Learn on PengienVision, Algebra 2Chapter 3: Polynomial Functions

Lesson 7: Transformations of Polynomial Functions

In this Grade 11 enVision Algebra 2 lesson, students learn to identify even and odd functions by testing for y-axis symmetry and rotational symmetry about the origin using both graphs and equations. The lesson also covers how transformations such as vertical stretches, horizontal shifts, and vertical translations affect the graphs of cubic and quartic parent functions. Students practice writing and interpreting transformed equations of the form g(x) = a(x − h)³ + k and similar quartic expressions.

Section 1

Classifying Polynomial Functions as Even, Odd, or Neither

Property

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis).

A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

Section 2

Square and Cube Functions

Property

Function	Definition	Domain	Range
Square Function	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cube Function	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$

Examples

For the square function $f(x) = x^2$ , both $x=4$ and $x=-4$ produce the same output, $f(x)=16$ , showing its symmetry.

Section 3

Distinguishing Function Symmetry from Degree

Property

A polynomial's degree does not determine its symmetry type. Even degree polynomials are not automatically even functions, and odd degree polynomials are not automatically odd functions. Symmetry is determined by testing: $f(-x) = f(x)$ for even functions, $f(-x) = -f(x)$ for odd functions.

Examples

Section 4

Transformations of Cubic and Quartic Functions

Property

Cubic and quartic functions can be transformed using the general form:

f(x) = a(x - h)^n + k

where $n = 3$ for cubic or $n = 4$ for quartic, $a$ controls vertical stretch/compression and reflection, $h$ represents horizontal shift, and $k$ represents vertical shift.

Examples

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Chapter 3: Polynomial Functions

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Lesson 1
Lesson 1: Graphing Polynomial Functions
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Lesson 2
Lesson 2: Adding, Subtracting, and Multiplying Polynomials
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Lesson 3
Lesson 3: Polynomial Identities
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Lesson 4
Lesson 4: Dividing Polynomials
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Lesson 5
Lesson 5: Zeros of Polynomial Functions
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Lesson 6
Lesson 6: Theorems About Roots of Polynomial Equations
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Lesson 7Current
Lesson 7: Transformations of Polynomial Functions
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Section 1

Classifying Polynomial Functions as Even, Odd, or Neither

Property

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis).

A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

Section 2

Square and Cube Functions

Property

Function	Definition	Domain	Range
Square Function	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cube Function	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$

Examples

For the square function $f(x) = x^2$ , both $x=4$ and $x=-4$ produce the same output, $f(x)=16$ , showing its symmetry.

Section 3

Distinguishing Function Symmetry from Degree

Property

Examples

Section 4

Transformations of Cubic and Quartic Functions

Property

Cubic and quartic functions can be transformed using the general form:

f(x) = a(x - h)^n + k

where $n = 3$ for cubic or $n = 4$ for quartic, $a$ controls vertical stretch/compression and reflection, $h$ represents horizontal shift, and $k$ represents vertical shift.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

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Chapter 3: Polynomial Functions

Back to enVision, Algebra 2

Lesson 1
Lesson 1: Graphing Polynomial Functions
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Lesson 2
Lesson 2: Adding, Subtracting, and Multiplying Polynomials
Learn Practice
Lesson 3
Lesson 3: Polynomial Identities
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Lesson 4
Lesson 4: Dividing Polynomials
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Lesson 5
Lesson 5: Zeros of Polynomial Functions
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Lesson 6
Lesson 6: Theorems About Roots of Polynomial Equations
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Lesson 7Current
Lesson 7: Transformations of Polynomial Functions
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Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Classifying Polynomial Functions as Even, Odd, or Neither

Property

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis).

A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

Section 2

Square and Cube Functions

Property

Function	Definition	Domain	Range
Square Function	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cube Function	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$

Examples

For the square function $f(x) = x^2$ , both $x=4$ and $x=-4$ produce the same output, $f(x)=16$ , showing its symmetry.

Section 3

Distinguishing Function Symmetry from Degree

Property

Examples

Section 4

Transformations of Cubic and Quartic Functions

Property

Cubic and quartic functions can be transformed using the general form:

f(x) = a(x - h)^n + k

where $n = 3$ for cubic or $n = 4$ for quartic, $a$ controls vertical stretch/compression and reflection, $h$ represents horizontal shift, and $k$ represents vertical shift.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 3: Polynomial Functions

Back to enVision, Algebra 2

Lesson 1
Lesson 1: Graphing Polynomial Functions
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Lesson 2
Lesson 2: Adding, Subtracting, and Multiplying Polynomials
Learn Practice
Lesson 3
Lesson 3: Polynomial Identities
Learn Practice
Lesson 4
Lesson 4: Dividing Polynomials
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Lesson 5
Lesson 5: Zeros of Polynomial Functions
Learn Practice
Lesson 6
Lesson 6: Theorems About Roots of Polynomial Equations
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Lesson 7Current
Lesson 7: Transformations of Polynomial Functions
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Lesson 7: Transformations of Polynomial Functions

Property

Property

Examples

Property

Examples

Property

Examples

Chapter 3: Polynomial Functions

Lesson 1: Graphing Polynomial Functions

Lesson 2: Adding, Subtracting, and Multiplying Polynomials

Lesson 3: Polynomial Identities

Lesson 4: Dividing Polynomials

Lesson 5: Zeros of Polynomial Functions

Lesson 6: Theorems About Roots of Polynomial Equations

Lesson 7: Transformations of Polynomial Functions

Lesson overview

Property

Property

Examples

Property

Examples

Property

Examples

Chapter 3: Polynomial Functions

Lesson 1: Graphing Polynomial Functions

Lesson 2: Adding, Subtracting, and Multiplying Polynomials

Lesson 3: Polynomial Identities

Lesson 4: Dividing Polynomials

Lesson 5: Zeros of Polynomial Functions

Lesson 6: Theorems About Roots of Polynomial Equations

Lesson 7: Transformations of Polynomial Functions

Lesson 1: Graphing Polynomial Functions

Lesson 2: Adding, Subtracting, and Multiplying Polynomials

Lesson 3: Polynomial Identities

Lesson 4: Dividing Polynomials

Lesson 5: Zeros of Polynomial Functions

Lesson 6: Theorems About Roots of Polynomial Equations

Lesson 7: Transformations of Polynomial Functions