Firefighter (pp. 1 of 2)

Firefighter (pp. 1 of 2) 

Algebra 2 

HS Mathematics 

Unit: 07 Lesson: 01 

A fire truck can get 12 feet from the base of a 

burning building. The extension ladder is used to 

reach the upper floors. 

x 

12 feet 

1. If the ladder is extended to a length of 13 

feet, the how far above the truck can the 

ladder reach on the building? (Draw a 

triangle to explain the problem, and show 

the work that leads to your answer.) 

y 

Because the ladder can be extended to various 

lengths, it can reach higher and higher up the side 

of the building. 

Answer the questions below to describe the 

relationship between these two quantities. 

2. What height can be reached above the truck if 

the ladder is extended to a length of 15 feet? 

(Draw a triangle to explain the problem, and 

show the work that leads to your answer.) 

3. In this problem, let x = the length (in feet) to which the ladder is extended, and y = the height (in 

feet) it can reach above the truck onto the building. Complete the table and graph of this 

relationship. 

Ladder 

length (ft) 

x 

12 

13 

15 

20 

25 

30 

Height 

above truck 

(ft) 

y 

Height above the truck (ft) 

40 

30 

20 

10 

37 

2 10 20 30 40 

Length of the Ladder (ft) 

©2010, TESCCC 08/01/10 page 7 of 18

Firefighter (pp. 2 of 2) 

Algebra 2 



A function to model the relationship between the length of the ladder (x) and the height it can extend 

2 

above the truck is f( x) x 144 . 

4. Where does this function come from? Or, how is it derived? 

5. To see the graph, you may have to change your 

calculator’s WINDOW. What WINDOW settings 

would match the graph on the preceding page? 

Xmin = ____ 

Xmax = ____ 

Xscl = ____ 

Ymin = ____ 

Ymax = ____ 

Yscl = ____ 

6. Type this function into your calculator’s Y = menu. Then go to the TABLE (2 nd , GRAPH). Scroll 

through the table until you see entries for X = 5, 6, 7, and 8. Record what you see below. 

X Y 1 

5 

6 

7 

8 

C) Explain what this means in context of the problem (with 

ladders and buildings). 

D) What does this mean in terms of the numbers? (In other 

words, why does the calculator do this?) 

7. What would you say is the domain of the function 

f x 

2 

( ) x 144 

? 

8. What is the range? 

9. How do the numerical domain and range relate to the original word problem? 


Extra Practice: Solving with x 2 

Algebra 2 



Multiple Choice. Read each question carefully. Show your work in the space provided. 

1. During a storm, a telephone pole broke 

and fell as shown. 

2. From her campsite, Wanda hiked 2 miles 

due east, then 1.5 miles due south, then 

walked directly back to the campsite. 

Camp 

12 feet 

2 miles 

16 feet 

1.5 miles 

What was the original height of the pole? 

A) 20 feet 

B) 28 feet 

C) 32 feet 

D) 36 feet 

Show your work: 

What was the total distance that Wanda 

hiked? 

E) 2.5 miles 

F) 3.5 miles 

G) 6 miles 

H) 6.25 miles 

Show your work: 

Solve. Solve each equation using the method described. 

3. Solve using 

square roots. 

2 

x 56 177 

4. Solve using 

square roots. 

2 

8x 35 63 

5. Solve by 

factoring. 

2 

x 5x24 0 

6. Solve using the 

quadratic 

formula. 

2 

2x 

9x5 0 


Squaring and Rooting: Inverses (pp. 1 of 4) 

Algebra 2 



Warm-Up 

Solve each equation for x. Remember: These equations usually have two solutions. 

A) 

2 

x 21 70 

B) 

2 

4x 12 13 

C) 

2 

( x 3) 4 40 

1. Review of Quadratic Functions 

2 

A. For the equation y x , complete 

the table. Then plot points to 

construct the graph. 

B. If the domain of the equation is {- 

3,-2,-1,0,1,2,3}, then what would 

be the range? 

C. Is this equation a function? 

Explain. 

x 

-3 

-2 

-1 

0 

1 

2 

3 

y 

x 

2 

y 

-4 

-4 

4 

4 

2. Inverse with Coordinates 

A. Recall that an inverse can be 

formed by switching the x- and y- 

coordinates in a relation. Use this 

method to make a table and graph 

2 

for the inverse of y x . 

B. Is this inverse a function? 

Explain. 

Inverse 

x 

y 

-3 

-2 

-1 

0 

1 

2 

3 

-4 

4 

4 

-4 

3. Inverse with Algebra 

Remember that another way 

to find an inverse is to switch 

the x and y variables in an 

equation, then solve for the 

“new” y. 

Equation: y x 

Inverse: ____ = ____ 2 

Solve: 

2 

B. Use a calculator to help you sketch 

the graph of this inverse equation 

in the box below. 

A. Use this algebraic method 

to find the inverse of 

2 

y x . 

Notation: 

f 

1 ( x 

) 

Does this graph match your answer 

from #2? Explain. 



Algebra 2 



These two relations are inverses of one another. 

Graph: 

y 

What are the properties of 

inverses? 

Graph: 

y 

 

Their graphs are 

x 

_________________ over the 

line ___________ 

x 

Equation: 

 

Their equations have the 

Equation: 

Shape: 

_______________ switched 

Shape: 

What’s the “PROBLEM” with this graph? 

This graph is ___________ a ________________! 

So, normally… 

We just graph the _______ ___________ and call it the _____________ _____________ function. 

Parent function: 

f( x) 

x 

Graph: 

Domain: 

__________________ 

Table: 

x y 

0 0 

1 1 

4 2 

9 3 

2 

2 4 6 8 

(You can only use ____________ 

values for _____.) 

Range: 

__________________ 

(You will only get ____________ 

answers for _____.) 



Algebra 2 



4. Review of Quadratic Transformations 

2 

y 2x 

2 

x 

-2 

A. For the equation above, 

complete the table. Then plot 

points to construct the graph. 

-1 

0 

1 

2 

y 

4 

B. This graph is a transformation of the parent graph 

2 

y x . Tell how each constant in the equation 

changes the graph of the parent function. 

 

 

What affect does the “2” have on the graph? 

What affect does the “2” coefficient have on the 

graph? 

-4 

-4 

4 


A. Again, recall that an inverse 

can be formed by switching 

the x- and y- coordinates in a 

relation. Use this method to 

make a table and graph for 

2 

the inverse of y 2x 

2. 

B. Is this inverse a function? Explain. 

x 

y 

-2 

-1 

0 

1 

2 

-4 

4 

4 

-4 






“new” y. 

Equation: 

Inverse: 

Solve: 

y 

 

2 

2x 

2 

B. Use a calculator to help you 

sketch the graph of this 

inverse equation in the box 

below. 


to find the inverse 

2 

of y 2x 

2. 

Notation: 

Does this graph match your 

answer from #5? Explain. 



Algebra 2 



7. Review of Quadratic Transformations 

2 

y ( x 3) 1 

x 

0 

y 

A. For the equation above, 

complete the table. Then plot 

points to construct the graph. 

B. This graph is a transformation of 

2 

the parent graph y x . Tell 

how each constant in the 

equation changes the graph of 

the parent function. 

 


1 

2 

3 

4 

5 

6 

-4 

-4 

4 

4 

 



A. Again, recall that an inverse 

can be formed by switching 

the x- and y- coordinates in a 

relation. Use this method to 

make a table and graph for 

2 

the inverse of y ( x3) 1. 

x 

y 

0 

1 

2 

3 

4 

B. Is this inverse a function? 

Explain. 

4 

5 

6 

-4 

-4 

4 






“new” y. 

Equation: 

Inverse: 

Solve: 

y 

 

2 

( x 3) 1 

B. Use a calculator to help you 

sketch the graph of this 

inverse equation in the box 

below. 


to find the inverse of 

2 

y ( x3) 1. 

Notation: 

Does this graph match your 

answer from #8? Explain. 


Transformations of f( x) 

x (pp. 1 of 3) 

Algebra 2 



WINDOW 

Xmin= -9.4 

Xmax= 9.4 

Xscl= 1 

Ymin= -6.2 

Ymax= 6.2 

Yscl= 1 

Use the given calculator settings to 

graph each function below. 

Then tell how the each new graph compares to the parent 

function f ( x) 

x (shown at the right). 

Parent Function: 

Endpoint: 

(0,0) 

4 

2 

f ( x) 

 

-2 2 4 6 

x 

1. Function: f( x) x 3 


4 

2 

Where is 

the 

Endpoint? 

4 

2 

Where is 

the 

Endpoint? 

-2 2 4 6 

(___,___) 

-2 2 4 6 

(___,___) 

How does the function compare to 

f ( x) 

x ? 


f ( x) 

x ? 



4 

Where is 

the 

Endpoint? 

4 

Where is 

the 

Endpoint? 

2 

(___,___) 

2 

(___,___) 

-2 2 4 6 

-2 2 4 6 


f ( x) 

x ? 


f ( x) 

x ? 




Algebra 2 



5. Function: f( x) ( x 4) 2 

6. Function: f( x) ( x 2) 1 

4 

2 

Where is 

the 

Endpoint? 

4 

2 

Where is 

the 

Endpoint? 

-2 2 4 6 

(___,___) 

-2 2 4 6 

(___,___) 


f ( x) 

x ? 


f ( x) 

x ? 

7. Function: f( x) 2.5 

x 

8. Function: f( x) 0.6 

x 

4 

2 

Where is 

the 

Endpoint? 

4 

2 

Where is 

the 

Endpoint? 

-2 2 4 6 

(___,___) 

-2 2 4 6 

(___,___) 


f ( x) 

x ? 


f ( x) 

x ? 

9. Function: f( x) 

 

x 

10. Function: f ( x) 

2 

x 

4 

2 

Where is 

the 

Endpoint? 

4 

2 

Where is 

the 

Endpoint? 

-2 2 4 6 

(___,___) 

-2 2 4 6 

(___,___) 


f ( x) 

x ? 


f ( x) 

x ? 



For square root functions, there are four basic transformations: 


Algebra 2 



Multiplying on the _____________ 

If a > 1, vertical ________________ 

If 0 < a < 1, ___________ _________ 

If a < 0, reflects over _____________ 

Adding on the _____________ 

Adding k translates the graph ______ 

Subtracting k ________________ the 

graph ________________ 

f ( x) 

a ( x h) 

 

k 

Adding on the _____________ 

Adding h translates the graph ______ 

Subtracting h ________________ the 

graph ________________ 

Write function rules for these graphs. Check them using a graphing calculator. 

A. Take the function f ( x) 

x , reflect it over 

the x-axis, and then translate it 5 units up. 

B. Take the function f ( x) 

x , stretch it to 

be twice as tall, then translate it 3 units to 

the left. 

4 

4 

2 

2 

-2 2 4 6 

-2 2 4 6 

Function rule: ________________________ 

Function rule: ________________________ 


Practice Problems: Transforming f( x) 

 

x 

Algebra 2 



1. Function: f( x) x3 2 

2. Function: 

y 

1 

x1 

2 

2 

4 

2 

-2 2 4 6 

Sketch 

the graph 

on the 

grid 

provided. 

4 

2 

-2 2 4 6 

Sketch 

the graph 

on the 

grid 

provided. 

What transformations of 

y x occurred? 


y x occurred? 

3. Function: f( x) x5 3 

4. Function: y 2 x3 4 

4 

2 

-2 2 4 6 

Sketch 

the graph 

on the 

grid 

provided. 

4 

2 

-2 2 4 6 

Sketch 

the graph 

on the 

grid 

provided. 


y x occurred? 


y x occurred? 

This function vertically stretches y x by a 

factor of 2, then translates it 3 units left. 

This function reflects y x across the x-axis, 

then translates it 2 units right and 5 units up. 

4 

2 

-2 2 4 6 

What is 

the 

function 

rule for the 

graph? 

4 

2 

-2 2 4 6 

What is 

the 

function 

rule for the 

graph? 

5. Function: ________________ 

6. Function: ________________ 


It’s What’s Inside That Counts (pp. 1 of 3) 

Algebra 2 



Each picture shows the graph of a function y=f(x) [dotted]. With the help of a calculator, you are 

asked to sketch the graph of y f (x) 

on the same grid. Look for similarities in the relationships 

between each set of graphs. 

1. Shown: y x 2 

y 

2. Shown: y x 3 

y 

4 

4 

2 

2 

-2 

2 4 

x 

-2 

2 4 

x 

You graph: y x 2 

3. Shown: y x 

4 

y 


4. Shown: y 2x 

6 

y 

4 

4 

2 

2 

-2 

2 4 

x 

-2 

2 4 

x 


5. Shown: y x 2 4 

y 

You graph: y 2x 

6 

6. Shown: y 3x 

x 

2 

y 

4 

4 

2 

2 

-2 

2 4 

x 

-2 

2 4 

x 

You graph: y x 2 4 

 

You graph: 

y 

3x 

x 

2 



Answer each question about the pairs of graphs shown in #1-6. 

Algebra 2 



7. In each set of graphs, do the original function (y=f(x), dotted) and its square root function 

( y f (x) 

) have the same y-intercept? Explain. 

8. In each set of graphs, do the original function (y=f(x), dotted) and its square root function 

( y f (x) 

) have the same x-intercepts? Explain. 

This square root 

function is “defined” 

over here 

y 

Don’t 

Go 

There! 

x 

The square root 

function is NOT 

defined over here 

Notice that square root functions “stop” on one 

side or the other because they have 

“endpoints” (like a ray in Geometry). 

As a result, there are certain parts of the 

graph where a square root function will not 

“go.” 

We say that the square root function is NOT 

defined in this area. 

Compare the pairs of graphs on the previous page to complete these statements. 

9. The square root function will go (or, is defined) only where the original function is _________. 

10. The square root function will not go (or, is NOT defined) where the original function is ________. 



Algebra 2 



Fill-in-the-Blank Notes 

Square root functions have a _____________________ ______________________. 

What does this mean? 

On the GRAPH… 

There are places where 

the function 

__________________. 

Don’t Go 

There! 

y 

y x On the TABLE… y x 

There are x-values that 

x y 

you 

x 

-3 ERROR 

-1 ERROR 

__________________. 

1 1 

3 1.73205 

 

Why? 

If you try to take the square root of a ________________ _________________, 

Your answers will be __________________________. 

Or, if you want answers that are ___________________, 

You can only take the square root of a ________________ _____________. 

To find the DOMAIN of a square root function, 

1) Take the expression inside the radical and 

f ( x) 

2x 

5 

What’s inside can’t 

be negative 

It must be positive 

(or zero) 

2) Then solve the 

Write: 2x 5 ____ _____ 

Solve: 

Find the domain of each of these square root functions. 

A) f ( x) 

5x 

14 

B) y 

3 x 12 

C) h( x) 

22 4x 

4 


Finding the Domain 

Algebra 2 



Use the graph to describe the domain of each square root function. 

1. y 2x 

8 

2. y 2 x 

y 

y 

-2 

2 4 

x 

-2 

2 4 

x 

Use the table to describe the domain of each square root function. 

3. y ( x 12) 

5 

4. y 2x 

7 

x y 

x y 

-14 ERROR 

2.5 1.4142 

-13 ERROR 

3.0 1 

-12 5 

3.5 0 

-11 6 

4.0 ERROR 

-10 6.4142 

4.5 ERROR 

Set up and solve an inequality to describe the domain of each square root function. 

5. f ( x) 

x 3 

6. f ( x) 

8x 

36 

7. f ( x) 

2x 

10 

8. g( x) 

18 3x 

9. h ( x) 

7x 

4 

10. p x) 

 

1 x 13 

( 

2 

11. f ( x) 

( x 4) 5 

12. y 2 x 7 

2 

Challenge: f ( x) 

x 2x 

15 


A Screeching Halt 

Algebra 2 



When a car must make a sudden stop, it often leaves skid marks on the 

pavement. 

Is there any way to tell from the skid mark how fast a car was going before it put on the brakes? 

The equation s 30fd 

can be used to estimate the speed (s, in miles per hour) a car must have 

been traveling in order to make a skid that measures d feet in length. The f in the equation is a 

coefficient of friction that depends on the road conditions. For a typical street, we can use f = 0.6. 

Or, here, 

s 30(0.6) 

d , which can be re-written as s = _________________ 

Skid Distance (ft) d 20 40 60 80 100 150 200 

Speed (mph) 

s 

60 

Use a graphing calculator to complete the 

table and sketch the graph for this function. 

Speed (mph) 

40 

20 

WINDOW 

Xmin = 0 

Xmax = 220 

Xscl = 20 

Ymin = 0 

Ymax = 70 

Yscl = 10 

20 40 100 200 

Skid Distance (ft) 

Use the TRACE and TABLE features to 

answer the questions that follow. 

1. A dog ran out in front of your car. You slammed on your brakes, and 

left a 15-ft skid mark. About how fast were you going? 

Distance (d) 

Speed (s) 

2. A second car almost ran into the back of you, but was able to stop 

after making a 108-ft skid mark. If the speed limit was 45 mph, was 

this driver speeding? 

3. A third car was going 48 miles per hour when it slammed on its 

breaks. How many feet will it take this car to skid to a stop? 

Distance (d) 

Distance (d) 

Speed (s) 

Speed (s) 


Solving Square Root Equations (pp. 1 of 2) 

Algebra 2 



48 18d 

How can this equation be solved algebraically? 

1) Isolate the radical 

2 

48 18d 

2) Square both sides 

2 

2304 18d 

3) Simplify and solve 

2304 18d 

 

18 18 

128 d 

? 

48 

? 

48 

48 48 

18(128) 

2304 

4) Check the solutions 

5 ( x 3) 9 

Square Root Equation x 5 3 x 

1) Isolate the radical 

2) Square both sides 

3) Simplify and Solve 

4) Check the solutions 

Solve each equation. 

A) 2 x 7 3 15 

B) x x 20 

C) 4x 

3 10 

3 


Solving Square Root Equations (pp. 2 of 2) 

Algebra 2 



Solve each square root equation. Remember to check your solutions. 

1. 3x 1 8 

2. 2x 3 5 

3. 

2 

x 

9 4 

4. x 13 

x 

5. 5 x 1 x 1 6. 3x 

6 x 4 

CHALLENGE: In this equation, you must isolate the radical and square both sides TWICE. 

x 5 x 1 


Getting Hot in Here (pp. 1 of 2) 

Algebra 2 



Suzy knows it will be hot inside her car, so before she 

starts her car, she rolls the windows down to let the 

car air out. Then, after starting the car and turning on 

her air conditioner, Suzy can feel the temperature in 

her car gradually start to drop. 

An equation to model this change is 

T 104 8 m 1 

, 

Where m = the number of minutes since Suzy turned on her air conditioner, and 

T = the temperature inside the car in degrees Fahrenheit. 

Use a graphing calculator to answer the questions that follow. 

Xmin= -1, Xmax= 11, Xscl= 1 

Ymin=60, Ymax=110, Yscl=10 

1. What was the temperature inside Suzy’s car at the exact moment she turned on her air 

conditioner (m=0)? (Tell two different ways to get the answer.) 

2. What was the temperature inside Suzy’s car 3 minutes after she turned on her air conditioner? 

(Tell two different ways to get the answer.) 

3. What was the temperature inside Suzy’s car 1 minute before she turned on her air conditioner? 

(Tell two different ways to get the answer.) 

4. Suzy will not feel comfortable in her car until the temperature reaches 84F. For what values 

of m is the temperature at or below this amount? 

5. However, if the temperature goes below 78F, Suzy says she is too cold. For what values of m 

is the temperature 78F or higher? 

6. During what interval of time will Suzy be comfortable in the car? How many minutes is this? 


Getting Hot in Here (pp. 2 of 2) 

Algebra 2 



 

For what values of m is the temperature at or above 84F? 

This question creates an 

inequality: 

Temperature 84 

Or, 104 8 m 1 84 

Which we can solve with the 

help of a graphing calculator: 

Y 1 = 104 8 x 1 

Y 2 = 84 

ON THE GRAPH WITH THE TABLE ALGEBRAICALLY 

First, find the function’s domain. 

How can you find the 

“intersection” on the graph? 

How can you make your table 

look like this? 

How does this table tell you the 

solution to the inequality? 

Then solve the inequality by 

isolating the radical and 

squaring both sides. 

How does this graph tell you the 

solution to the inequality? 

Solve these inequalities using a method of your choice. 

A) 3x 5 4 

B) 7 4x 15 

10 

C) x5 x5 6 


Solving Square Root Inequalities Practice 

Algebra 2 



Solve each square root inequality. Check using a graphing calculator. 

2 

1. 2x 3 9 

2, 7 5x 

8 

3. x 7 4 

4. x 13 x 

5. x 4 x 

6. 2x 1 

3

Firefighter (pp. 1 of 2)

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