Learn on PengienVision, Algebra 1Chapter 8: Quadratic Functions

Lesson 2: Quadratic Functions in Vertex Form

In this Grade 11 enVision Algebra 1 lesson, students learn to graph quadratic functions written in vertex form, f(x) = a(x − h)² + k, by identifying how the parameters a, h, and k control the vertex location, axis of symmetry, and width or direction of a parabola. The lesson covers vertical and horizontal translations of the parent function f(x) = x², as well as how the sign and absolute value of a determine whether the parabola opens upward or downward and whether it appears narrower or wider than the parent graph.

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 8: Quadratic Functions

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Lesson 1
Lesson 1: Key Features of Graphs of a Quadratic Function
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Lesson 2Current
Lesson 2: Quadratic Functions in Vertex Form
Learn Practice
Lesson 3
Lesson 3: Quadratic Functions in Standard Form
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Lesson 4
Lesson 4: Modeling with Quadratic Functions
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Lesson 5
Lesson 5: Comparing Linear, Exponential, and Quadratic Models
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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 8: Quadratic Functions

Back to enVision, Algebra 1

Lesson 1
Lesson 1: Key Features of Graphs of a Quadratic Function
Learn Practice
Lesson 2Current
Lesson 2: Quadratic Functions in Vertex Form
Learn Practice
Lesson 3
Lesson 3: Quadratic Functions in Standard Form
Learn Practice
Lesson 4
Lesson 4: Modeling with Quadratic Functions
Learn Practice
Lesson 5
Lesson 5: Comparing Linear, Exponential, and Quadratic Models
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 8: Quadratic Functions

Back to enVision, Algebra 1

Lesson 1
Lesson 1: Key Features of Graphs of a Quadratic Function
Learn Practice
Lesson 2Current
Lesson 2: Quadratic Functions in Vertex Form
Learn Practice
Lesson 3
Lesson 3: Quadratic Functions in Standard Form
Learn Practice
Lesson 4
Lesson 4: Modeling with Quadratic Functions
Learn Practice
Lesson 5
Lesson 5: Comparing Linear, Exponential, and Quadratic Models
Learn Practice

Lesson 2: Quadratic Functions in Vertex Form

Property

Examples

Property

Examples

Chapter 8: Quadratic Functions

Lesson 1: Key Features of Graphs of a Quadratic Function

Lesson 2: Quadratic Functions in Vertex Form

Lesson 3: Quadratic Functions in Standard Form

Lesson 4: Modeling with Quadratic Functions

Lesson 5: Comparing Linear, Exponential, and Quadratic Models

Lesson overview

Property

Examples

Property

Examples

Chapter 8: Quadratic Functions

Lesson 1: Key Features of Graphs of a Quadratic Function

Lesson 2: Quadratic Functions in Vertex Form

Lesson 3: Quadratic Functions in Standard Form

Lesson 4: Modeling with Quadratic Functions

Lesson 5: Comparing Linear, Exponential, and Quadratic Models

Lesson 1: Key Features of Graphs of a Quadratic Function

Lesson 2: Quadratic Functions in Vertex Form

Lesson 3: Quadratic Functions in Standard Form

Lesson 4: Modeling with Quadratic Functions

Lesson 5: Comparing Linear, Exponential, and Quadratic Models