Unit 2 Honors Algebra 2

FUNCTIONS AND THEIR INVERSES (6-6 and 4-2)

Learning Targets

1. Use the horizontal line test to determine whether a function has an inverse that is also a function.

2. Find the inverse function when given a function. (numerically, graphically, and algebraically)

An inverse in mathematics is used to “undo” things. The inverse of addition is subtraction, the inverse of multiplication

is division. Functions are also capable of undoing each other. Two functions that undo each other are called inverse

functions.

The “inverse of a function” undoes the function.

The graph of a function and its inverse are reflections across the “diagonal” line y x .

Sometimes the graph of an inverse is a function and sometimes it is not.

Do you remember what test was used to determine whether or not a graph represented a function?

Sometimes we would like to determine whether or not the inverse of a function is a function itself. How do we do this?

Example 1: Determine whether the inverse of each relation is a function. Explain your reasoning.

Vocabulary:

Relation: any set of ordered pairs … (a bunch of points on graph paper)

Function: a relation where each x-value only has one y-value … (a graph passing the Vertical Line Test)

Inverses can be found numerically, graphically, and algebraically.

Numerically … To find the inverse of a relation numerically, ______________________________________________.

Example 2: Use the relation below and find the inverse numerically.

Original Relation: Inverse:

x 0 1 5 8

y 2 5 6 9

Unit 2 Honors Algebra 2

Graphically … To find the inverse graphically, _________________________________________________________.

Example 3: Plot the original relation and connect the points from example 2 on the grid below.

a) Graph the line y = x.

b) Reflect the points over the line y = x and connect the points.

Compare it to the original.

c) What is the domain and range of each relation?

Algebraically … To find the inverse algebraically, _______________________________________________________.

When finding the inverse of f (x), label the inverse function __________.

Example 4: If 4( 9)f x x , find 1f x .

Example 5: If 2 7

3

xg x , find

1g x .

Example 6: If 3

68

h x x , find 1h x .

Unit 2 Honors Algebra 2

Example 7: Graph 12

5f x x , then write and graph the inverse.

Example 8: Given the function 9

325

F C , where F is temperature in Fahrenheit degrees and C is the temperature in

Celsius degrees. Complete the table of values below.

Celsius Fahrenheit

– 6°

42°

81°

Example 9: The period of a pendulum, T, in seconds is given by the function ( ) 29.8

LT L , where L is the length of

the pendulum in meters.

a) Find the inverse of ( )T L .

b) Explain what the inverse represents.

c) Determine the length of a pendulum with a period of 1.57 seconds.

x

y

Unit 2 Honors Algebra 2

Unit 2 Honors Algebra 2

SQUARE ROOT FUNCTIONS (5-7)

Learning Targets

1. Graph the parent function f x x .

2. Identify the anchor points on the parent square root function.

3. Apply transformations to f x x to draw the graph of the transformed function.

4. Describe the transformations when given the transformed equation of a square root function.

5. Write the equation when given a description of the transformation to the parent square root function.

Example 1: Given 2f x x , graph

1f x

and find

1f x

. What do you notice?

Example 2: Graph the parent function f x x by completing the table of values.

a) What is the domain?

b) What is the range?

The transformation rules we have used in the past continue to work the same way.

Example 3: For each function below, describe the transformation of the square root function and then graph it.

State the domain and range.

a) 4 3 7g x x b) 1

52

h x x

Domain: __________ Domain: __________ Range: __________ Range: __________

x y

0

1

4

9

x y

x y

x

y

x

y

x

y

x

y

Unit 2 Honors Algebra 2 Example 4: Use the following descriptions to write the transformation rule and equation of the new function.

a) The parent function f x x is reflected across the x-axis, vertically compressed by a factor of 15

, and

translated right 8 units and down 7 units.

b) The parent function f x x is horizontally stretched by a factor of 6, translated left 4 units and up 3 units.

Example 5: A juice drink manufacturer is designing an advertisement for a national sports event on its cans. The lateral

surface area of the cans is given by the function ( ) 2.5L h h , where h is the height of the can. The total surface area of

the can is given by the function ( ) 2.5 ( 1.25)T h h . The graphic designer needs to know how the height of the can

varies as a function of the lateral surface area.

a) Find the inverse, h(L), of the function L(h).

b) Explain the meaning of the inverse function.

c) If the lateral surface area of one can is 35.34 in2, determine the height of this can.

Example 6: The area of a regular octagon can be found by using the formula 22 2 1A s s , where s is the length

of each side.

a) Find the inverse of A(s).

b) What does the inverse represent?

c) Determine the side length of a regular octagon whose area is 9.68 2 9.68 meters.

Example 7: Consider the parent function of the exponential, 3x

f x .

a) Find the inverse of the function graphically.

b) State the domain and range of each function.

c) What else changed?

x

y

Unit 2 Honors Algebra 2

KEY FEATURES OF GRAPHS OF FUNCTIONS (3-7 and 6-2)

Learning Targets

1. Identify domain and range of functions.

2. Identify an absolute maximum and minimum.

3. Identify a local maximum and minimum.

4. Identify asymptotes of a graph.

5. State the end behavior for functions.

6. Describe the rate of change for a graph.

7. Compare the key features between two or more graphs.

Example 1: State everything you can about the graphs below. What parts of the graph are the “key features”?

a) b)

c) d)

Unit 2 Honors Algebra 2

The end behavior of a graph is a description of the values of the function (the y-values) as x approaches positive or

negative infinity. The end behavior of a function can change as the function is transformed.

Example 2: For each function: a) sketch a graph of the function, b) state the domain and range, c) state the

absolute maximum and minimum, d) state the local maximum and minimum, e) write the equation of any

asymptotes, and f) state the end behavior.

( ) 1 5f x x

2( ) 2 3f x x

3( ) 4 2f x x x

x

y

x

y

x

y

Unit 2 Honors Algebra 2

( ) 2 6f x x

( ) 5(3) 4xf x

Example 3: Compare and contrast the key features of each function in the graphs below. Include rate of change.

a) b)

x

y

x

y

Unit 2 Honors Algebra 2

Unit 2 Honors Algebra 2

SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY (12-7)

Learning Targets

1. Solve systems of equations in two variables by graphing.

2. Check the solutions to systems of equations algebraically.

A __________________________________ is a collection of two or more equations with a same set of variables.

The __________________________ of a system of equations is the set of points that make all of the equations in

the system true, or where the graphs ________________________.

What are the possibilities for solutions of a system? Sketch a picture to match each.

Example 1: The following graph is from a system of equations. Use the graph to determine the solution(s) to the system

of equations.

Example 2: Decide which of the following coordinate points are solutions to . Be sure to

show your work!

a) (4, -2) b) (2, -1) c) (12, 6) d) (8, 2)

212 ( 4)

46{y x

x y

Unit 2 Honors Algebra 2

Example 3: Graph the system on the grid below.

Example 4: Solve each of the following systems of equations graphically using your graphing calculator. Include a sketch

of your graph. Verify your answers algebraically.

a) b)

c)

3

2 5

16 ( 1)

xy

y x

d)

2 2

2 2

4 2 745

2 3 673

x y

x y

Use the graph to find the solutions to the system of equations.

Then check your answers by plugging them into the system of

equations.

213 5

44

y x x

y

22

8

xy

y x

2

2 9 6

( 3) 9

y x

y x

{ {

{

{ {