Transforming Functions y = a f( bx c) ± d Goals and Objectives

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Algebra II

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Analyzing and Working with Functions

Part 2

2015-04-21

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Table of Contents

The 12 Basic Functions (Parent Functions)

Transforming Functions

Operations with FunctionsComposite Functions

Inverse FunctionsPiecewise Functions

Function Basics

click on the topic to go to that sectionPart 1

Part 2

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Transforming Functions

y = a f( bx ∓ c) ± d

Return toTable ofContents

(Or making parent functions match data.)

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Goals and ObjectivesStudents will be able to transform any function,

including parent functions, algebraically and graphically.

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Why do we need this?Many different factors in life affect data and

information. These situations can be modeled by functions, but not all are the same. The 12 basic

functions represent common graphs of information. Transformations of these functions get us closer to a result and even apply to situations that are not easily

representable in a common form.

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Explore this on your own! Pull out a graphing calculator!y = a f( bx ∓ c) ± d

Choose one of the Basic Functions. Try adding and subtracting numbers inside and outside of the function. Then multiply and divide different functions by numbers in different places. What happens to the graph? Make a list:

a makes the function _____________________ b makes the function _____________________

c makes the function _____________________

d makes the function _____________________

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Now, we are going to formalize your results and explore each part individually.

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Vertical Shifts

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Vertical Shifts occur when a constant is added to or subtracted from OUTSIDE

of the function.

The function will be translated UP d units if d is added.

The function will be translated DOWN d units if d is subtracted. <

+ d - d

<

Vertical Shifts

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Try it on your calculator!

Use the function y = x2. Write the new equation and then enter it in your calculator to check your work.

a) up 4

b) down 5

c) down 3

d) up 2

Vertical Shifts

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Let the graph of f(x) be: Graph y = f(x) + 2:

Vertical Shifts

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1 Given the graph of h(x), which of the following graphs is y = h(x) - 1?

A B

C D

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2 Given the graph of h(x), which of the following is h(x) + 2?

AB

C D

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Horizontal Shifts

y = f(x ∓ c)

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Horizontal Shifts occur when a constant is added to or subtracted

INSIDE of the function.

The function will be translated LEFT c if c is added INSIDE.

The function will be translated RIGHT c if c is subtracted

INSIDE. <

+ c - cNotice the direction is opposite the sign of c.

y = f(x ∓ c)

<

Horizontal Shifts

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Use the function y = x3. Write the new equation and then enter it in the calculator to check your work.

a) right 4

b) left 5

c) left 3

d) right 2

Horizontal Shifts

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Let the graph of f(x) be: Graph y = f(x + 2):

Horizontal Shifts

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3 Given the graph of h(x), which of the following graphs is y = h(x - 1)?

A B

C D

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4 Given the graph of h(x), which of the following graphs is h(x + 3)?

AB

C D

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Transform g(x) as indicated:

1. h(x) = g(x) + 3

2. r(x) = g(x) - 5

3. p(x) = g(x - 4) + 2

4. m(x) = g(x + 3)

5. b(x) = g(x - 5) + 4

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Reflections

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Reflection over the x-axis: Reflection over the y-axis:

x

y

x

y

Reflections

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What happens when you use negatives? Look at the following...

a) e)

b) f)

c) g)

d) h)

Reflections

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Let the graph of f(x) be: Graph y = f(-x):

ReflectionsSlide 26 / 113

Let the graph of f(x) be: Graph y = -f(x): Reflections

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Transform g(x) as indicated:

1. h(x) = -g(x) + 4

2. r(x) = g(-x) - 2

3. p(x) = -g(x - 1) + 2

4. m(x) = -g(x + 2)

5. b(x) = g(-x) - 4

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5 Given the graph of h(x), which of the following is -h(x)?

A B

C D

Slide 29 / 1136 Given the graph of h(x), which of the following graphs is y = h(-x)?

A B

C D

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Vertical Stretch & Shrink

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Vertical Stretches and Shrinks occur when a constant is multiplied OUTSIDE of a function.

The parent function y = f(x) is :stretched vertically if |a| > 1

shrunk vertically if 0 < |a| < 1

Stretches and shrinks are the first transformation that do not yield

congruent figures.

Note: Notice how the x-intercepts DO NOT change.


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Stretch or shrink the following functions as indicated. Graph the parent function first, and then the transformed function in the same window.


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Let the graph of f(x) be:

Graph y = 2f(x):


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Graph y =(1/3 )f(x):


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7 Given the graph of h(x) at right, which of the following is the graph of y = 2h(x)?

A B

CD

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8 Given the graph of h(x) at right, which of the following is the graph of y = h(x)?

A B

CD

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Horizontal Stretch & Shrink

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Horizontal Stretches and Shrinks occur when a constant is multiplied to x INSIDE a function.

The parent function y = f(x) is:shrunk horizontally if |b| > 1

stretched horizontally if 0 < |b| < 1

Note: Notice how the y-intercepts DO NOT change.


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Try it on your calculator!Stretch or shrink the following functions as indicated. Graph the parent function first, and then the transformed function in the same window.


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Graph y = f(2x):


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Graph y = f(.5x):


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9 Given the graph of h(x) at right, which of the following is the graph of y = h(2x)?

A B

CD

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10 Given the graph of h(x) at right, which of the following is the graph of y = h(0.5x)?

A B

CD

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Combining Transformations

What goes first? Any thoughts?

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To combine transformations, follow order of operations:

Horizontal reflection over y-axis

Horizontal Stretch of b

Horizontal Slide of c

Vertical Reflection over x-axis

Vertical Stretch of a

Vertical Shift of d

y = a f( bx ∓ c) ± d


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Graph y = 2f(.5x+1) - 2

Remember...

Work slowly and use a different colored pencil for each transformation to help!






Vertical Shift of d


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Remember...Let the graph of f(x) be:

Graph y =(-1/3 )f(2x + 1) + 2:






Vertical Shift of d


Combining TransformationsSlide 48 / 113


Graph y = (-1/2)f(-x + 2) +1:

Remember...Horizontal reflection over y-axis





Vertical Shift of d



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11 Given the graph of h(x) at right, which of the following is the graph of y = 2h(-x + 1) - 3?

A B

C D

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12 Given the graph of h(x) at right, which of the following is the graph of y = -0.5h(2x - 1) + 2?

A

B

C

D

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Transforming Parent Functions

y = s inxy = 5s in(6x) + 2

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Remember the 12 basic functions...The Identity Function

y = xThe Squaring Function

y = x2The Cubing Function

y = x3

The Reciprocal Functiony = 1/x

The Absolute Value Functiony = ΙxΙ

The Square Root Function

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continued...The Exponential Function

y = exThe Natural Log Function

y = lnxThe Logistic Function

The Sine Functiony = sinx

The Cosine Functiony = cosx

The Greatest Integer Functiony = [x]

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Transform y = x2 into the following:As the function changes, what happens to the domain and range?

Note: when studying parabolas, af(bx + c) + d becomes af(bx + h) + k, where (h, k) is the vertex of the parabola.

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Transform y = x3 into the following:As the function changes, what happens to the domain and range?

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Transform y = ΙxΙ into the following:As the function changes, what happens to the domain and range?

Note: when studying the absolute value function, af(bx + c) + d becomes af(bx +h) + k, where (h, k) is the vertex of the absolute value function.

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Transform into the following:As the function changes, what happens to the domain and range?

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Transform into the following:Graph on the board and then check with your calculator! What happened?

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13 Which of the following is the equation for the graph shown?

A

B

C

D

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A

B

C

D

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A

B

C

D

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16 Which of the following is the eqution for the graph shown?

A

B

C

D

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Inverse Functions


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Goals and ObjectivesStudents will be able to recognize and find an inverse function: a) using coordinates, b) graphically and c) algebraically.

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Why do we need this?Sometimes, it is important to look at a problem from

the inside out. Addition undoes subtraction. Multiplication undoes division. In order to look deeply into different problems, we must try to see things from

the inside out. Inverse functions undo original functions.

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You can also prove that two functions are inverses by GRAPHING. The graph of an inverse is the reflection of the function over y = x.

The following functions are inverses of each other. Look at their graphs and make a conjecture about the x and y values of inverse functions.

Inverse Function

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The inverse of a function is the reflection over y = x. As you observed, this will result in the switching of x and y values.

Examples:

a) Find the inverse of: b) Find the inverse of:

f(x)= {(1, 2), (3, 5), (-7, 6)} X Y

3 2

4 4

5 -5

6 7

Inverse FunctionA

nsw

er

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17 What is the inverse of {(1, 4), (5, 3), (2, -1)}?

A {(4,1), (3,5), (2,-1)}

B {(-1,-4), (-5,-3), (-2,1)}

C {(4,1), (3,5),(-1,2)}

D {(-4,-1), (-3,-5), (1,-2 )}

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18 If the inverse of a function is {(1, 0), (3, 3), (-4, -5)}, what was the original function?

A {(0,1), (3,3), (-5,-4)}

B {(-1,-4), (-5,-3), (-2,1)}

C {(0,-1), (-3,-3),(4,5)}

D {(0,1), (3,3), (-4,-5)}

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19 What is the inverse of:A B

C D

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20 Will the inverse of the following points be a function? Why or why not?

Yes

No

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Draw the inverse of the given function.

Is the inverse a function? Why or why not?

Inverse

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Draw the inverse of the given function.

Is the inverse a function? Why or why not?

InverseSlide 76 / 113

Just like the Vertical Line Test, there is a simple way to determine if a function's inverse is also a function, just from looking at its graph.Def: The Horizontal Line Test is used to determine if a function has an inverse. If a horizontal line crosses the function more than once, it WILL NOT have an inverse.

Move this line to check:

Horizontal Line Test

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Example: Will the inverse of the given function be a function?

Move this line to check:

Horizontal Line Test

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21 Which graph is the inverse of the function at right?

A BCD

A B C D

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A B

C D

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C DBA

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24 Will the inverse of be a function?

Yes

No

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Knowing that the inverse of a function switches x and y values, we can take this concept further when given an equation.

Given: Switch x and y. Solve for y: Inverse function:

Finding the Inverse of a Function Algebraically

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InversePractice: Find the inverse of the following functions.

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InversePractice: Find the inverse of the following function.

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25 Which of the following choices is the inverse of

A

B

C

D

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26 Which of the following is the inverse of

A

B

C

D

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27 Find the inverse of

A

B

C

D

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Piecewise Functions


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Goals and ObjectivesStudents will be able to recognize Piecewise

Functions, graph them and correctly evaluate them at given points.

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Why do we need this?Recognize this problem?

A cell phone carrier charge $70 for the first 1000

minutes and $.25 for each minute after the first 1000.

We all deal with piecewise functions and don't know it. Many common situations have limits, overages

and maybe several parts. Knowing how to analyze these problems makes us smart consumers.

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Let's examine the previous situation...A cell phone carrier charge $70 for the first 1000 minutes and $.25 for each minute after the first 1000.

Time in Minutes

1000

Dol

lars c

harg

ed 70

Graphic Representation: In Mathematical Notation:

In words:

The graph starts as a constant graph of y = 70 and after x = 1000 the graph becomes y = .25(x - 1000) + 70

Piecewise FunctionSlide 92 / 113

a combination of other functions, each defined on a specific interval.

Notation:

Note: There can be as many functions included as necessary.

Piecewise Function

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Create the piecewise equation for the following graph.

if

if

Piecewise Function

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Piecewise FunctionWhat is the domain and range of the function?

Domain:

Range:

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Piecewise FunctionCreate the piecewise notation for the following graph.

if

if

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Piecewise FunctionWhat is the domain and range of the function?

Domain:

Range:

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Piecewise FunctionCreate the piecewise notation for the following graph.

if

if

if

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Piecewise FunctionState the domain and range of the function.

Domain:

Range:

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· Visualize the entire graph of each individual piece

· Graph only the parts defined by x (Be aware of endpoints being either open or closed.)

· Repeat for each part.

Graphing a piecewise function

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Domain and RangeWhat is the domain and range of the function?

Domain:

Range:

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Graph:

Piecewise Functions

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Graph:

Piecewise Functions

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Evaluating Piecewise Functions

Piecewise Functions are evaluated exactly like we do regular functions. The only difference is that you will

need to decide which part of the function to use depending on the value of x you are given.

Find: a) f(-3)b) f(5)c) f(1)d) f(-10)

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28 Find f(0) given:

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29 Find f(1) given:

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30 Find f(-11) given:

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31 Find f(7) given:

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32 Find f(2) given:

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33 Find f(3) given:

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Making Piecewise Functions ContinuousDef: A continuous function is one that has no breaks, jumps, holes and will have a value for each point in the domain.

Is this function continuous?

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We can make a piecewise function continuous by getting the open endpoint of the first equation to match the closed endpoint of the second.

The critical point is when x = 1.

1. Set the pieces equal to each other.2. Plug in x.3. Solve for b.4. Write the new, continuous function.

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Find the value of b that makes the function continuous. Rewrite the function.

1) What is the critical point?

2) Find b.

3) Rewrite the function.

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Find the value of a that makes the function continuous. Rewrite the function.

1) What is the critical point?

2) Find a.

3) Rewrite the function.

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Transforming Functions y = a f( bx c) ± d Goals and Objectives