Learn on PengiBig Ideas Math, Algebra 2Chapter 4: Polynomial Functions

Lesson 7: Transformations of Polynomial Functions

In this Grade 8 lesson from Big Ideas Math Algebra 2, students learn how to describe and write transformations of polynomial functions, including horizontal and vertical translations, reflections over the x- and y-axes, and horizontal and vertical stretches and shrinks. Using cubic and quartic parent functions such as f(x) = x³ and f(x) = x⁴, students practice identifying transformation rules expressed in function notation like f(x − h), f(x) + k, a·f(x), and f(ax). The lesson connects polynomial transformations to previously learned techniques applied to linear, absolute value, and quadratic functions.

Section 1

Graph Quadratic Functions of the form f(x) = x^2 + k

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

If $k > 0$ , shift the parabola vertically up $k$ units.
If $k < 0$ , shift the parabola vertically down $|k|$ units.

Examples

To graph $f(x) = x^2 + 4$ , you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$ .

To graph $f(x) = x^2 - 5$ , you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$ .

Section 2

Graph Quadratic Functions of the form f(x) = (x - h)^2

Property

The graph of $f(x) = (x - h)^2$ shifts the graph of $f(x) = x^2$ horizontally $h$ units.

If $h > 0$ , shift the parabola horizontally right $h$ units.
If $h < 0$ , shift the parabola horizontally left $|h|$ units.

Examples

To graph $f(x) = (x - 3)^2$ , you shift the graph of $f(x) = x^2$ to the right 3 units. The vertex moves from $(0, 0)$ to $(3, 0)$ .

To graph $f(x) = (x + 4)^2$ , you rewrite it as $f(x) = (x - (-4))^2$ . This means you shift the graph of $f(x) = x^2$ to the left 4 units.

Book overview

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Chapter 4: Polynomial Functions

Back to Big Ideas Math, 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: Dividing Polynomials
Learn Practice
Lesson 4
Lesson 4: Factoring Polynomials
Learn Practice
Lesson 5
Lesson 5: Solving Polynomial Equations
Learn Practice
Lesson 6
Lesson 6: The Fundamental Theorem of Algebra
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Lesson 7Current
Lesson 7: Transformations of Polynomial Functions
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Lesson 8
Lesson 8: Analyzing Graphs of Polynomial Functions
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Lesson 9
Lesson 9: Modeling with Polynomial Functions
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Lesson overview

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Section 1

Graph Quadratic Functions of the form f(x) = x^2 + k

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

If $k > 0$ , shift the parabola vertically up $k$ units.
If $k < 0$ , shift the parabola vertically down $|k|$ units.

Examples

To graph $f(x) = x^2 + 4$ , you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$ .

To graph $f(x) = x^2 - 5$ , you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$ .

Section 2

Graph Quadratic Functions of the form f(x) = (x - h)^2

Property

The graph of $f(x) = (x - h)^2$ shifts the graph of $f(x) = x^2$ horizontally $h$ units.

If $h > 0$ , shift the parabola horizontally right $h$ units.
If $h < 0$ , shift the parabola horizontally left $|h|$ units.

Examples

To graph $f(x) = (x - 3)^2$ , you shift the graph of $f(x) = x^2$ to the right 3 units. The vertex moves from $(0, 0)$ to $(3, 0)$ .

To graph $f(x) = (x + 4)^2$ , you rewrite it as $f(x) = (x - (-4))^2$ . This means you shift the graph of $f(x) = x^2$ to the left 4 units.

Book overview

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

Continue this chapter

Chapter 4: Polynomial Functions

Back to Big Ideas Math, Algebra 2

Lesson 1
Lesson 1: Graphing Polynomial Functions
Learn Practice
Lesson 2
Lesson 2: Adding, Subtracting, and Multiplying Polynomials
Learn Practice
Lesson 3
Lesson 3: Dividing Polynomials
Learn Practice
Lesson 4
Lesson 4: Factoring Polynomials
Learn Practice
Lesson 5
Lesson 5: Solving Polynomial Equations
Learn Practice
Lesson 6
Lesson 6: The Fundamental Theorem of Algebra
Learn Practice
Lesson 7Current
Lesson 7: Transformations of Polynomial Functions
Learn Practice
Lesson 8
Lesson 8: Analyzing Graphs of Polynomial Functions
Learn Practice
Lesson 9
Lesson 9: Modeling with Polynomial Functions
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

Graph Quadratic Functions of the form f(x) = x^2 + k

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

If $k > 0$ , shift the parabola vertically up $k$ units.
If $k < 0$ , shift the parabola vertically down $|k|$ units.

Examples

To graph $f(x) = x^2 + 4$ , you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$ .

To graph $f(x) = x^2 - 5$ , you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$ .

Section 2

Graph Quadratic Functions of the form f(x) = (x - h)^2

Property

The graph of $f(x) = (x - h)^2$ shifts the graph of $f(x) = x^2$ horizontally $h$ units.

If $h > 0$ , shift the parabola horizontally right $h$ units.
If $h < 0$ , shift the parabola horizontally left $|h|$ units.

Examples

To graph $f(x) = (x - 3)^2$ , you shift the graph of $f(x) = x^2$ to the right 3 units. The vertex moves from $(0, 0)$ to $(3, 0)$ .

To graph $f(x) = (x + 4)^2$ , you rewrite it as $f(x) = (x - (-4))^2$ . This means you shift the graph of $f(x) = x^2$ to the left 4 units.

Book overview

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

Continue this chapter

Chapter 4: Polynomial Functions

Back to Big Ideas Math, Algebra 2

Lesson 1
Lesson 1: Graphing Polynomial Functions
Learn Practice
Lesson 2
Lesson 2: Adding, Subtracting, and Multiplying Polynomials
Learn Practice
Lesson 3
Lesson 3: Dividing Polynomials
Learn Practice
Lesson 4
Lesson 4: Factoring Polynomials
Learn Practice
Lesson 5
Lesson 5: Solving Polynomial Equations
Learn Practice
Lesson 6
Lesson 6: The Fundamental Theorem of Algebra
Learn Practice
Lesson 7Current
Lesson 7: Transformations of Polynomial Functions
Learn Practice
Lesson 8
Lesson 8: Analyzing Graphs of Polynomial Functions
Learn Practice
Lesson 9
Lesson 9: Modeling with Polynomial Functions
Learn Practice

Lesson 7: Transformations of Polynomial Functions

Property

Examples

Property

Examples

Chapter 4: Polynomial Functions

Lesson 1: Graphing Polynomial Functions

Lesson 2: Adding, Subtracting, and Multiplying Polynomials

Lesson 3: Dividing Polynomials

Lesson 4: Factoring Polynomials

Lesson 5: Solving Polynomial Equations

Lesson 6: The Fundamental Theorem of Algebra

Lesson 7: Transformations of Polynomial Functions

Lesson 8: Analyzing Graphs of Polynomial Functions

Lesson 9: Modeling with Polynomial Functions

Lesson overview

Property

Examples

Property

Examples

Chapter 4: Polynomial Functions

Lesson 1: Graphing Polynomial Functions

Lesson 2: Adding, Subtracting, and Multiplying Polynomials

Lesson 3: Dividing Polynomials

Lesson 4: Factoring Polynomials

Lesson 5: Solving Polynomial Equations

Lesson 6: The Fundamental Theorem of Algebra

Lesson 7: Transformations of Polynomial Functions

Lesson 8: Analyzing Graphs of Polynomial Functions

Lesson 9: Modeling with Polynomial Functions

Lesson 1: Graphing Polynomial Functions

Lesson 2: Adding, Subtracting, and Multiplying Polynomials

Lesson 3: Dividing Polynomials

Lesson 4: Factoring Polynomials

Lesson 5: Solving Polynomial Equations

Lesson 6: The Fundamental Theorem of Algebra

Lesson 7: Transformations of Polynomial Functions

Lesson 8: Analyzing Graphs of Polynomial Functions

Lesson 9: Modeling with Polynomial Functions