transformations and symmetry of polynomial...

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4.3 Transformations and Symmetry of Polynomial Functions 327

327

LEARNING GOALS

M.C. Escher is a well-known artist with a unique visual perspective. Many of his

works display elusive connections, peculiar symmetry, and tessellations.

Tessellations are symmetric designs with a repeated pattern.

You can find many images of Eschers work on the World Wide Web. Take a look and

enjoy! Make sure to take a close look, because things may not be as straightforward

as they seem.

In this lesson, you will:

Dilate, re!ect, and translate cubic and quartic functions.

Understand that not all polynomial functions can be formed through transformations.

Explore differences between even and odd functions, and even and odd degree functions.

Use power functions to build cubic, quartic, and quintic functions.

Explore the possible graphs of cubic, quartic, and quintic functions, and extend graphical properties to higher-degree functions.

KEY TERMS

polynomial function

quartic function

quintic function

Function MakeoverTransformations and Symmetry of Polynomial Functions

4.3

600005_A2_TX_Ch04_293-402.indd 327 14/03/14 2:44 PM

328 Chapter 4 Polynomial Functions

4

Problem 1

Students will complete a table

of values given three reference

points for a cubic function.

They then use the property of

symmetry to determine other

reference points and then graph

the cubic function.

How did you determine the coordinates of the point that corresponds to

point (1, 1)?

If this basic cubic function is symmetric about the origin, what are the

coordinates of the point that corresponds to point (0, 0)?

If the x-coordinate is 3, what is the y-coordinate of a point on this basic

cubic function?

If the x-coordinate is 23, what is the y-coordinate of a point on this basic

cubic function?

PROBLEM 1 Refer to the Reference Points

Recall that reference points are a set of points that are used to graph a basic function.

Previously, you used reference points and the key characteristics of a parabola to graph the

basic quadratic function. You learned that the reference points for the basic quadratic

function are (0, 0), (1, 1), and (2, 4). The basic quadratic function is symmetric about the

y-axis; that is, f(x) 5 f(2x). Therefore, you can use symmetry to graph two other points of the

basic function, (21, 1), (22,4).

Lets consider a set of reference points and the property of symmetry to graph the basic

cubic function.

To complete Questions 1 and 2, consider the basic cubic function, f(x) 5 x3.

1. Complete the table for the given reference points. Then graph the points on the

coordinate plane shown.

x f(x) 5 x3

0 0

1 1

2 8

x

2

4

22

42024 22 8628 26

y

24

26

28

6

8

2. The graph of the basic cubic function is symmetric about the origin. So, f(x) 5 2f(2x).

Use the property of symmetry to determine 2 other points from the reference points.

Then, use these points to graph the basic cubic function on the coordinate plane

shown.

I can determine the points (21, 21) and (22, 28).

See the graph in Question 1.

Thepattern for a basic cubic function is to cube

the input value to get the output value. So,

from the origin, move over 1 unit and up 1 unit.

For the next point, start at the origin, move over 2 units and up 8 units.

600005_A2_TX_Ch04_293-402.indd 328 14/03/14 2:44 PM

Give the de!nition

of symmetry. Ask

students to give

examples of things that

are symmetrical (the

number 8, a human

body, a star!sh). Explain

that functions can be

symmetrical too. Draw

several different functions

on the board. For each

function, have a student

state whether the function

is symmetric and identify

its axis of symmetry.

Give the d

Grouping

Ask a student to read the

introduction. Discuss as

a class.

Have students complete

Questions 1 and 2 with a

partner. Then have students

share their responses as

a class.

Guiding Questions for Share Phase, Questions 1 and 2

If this basic cubic function is

symmetric about the origin,

what are the coordinates of

the point that corresponds to

point (1, 1)?

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Problem 2

The transformation function

g(x) 5 Af(B(x 2 C) 1 D is given

and a table that describes

the effect of each constant in

the equation with regard to

the graph of the function is

provided. Students complete a

table to show the coordinates

of g(x) 5 Af(B(x 2 C) 1 D

after each type of speci!ed

transformation performed on

f(x). In the next activity, the

graph of a basic cubic function

is given and each constant in

the transformation function is

altered to create new functions.

Students will use reference

points provided to determine

their corresponding points

on each transformed graph.

They answer questions about

the symmetry of the graphs

and determine algebraically if

the transformed functions are

even, odd, or neither. In the last

activities, students complete

tables to summarize the effects

that transformations have

on the base cubic function

c(x) 5 x 3 and the basic quartic

function q(x) 5 x 4 .

Grouping

Ask a student to read the

introduction. Discuss as a class.

PROBLEM 2 Will Symmetry Prevail?

Transformations performed on a function f(x) to form a new function g(x) can be described by

the transformational function:

g(x) 5 Af(B(x 2 C)) 1 D.

Previously, you graphed quadratic functions using this notation. You can use this notation

to identify the transformations to perform on any function.

Recall that the constants A and D affect the outside of the function (the output values). For

instance, if A 5 2, then you can multiply each y-coordinate of f(x) by 2 to determine the

y-coordinates of g(x).

The constants B and C affect the inside of the function (the input values). For instance,

if B 5 2, then you can multiply each x-coordinate of f(x) by 1 __ 2 to determine the x-coordinates

of g(x).

Function

Form

Equation

InformationDescription of Transformation of Graph

y 5 Af(x)

|A| . 1 vertical stretch of the graph by a factor of A units

0 , |A| , 1 vertical compression of the graph by a factor of A units

A , 0 re!ection across the x-axis

y 5 f(Bx)

|B| . 1 compressed horizontally by a factor of 1 ___ |B|

0 , |B| , 1 stretched horizontally by a factor of 1 ___ |B|

B , 0 re!ection across the y-axis

y 5 f(x 2 C)

C . 0 horizontal shift right C units

C , 0 horizontal shift left C units

y 5 f(x) 1 D

D . 0 vertical shift up D units

D , 0 vertical shift down D units

Pair students to read the table. Instruct partners to take turns reading

the descriptions of transformations. After reading a description aloud,

have students restate the description in their own words to demonstrate

understanding. Circulate among the pairs to ensure correct descriptions

are given.

Pair stude


4

Grouping


Question 1 with a partner.

Then have students share their

responses as a class.

Guiding Questions for Share Phase, Question 1

A vertical dilation by

a factor of A affects

which coordinate?

Does a vertical dilation by

a factor of A compress the

graph or stretch the graph?

A horizontal dilation

by a factor of B affects

which coordinate?

Does a horizontal dilation by

a factor of B compress the

graph or stretch the graph?

A horizontal translation

of C units affects

which coordinate?

Does the horizontal

translation of C units shift the

graph to the left or shift the

graph to the right?

A vertical translation

of D units affects

which coordinate?

Does the vertical translation

of D units shift the graph up

or shift the graph down?

1. Complete the table to show the coordinates of g(x) 5 Af(B(x 2 C)) 1 D after each type

of transformation performed on f(x).

Type of Transformation Performed on f(x) Coordinates of f(x) Coordinates of g(x)

Vertical Dilation by a Factor of A (x, y) ( x , Ay )

Horizontal Dilation by a Factor of B (x, y) ( 1 __ B

x, y )

Horizontal Translation of C units (x, y) ( x 1 C , y )

Vertical Translation of D units (x, y) ( x , y 1 D )

All four transformations: A, B, C, and D (x, y) ( 1 __ B

x 1 C, Ay 1 D )

You are now ready to transform any

function!

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Grouping


Questions 2 through 8 with a



a class.

Guiding Questions for Share Phase, Question 2

Is the graph of g(x) closer

to or further away from the

y-axis with respect to c(x)?

How did you determine the

point that corresponds to the

point (1, 1) when graphing

the function g(x)?

Is the graph of h(x) closer






the function h(x)?

Are both g(x) and h(x)

symmetric about the origin?

Is the original function, c(x)


If a function is symmetric

about the origin, is it

considered an even or an

odd function?

Does 2g(x) 5 g(2x)?

Does 2h(x) 5 h(2x)?

2. The graph of the basic cubic function c(x) 5 x3 is shown.

x

4

8

24

21022 21 4324 23

y

28

212

216

12

16g(x)

h(x)

c(x)

a. Suppose that g(x) 5 2c(x). Use reference

points and properties of symmetry to

complete the table of values for g(x).

Then, graph and label g(x) on the same

coordinate plane as c(x).

See graph and table.

b. Suppose that h(x) 5 1 __ 2 c(x). Use reference


complete the table of values for h(x).

Then, graph and label h(x) on the same

coordinate plane as c(x) and g(x).


c. Describe the symmetry of g(x) and h(x). How does the symmetry of g(x) and h(x)

compare to the symmetry of c(x)?

Both g(x) and h(x) are symmetric about the origin. The original function c(x) is also

symmetric about the origin.

d. Determine whether g(x) and h(x) are even functions, odd functions, or neither.

Verify your answer algebraically.

Both g(x) and h(x) are odd functions. I know they are odd functions because they

are both symmetric about the origin. I can prove this algebraically by showing

that 2g(x) 5 g(2x) and 2h(x) 5 h(2x).

2g(x) 5 2(2x3) 5 22x3 2h(x) 5 2(0.5x3) 5 20.5x3

g(2x) 5 2(2x)3 5 22x3 h(2x) 5 0.5(2x)3 5 20.5x3

Reference

Points on c(x)

Corresponding

Points on g(x)

(0, 0) (0, 0)

(1, 1) (1, 2)

(2, 8) (2, 16)

Reference

Points on c(x)

Corresponding

Points on h(x)

(0, 0) (0, 0)

(1, 1) (1, 1 __ 2 )

(2, 8) (2, 4)


4


x

4

8

24

21022 21 4324 23

y

28

212

216

12

16u(x)

v(x)

c(x)

a. Suppose that u(x) 5 c(2x). Use reference


complete the table of values for u(x).

Then, graph and label u(x) on the

same coordinate plane as c(x).


b. Suppose that v(x) 5 c ( 1 __ 2 x ) . Use reference points and properties symmetry to

complete the table of values for v(x).

Then, graph and label v(x) on the

same coordinate plane as c(x) and u(x).


c. Describe the symmetry of u(x) and v(x). How does the symmetry of u(x) and v(x)


Both u(x) and v(x) are symmetric about the origin. The original function c(x) is also


d. Determine whether u(x) and v(x) are even functions, odd functions, or neither.


Both u(x) and v(x) are odd functions. I know they are odd functions because they


that 2u(x) 5 u(2x) and 2v(x) 5 v(2x).

2u(x) 5 2(2x3) 5 22x3 2v(x) 5 2(0.5x3) 5 20.5x3

u(2x) 5 2(2x)3 5 22x3 v(2x) 5 0.5(2x)3 5 20.5x3

Reference

Points on c(x)

Corresponding

Points on u(x)

(0, 0) (0, 0)

(1, 1) ( 1 __ 2 , 1)

(2, 8) (1, 8)

Reference

Points on c(x)

Corresponding

Points on v(x)

(0, 0) (0, 0)

(1, 1) (2, 1)

(2, 8) (4, 8)


Is the graph of u(x) closer






the function u(x)?

Is the graph of v(x) closer






the function v(x)?

Are both u(x) and v(x)


Does 2u(x) 5 u(2x)?

Does 2v(x) 5 v(2x)?

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x

4

8

24

21022 21 4324 23

y

28

212

216

12

16

c(x)a(x) = b(x)

a. Suppose that a(x) 5 2c(x). Use reference


complete the table of values for a(x).

Then, graph and label a(x) on the



b. Suppose that b(x) 5 c(2x). Use reference

points and properties symmetry to

complete the table of values for b(x).

Then, graph and label b(x) on the

same coordinate plane as c(x) and a(x).


c. Describe the symmetry of a(x) and b(x). How does the symmetry of a(x) and b(x)


Both a(x) and b(x) are symmetric about the origin. The original function c(x) is also


d. Determine whether a(x) and b(x) are even functions, odd functions, or neither.


Both a(x) and b(x) are odd functions. I know they are odd functions because they


that 2a(x) 5 a(2x) and 2b(x) 5 b(2x).

2a(x) 5 2(2(x3)) 5 x3 2b(x) 5 2(2x3) 5 x3

a(2x) 5 2(2x)3 5 x3 b(2x) 5 (2(2x))3 5 x3

Reference

Points on c(x)

Corresponding

Points on a(x)

(0, 0) (0, 0)

(1, 1) (1, 21)

(2, 8) (2, 28)

Reference

Points on c(x)

Corresponding

Points on b(x)

(0, 0) (0, 0)

(1, 1) (1, 21)

(2, 8) (2, 28)

Guiding Questions for Share Phase, Question4




the function a(x)?




the function b(x)?

Are both a(x) and b(x)





the function a(x)?




the function b(x)?

Are both a(x) and b(x)


Does 2a(x) 5 a(2x)?

Does 2b(x) 5 b(2x)?

Why are the graphs of a(x)

and b(x) the same?


4


Is the graph of m(x) located

to the left or to the right of

the graph of c(x)?




the function m(x)?

Is the graph of n(x) located to

the left or to the right of the

graph of c(x)?




the function n(x)?

Is the point of symmetry

translated the same way the

functions are translated?

Why is the function m(x)

symmetric about the point

(25, 0)?

Why is the function n(x)


(5, 0)?

If a function is not symmetric

about the y-axis or the origin,

is it considered an even or an

odd function?


x

4

8

24

42024 22 8628 26

y

28

212

216

12

16

c(x)n(x) m(x)

a. Suppose that m(x) 5 c(x 2 5). Use reference


complete the table of values for m(x).

Then, graph and label m(x) on the



b. Suppose that n(x) 5 c(x 1 5). Use reference


complete the table of values for n(x).

Then, graph and label n(x) on the

same coordinate plane as c(x) and m(x).


c. Describe the symmetry of m(x) and n(x). How does the symmetry of a(x) and b(x)


The function m(x) is symmetric about the point (5, 0). The function n(x) is

symmetric about the point (25, 0). The original function c(x) is symmetric about

the origin. The point of symmetry is translated in the same way the functions are

translated.

d. Determine whether m(x) and n(x) are even functions, odd functions, or neither.


Both m(x) and n(x) are neither even nor odd. I know they are neither even or odd

functions because they are not symmetric about the y-axis or the origin.

Reference

Points on c(x)

Corresponding

Points on m(x)

(0, 0) (5, 0)

(1, 1) (6, 1)

(2, 8) (7, 8)

Reference

Points on c(x)

Corresponding

Points on n(x)

(0, 0) (25, 0)

(1, 1) (24, 1)

(2, 8) (23, 8)

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x

4

8

24

42024 22 8628 26

y

28

212

216

12

16j(x)

k(x)

c(x)

a. Suppose that j(x) 5 c(x) 1 5. Use reference


complete the table of values for j(x).

Then, graph and label j(x) on the same

coordinate plane as c(x).


b. Suppose that k(x) 5 c(x) 2 5. Use reference


complete the table of values for k(x).

Then, graph and label k(x) on the same

coordinate plane as c(x) and j(x).


c. Describe the symmetry of j(x) and k(x). How does the symmetry of j(x) and k(x)


The function j(x) is symmetric about the point (0, 5). The function k(x) is symmetric

about the point (0, 25). The original function c(x) is symmetric about the origin.

The point of symmetry is translated in the same way the functions are translated.

d. Determine whether j(x) and k(x) are even functions, odd functions, or neither.


Both j(x) and k(x) are neither even nor odd. I know they are neither even or odd

functions because they are not symmetric about the y-axis or the origin.

Reference

Points on c(x)

Corresponding

Points on j(x)

(0, 0) (0, 5)

(1, 1) (1, 6)

(2, 8) (2, 13)

Reference

Points on c(x)

Corresponding

Points on k(x)

(0, 0) (0, 25)

(1, 1) (1, 24)

(2, 8) (2, 3)


Is the graph of j(x) located

above or below the graph

of c(x)?




the function j(x)?

Is the graph of k(x) located

above or below the graph

of c(x)?




the function k(x)?

Is the point of symmetry

translated the same way the

functions are translated?

Why is the function j(x)


(0, 5)?

Why is the function k(x)


(0, 25)?

If a function is not symmetric

about the y-axis or the origin,

is it considered an even or an

odd function?


4


Under what circumstances

was the transformed cubic

function not symmetric about

the origin?

What rigid motions resulted

in transforming a basic cubic

function into a function that

is neither odd nor even?

7. Complete the table to summarize the effects that transformations have on the basic

cubic function c(x) 5 x3. The !rst row has been completed for you.

Effects of Rigid Motions on the Basic Cubic Function c(x) 5 x3

Rigid MotionNew Transformed

Function p(x) in

Terms of c(x)

Description of

Symmetry of p(x)

Is p(x) Even, Odd,

or Neither?

Vertical

Stretch Dilation

p(x) 5 Ac(x),

|A| . 1

Symmetric about

the point (0, 0)Odd

Vertical

Compression Dilation

p(x) 5 Ac(x),

0 , |A| , 1

Symmetric about

the point (0, 0)Odd

Horizontal

Stretch Dilation

p(x) 5 c(Bx),

0 , |B| , 1

Symmetric about

the point (0, 0)Odd

Horizontal


p(x) 5 c(Bx),

|B| . 1

Symmetric about

the point (0, 0)Odd

Reflection across

x-axisp(x) 5 2c(x)

Symmetric about

the point (0, 0)Odd

Reflection across

y-axisp(x) 5 c(2x)

Symmetric about

the point (0, 0)Odd

Vertical

Translationp(x) 5 c(x) 1 D

Symmetric about

the point (0, D)Neither

Horizontal

Translationp(x) 5 c(x 2 C)

Symmetric about

the point (C, 0)Neither

8. Do you think that your results in Question 7 would be the same for any odd power

function? Explain your reasoning.

Yes. Because the general shape and end behavior is the same for any odd power

function, I think that the results in the table would be the same.

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9. The graph of the basic quartic function q(x) 5 x4 and its reference points are shown.

Use the graph to sketch the function after dilations, re!ections, and translations. Pay

special attention to the symmetry after the transformations. Record your conclusions by

completing the table that follows. The "rst row has been completed for you.

x(0, 0)

(1, 1)

(2, 16)

q(x)

y

Effects of Rigid Motions on the Basic Cubic Function q(x) 5 x4

Rigid Motion

New Transformed

Function p(x) in

Terms of q(x)

Description of

Symmetry of p(x)

Is p(x) Even, Odd,

or Neither?

Vertical

Stretch Dilation

p(x) 5 Ac(x),

|A| . 1

Symmetric about

the y-axisEven

Vertical


p(x) 5 Aq(x),

0 , |A| , 1

Symmetric about

the y-axisEven

Horizontal

Stretch Dilation

p(x) 5 q(Bx),

0 , |B| , 1

Symmetric about

the y-axisEven

Horizontal


p(x) 5 q(Bx),

|B| . 1

Symmetric about

the y-axisEven

Reflection across

x-axisp(x) 5 2q(x)

Symmetric about

the y-axisEven

Reflection across

y-axisp(x) 5 q(2x)

Symmetric about

the y-axisEven

Vertical

Translationp(x) 5 q(x) 1 D

Symmetric about

the y-axisEven

Horizontal

Translationp(x) 5 q(x 2 C)

Symmetric about

the line x 5 CNeither

10. Do you think that your results in Question 9 would be the same for any even power

function? Explain your reasoning.

Yes. Because the general shape and end behavior is the same for any even power

function, I think that the results in the table would be the same.

600005_A2_TX_Ch04_293-402.indd 337 14/03/14 2:44 PM

Grouping


Questions 9 through 11 with

a partner. Then have students


a class.

Guiding Questions for Share Phase, Questions 9 through 11

Under what circumstances

was the transformed quartic

function not symmetric about

the y-axis?

What rigid motions resulted

in transforming a basic

quartic function into a

function that is neither odd

nor even?


4

Problem 3

Students will analyze the graphs

of f(x) and g(x), describe the

transformations performed on

f(x) to create g(x), then write

an equation for g(x) in terms

of f(x).

Grouping


Questions 1 and 2 with a



a class.


Was the graph of the

function translated vertically

or horizontally? If so, how

many units?

Was the graph of the function

compressed or stretched? If

so, how many units?

Was the graph of the function re"ected across the axis? How does this affect

the equation for the function?

Was the graph of the function compressed vertically? If so, by what scale

factor? How do you know?

Was the graph of the function stretched? If so, by what scale factor? How do

you know?

Was the graph of the function re"ected across any line? If so, what is the

equation of the line?

11. Use the appropriate word from the box to complete each statement.

always sometimes never

a. If a dilation is performed on an odd function f(x) to produce g(x), then g(x) will

always be an odd function.

b. If a re!ection is performed on an even function f(x) to produce g(x), then g(x) will

always be an even function.

c. If a translation is performed on an odd function f(x) to produce g(x), then g(x) will

never be an odd function.

d. If a translation is performed on an even function f(x) to produce g(x), then g(x) will

sometimes be an even function.

PROBLEM 3 Multiple Transformations

1. Analyze the graphs of f(x) and g(x). Describe the transformations performed on f(x) to

create g(x). Then, write an equation for g(x) in terms of f(x). For each set of points shown

on f(x), the corresponding points after the rigid motions are shown on g(x).

a. g(x) 5 3f(x) 2 4

x

4

8

24

424 22 8628 26

y

28

212

12

16

20

(1, 1)

(0, 0)

(2, 8)

f(x)

x

4

0

8

24

424 22 8628 26

y

28

212

12

16

20(2, 20)

g(x)

(0, 24)

(1, 21)

600005_A2_TX_Ch04_293-402.indd 338 14/03/14 2:44 PM

transformations and symmetry of polynomial...

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