛ f ⎞ 

4) Quotient: ⎜ 

⎟(x) 

= 

⎝ g ⎠ 

f(x) ÷ g(x), g(x) ≠ 0 

To use a graphing utility to graph the sum of two functions, . . . 

enter the first function as y 1 and enter the second function as y 2 . 

Define y 3 as y 1 + y 2 , then graph y 3 . 


= 7x 

− 5 and g( x) 

= 3 − 2x 

. Find 

( f − g)(4) . 

28 




II. Compositions of Functions (Pages 53−56) 

The composition of the function f with the function g is 

( f g)( x ) = f(g(x)) . 


How to find 

compositions of one 

function with another 

function 

For the composition of the function f with g, the domain of 

f g is . . . the set of all x in the domain of g such that 

g(x) is in the domain of f. 


= 3x 

+ 4 and let g ( x) 

= 2x 2 −1. Find 

(a) ( f g)( 

x) 

and (b) ( g f )( x) 

. 

(a) 6x 2 + 1 (b) 18x 2 + 48x + 31 

III. Applications of Combinations of Functions (Page 57) 

The function f ( x) 

= 0. 06x 

represents the sales tax owed on a 

purchase with a price tag of x dollars and the function 

g( x) 

= 0. 75x 

represents the sale price of an item with a price tag 

of x dollars during a 25% off sale. Using one of the combinations 

of functions discussed in this section, write the function that 

represents the sales tax owed on an item with a price tag of x 

dollars during a 25% off sale. 


How to use combinations 

of functions to model and 

solve real-life problems 

(f g)(x) = (g f)(x) = 0.045x 



Page(s) 

Exercises 



Section 1.6 Inverse Functions 19 

Section 1.6 Inverse Functions 

Objective: In this lesson you learned how to find inverse functions 

graphically and algebraically. 

Course Number 

Instructor 

Date 



Inverse function Let f and g be two functions. If f(g(x)) = x for every x in the domain 

of g and g(f(x)) = x for every x in the domain of f, then g is the inverse function of the 

function f. The function g is denoted by f −1 . 

One-to-one A function f is one-to-one if, for a and b in its domain, f(a) = f(b) implies 

a = b. 

Horizontal Line Test A function is one-to-one if every horizontal line intersects the 

graph of the function at most once. 

I. Inverse Functions (Pages 62−64) 

For a function f that is defined by a set of ordered pairs, to form 

the inverse function of f, . . . interchange the first and second 

coordinates of each of these ordered pairs. 


How to find inverse 

functions informally and 

verify that two functions 

are inverse functions of 

each other 

For a function f and its inverse f −1 , the domain of f is equal to 

the range of f −1 , and the range of f is equal to 

the domain of f −1 . 

To verify that two functions, f and g, are inverses of each other, 

. . . find f(g(x)) and g(f(x)). If both of these compositions are 

equal to the identity function, then the functions are inverses of 

each other. 

Example 1: Verify that the functions f ( x) 

= 2x 

− 3 and 

+ 3 

g ( x) 

= x are inverses of each other. 

2 

II. The Graph of an Inverse Function (Page 65) 

If the point (a, b) lies on the graph of f, then the point 

( b , a ) lies on the graph of f −1 and vice versa. The 

graph of f −1 is a reflection of the graph of f in the line 

y = x . 


How to use graphs of 

functions to decide 

whether functions have 

inverse functions 




III. The Existence of an Inverse Function (Page 66) 

If a function is one-to-one, that means . . . that no two 

elements in the domain of the function correspond to the same 


How to determine if 

functions are one-to-one 

element in the range of the function. 

A function f has an inverse f −1 if and only if . . . 

f is one-to-one. 

To tell whether a function is one-to-one from its graph, . . . 

simply use the Horizontal Line Test, that is, check to see that 

every horizontal line intersects the graph of the function at most once. 

Example 2: 

Does the graph of the function at the right have an 

inverse function? Explain. 

No, it doesn’t pass the Horizontal Line Test. 

IV. Finding Inverse Functions Algebraically (Pages 67−68) 

To find the inverse of a function f algebraically, . . . 

1) Use the Horizontal Line Test to decide whether f has an 


How to find inverse 

functions algebraically 

inverse function. 

2) In the equation for f(x), replace f(x) by y. 

3) Interchange the roles of x and y, and solve for y. 

4) Replace y by f −1 (x) in the new equation. 

5) Verify that f and f −1 are inverse functions of each other by showing 

that the domain of f is equal to the range of f −1 , the range of f is equal 

to the domain of f −1 , and f(f −1 (x)) = x and f −1 (f(x)) = x. 

Example 3: Find the inverse (if it exists) of f ( x) 

= 4x 

− 5 . 

f −1 (x) = 0.25x + 1.25 


Page(s) 

Exercises 



Section 1.7 Linear Models and Scatter Plots 21 

Section 1.7 Linear Models and Scatter Plots 

Objective: In this lesson you learned how to use scatter plots and a 

graphing utility to find linear models for data. 

Course Number 

Instructor 

Date 



Fitting a line to data The process of finding a linear model to represent the 

relationship described by a scatter plot. 

I. Scatter Plots and Correlation (Pages 73−74) 

Many real-life situations involve finding relationships between 

two variables. If data is collected and written as a set of ordered 

pairs, the graph of such a set is called a scatter plot . 


How to construct scatter 

plots and interpret 

correlation 

For a collection of ordered pairs of the form (x, y), if y tends to 

increase as x increases, the collection is said to have a(n) 

positive correlation . If y tends to decrease as x 

increases, the collection is said to have a(n) negative 

correlation . 

II. Fitting a Line to Data (Pages 74−77) 

To fit a line to data represented in a scatter plot, . . . sketch 

the line that appears to fit the points, find two points on the line, 

and then find the equation of the line that passes through the two 

points, OR use the linear regression feature of a graphing utility 

to find the linear model. 


How to use scatter plots 

and a graphing utility to 

find linear models for 

data 

To measure how well a linear model fits the data used to find the 

model, . . . compare the actual values with the values given 

by the model. 




The correlation coefficient r of a set of data varies between 

− 1 and 1 . The closer | r | is to 1, the better . . . 

the points can be described by a line. 

Example 1: The numbers of U.S. Navy personnel p in 

thousands on active duty for the years 2002 

through 2006 are shown in the table. Use the 

regression capabilities of a graphing utility to find 

a linear model for the data. Let t represent the year 

with t = 2 corresponding to 2002. 

Year 2002 2003 2004 2005 2006 

p 383 381 376 364 353 

(Source: U.S. Department of Defense) 

p = − 7.7t + 402.2 

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chapter 1 student notes

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