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COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION – PART 2

COMMON CORE ALGEBRA II

You will recall from unit 1 that in order to find the inverse of a function, you must switch x and y and solve for

y. Also, for a function, f(x), the inverse function is written as 1( )f x .

Exercise #2: Find 1( )f x .

(a) f (x) = -3x + 2

(b) 1

( ) 5 2

f x x 

The inverses of polynomial and rational functions can also be found.

Exercise #3: Find formulas for the inverse of each of the following simple rational or polynomial functions.

(a) ( ) 2

x g x

x 

 (b)

3 ( )

2

x h x

x

 

(c) 32( 2)y x  (d) 34 4

( ) 2

x f x

 

(e) 4

2

x y

x

 

 (f)

5( ) ( 4) 6f x x  

• 2

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

In the next lesson, we will be working on transforming functions. For this lesson, we will practice writing new

functions that are related to these transformations.

Exercise #4: The following table includes two parent functions and a quadratic function. Complete each of the

following function changes. You do not need to simplify the resulting function. Do not do the last column.

Each function change is written as if the function were g(x). Make the same change to m(x) and h(x).

Function

Change

A. Cubic

Parent

Function 3( )g x x

B. Square Root

Parent

Function

( )m x x

C. Quadratic Function 2( ) 4 3h x x x  

Transformation

f(-x)

-f(x)

f(x+2)

f(x)-2

f(x-2)+5

3f(x)

f(2x)

*****

1 ( )

2 f x

1

3 f x      

*****

2 ( 3)f x

*****

( ) 5f x 

*****

• 3

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION – PART 2

COMMON CORE ALGEBRA II HOMEWORK

1. Find formulas for the inverse of each of the following functions.

(a) 5

2

x y

x 

 (b)

3 2

4

x y

x

 

(c) f (x) = -2x

3 + 3 (d) g(x) = -(x +1) 3 + 2

(e) h(x) = x 3 - 3 (f)

g(x) =

1

x -1

(g) 52( 4) 5y x   (h) 4 2

3

x y

x

 

• 4

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

2. Complete the following table. Column A and B are parent functions, and Column C is a cubic function.

NOTE: Each function change is written as if the function were f(x). Make the change same change to

j(x) and g(x). You do not need to simplify.

Function Change A. Absolute Value

Parent Function

( )f x x

B. Quartic Parent

Function 4( )j x x

C. Cubic Function 3 2( ) 4 3 6g x x x x   

f(-x)

-f(x)

f(x-4)

f(x)+1

f(x+3)-2

2f(x)

f(3x)

1 ( )

3 f x

1

2 f x      

4 ( 1)f x

( ) 6f x 

• 5

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

LESSON #43 – FUNCTION TRANSFORMATIONS

COMMON CORE ALGEBRA II

In the previous lesson, we worked with function changes such as f(-x) and f(x+3). Each of these changes causes

a predictable change in the graph of the function known as a transformation. For example, making the x

negative, f(-x), will always cause the graph to change in the same way. The easiest function to see all of the

transformations is the square root parent function.

Exercise #1: Go back to lesson #72, graph the square root parent function as Y1 and each of the changes

(transformations) as Y2. Determine the transformation that occurred for each problem, and write it in the last

column.

Exercise #2: Summarize your findings in the table below.

Function Notation Transformation Groups of Transformations

f(-x)

-f(x)

f(x+a)

f(x-a)

f(x)+a

f(x)-a

af(x) where 1a 

af(x) where 0 1a 

f(ax) where 1a 

f(ax) where 0 1a 

Note: We will be working with horizontal stretches and compressions more extensively in the next unit.

• 6

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

Exercise #3: What is the equation of the absolute value parent function, a(x)= x , after each of the following

transformations.

a. Shift Right 3

b. Reflection in the x-axis.

c. Shift Down 2

d. Vertical Stretch of 3.

e. Reflection in the y-axis

f. Shift left 3, up 4.

g. Vertical Compression of

1

2 .

Exercise #4: The function  f x is shown on the grid below. A second

function, g, is defined by    3 1g x f x   .

(a) Identify how the graph of f has been transformed to produce the graph

of g and sketch it on the grid.

(b) A third function, h, is defined by h(x)= 2f (x). Identify how the graph of f has been transformed to produce

the graph of h and sketch it on the grid. (Note: The maximum of h will be off the grid).

Exercise #5: If the parabola y = x 2 were shifted 6 units left and 2 units down, its resulting equation would be

which of the following? Verify by graphing your answer and seeing if its turning point is at (-6,-2).

(1) y = (x + 6)

2 + 2 (3) y = (x - 6) 2 - 2

(2) y = (x + 6)

2 - 2 (4) y = (x - 6) 2 + 2

y

x

• 7

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

Exercise #6: The graph of a function  f x is shown below on two grids. Sketch (a) the graph of  f x and

(b) the graph of  f x .

Exercise #7: Determine an equation for the linear function   5 7g x x  both after a reflection in the x-axis and

Exercise #8: If the point  3,12 lies on the graph of the function  y f x , which of the following points must

lie on the graph of  3y f x ?

(1)  9, 36 (3)  3, 4

(2)  3, 36 (4)  9,12

Exercise #9: If   22 5 3f x x x    and  g x is the reflection of  f x across the y-axis, then an equation of g

is which of the following?

(1)   22 5 3g x x x    (3)   22 5 3g x x x  

(2)   22 5 3g x x x    (4)   22 5 3g x x x  

y

x

(a) Graph and label .

y

x

(b) Graph and label

• 8

COMMON CORE ALGEBRA II, UNIT #5 – WORKING WITH FUNCTIONS AND RELATIONS

Modified from eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

LESSON #43 - FUNCTION TRANSFORMATIONS

COMMON CORE ALGEBRA II HOMEWORK

1. What is the equation of the quintic parent function, f (x)= x 5, after each of the following transformations?

You do not need to simplify.

a. Shift left 4

b. Reflection in the y-axis

c. Shift Up 1

d. Vertical Compression of

1

4

e. Reflection in the x-axis

f. Shift right 5, down 2

g. Vertical Stretch of 3

2. Which of the following represents the turning point of f (x)=(x -8) 2 -4?

(1) (-8,-4)

(3) (-8,4)

(2) (8,4)

(4) (8,-4)

3. Consider the quadratic function f (x)= x 2 -4x -5. The quadratic functions g and h are defined by the

formula, g(x)= 2f (x) and h(x)=

1

2 f (x) . Determine formulas for both g and h in standard form.

4. Which of the following equations would represent the graph of the parabola 23 4 1y x x   after a

reflection in the x-axis?

(1) 23 4 1y x x    (3) 23 4 1y x x  

(2) 23 4 1y x x  