Learn on PengiBig Ideas Math, Algebra 2Chapter 2: Quadratic Functions

Lesson 1: Transformations of Quadratic Functions

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 2, students learn how the constants a, h, and k in the vertex form g(x) = a(x − h)² + k determine horizontal and vertical translations, reflections across the x-axis, and vertical or horizontal stretches and shrinks of the parent function f(x) = x². Students practice identifying and describing these transformations by matching equations to parabola graphs and writing equations from graphs. The lesson builds fluency with quadratic functions in vertex form as a foundation for deeper work with parabolas throughout the chapter.

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.

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Chapter 2: Quadratic Functions

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Lesson 1Current
Lesson 1: Transformations of Quadratic Functions
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Lesson 2
Lesson 2: Characteristics of Quadratic Functions
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Lesson 3
Lesson 3: Focus of a Parabola
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Lesson 4
Lesson 4: Modeling with Quadratic Equations
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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 2: Quadratic Functions

Back to Big Ideas Math, Algebra 2

Lesson 1Current
Lesson 1: Transformations of Quadratic Functions
Learn Practice
Lesson 2
Lesson 2: Characteristics of Quadratic Functions
Learn Practice
Lesson 3
Lesson 3: Focus of a Parabola
Learn Practice
Lesson 4
Lesson 4: Modeling with Quadratic Equations
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 2: Quadratic Functions

Back to Big Ideas Math, Algebra 2

Lesson 1Current
Lesson 1: Transformations of Quadratic Functions
Learn Practice
Lesson 2
Lesson 2: Characteristics of Quadratic Functions
Learn Practice
Lesson 3
Lesson 3: Focus of a Parabola
Learn Practice
Lesson 4
Lesson 4: Modeling with Quadratic Equations
Learn Practice

Lesson 1: Transformations of Quadratic Functions

Property

Examples

Property

Examples

Chapter 2: Quadratic Functions

Lesson 1: Transformations of Quadratic Functions

Lesson 2: Characteristics of Quadratic Functions

Lesson 3: Focus of a Parabola

Lesson 4: Modeling with Quadratic Equations

Lesson overview

Property

Examples

Property

Examples

Chapter 2: Quadratic Functions

Lesson 1: Transformations of Quadratic Functions

Lesson 2: Characteristics of Quadratic Functions

Lesson 3: Focus of a Parabola

Lesson 4: Modeling with Quadratic Equations

Lesson 1: Transformations of Quadratic Functions

Lesson 2: Characteristics of Quadratic Functions

Lesson 3: Focus of a Parabola

Lesson 4: Modeling with Quadratic Equations