Learn on PengienVision, Algebra 2Chapter 2: Quadratic Functions and Equations

Lesson 1: Vertex Form of a Quadratic Function

In this Grade 11 enVision Algebra 2 lesson, students learn to identify key features of quadratic functions using the vertex form f(x) = a(x − h)² + k, including the vertex, axis of symmetry, domain, range, and direction of opening. Students explore how the parameters a, h, and k produce vertical stretches or compressions and horizontal or vertical translations of the parent function f(x) = x². The lesson also covers writing quadratic equations in vertex form given a vertex and an additional point or graph.

Section 1

Vertex Form for a Quadratic Equation

Property

A quadratic equation $y = ax^2 + bx + c$ , $a \neq 0$ , can be written in the vertex form

y = a(x - x_v)^2 + y_v

where the vertex of the graph is $(x_v, y_v)$ .

Examples

The equation $y = -2(x - 1)^2 + 5$ is in vertex form. The vertex is at $(1, 5)$ , and because $a = -2$ is negative, the parabola opens downward.

To write $y = x^2 - 8x + 10$ in vertex form, find the vertex. $x_v = \frac{-(-8)}{2(1)} = 4$ . Then $y_v = 4^2 - 8(4) + 10 = -6$ . The vertex form is $y = (x - 4)^2 - 6$ .

Section 2

Graphing in Vertex Form Using Properties

Property

How To Graph a quadratic function in the form $f(x) = a(x - h)^2 + k$ using properties

Determine if the parabola opens upward ( $a > 0$ ) or downward ( $a < 0$ ).
Find the axis of symmetry, $x = h$ .
Find the vertex, $(h, k)$ .
Find the $y$ -intercept by calculating $f(0)$ .
Find the $x$ -intercepts by solving $f(x) = 0$ .
Graph the parabola using these key points.

Examples

For $f(x) = 3(x-2)^2 + 5$ , the parabola opens upward ( $a=3$ ), the vertex is $(2, 5)$ , and the axis of symmetry is $x=2$ .

For $f(x) = -(x+1)^2 - 4$ , the parabola opens downward ( $a=-1$ ), the vertex is $(-1, -4)$ , and the axis of symmetry is $x=-1$ .

Book overview

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

Continue this chapter

Chapter 2: Quadratic Functions and Equations

Back to enVision, Algebra 2

Lesson 1Current
Lesson 1: Vertex Form of a Quadratic Function
Learn Practice
Lesson 2
Lesson 2: Standard Form of a Quadratic Function
Learn Practice
Lesson 3
Lesson 3: Factored Form of a Quadratic Function
Learn Practice
Lesson 4
Lesson 4: Complex Numbers and Operations
Learn Practice
Lesson 5
Lesson 5: Completing the Square
Learn Practice
Lesson 6
Lesson 6: The Quadratic Formula
Learn Practice
Lesson 7
Lesson 7: Linear-Quadratic Systems
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Vertex Form for a Quadratic Equation

Property

A quadratic equation $y = ax^2 + bx + c$ , $a \neq 0$ , can be written in the vertex form

y = a(x - x_v)^2 + y_v

where the vertex of the graph is $(x_v, y_v)$ .

Examples

The equation $y = -2(x - 1)^2 + 5$ is in vertex form. The vertex is at $(1, 5)$ , and because $a = -2$ is negative, the parabola opens downward.

To write $y = x^2 - 8x + 10$ in vertex form, find the vertex. $x_v = \frac{-(-8)}{2(1)} = 4$ . Then $y_v = 4^2 - 8(4) + 10 = -6$ . The vertex form is $y = (x - 4)^2 - 6$ .

Section 2

Graphing in Vertex Form Using Properties

Property

How To Graph a quadratic function in the form $f(x) = a(x - h)^2 + k$ using properties

Determine if the parabola opens upward ( $a > 0$ ) or downward ( $a < 0$ ).
Find the axis of symmetry, $x = h$ .
Find the vertex, $(h, k)$ .
Find the $y$ -intercept by calculating $f(0)$ .
Find the $x$ -intercepts by solving $f(x) = 0$ .
Graph the parabola using these key points.

Examples

For $f(x) = 3(x-2)^2 + 5$ , the parabola opens upward ( $a=3$ ), the vertex is $(2, 5)$ , and the axis of symmetry is $x=2$ .

For $f(x) = -(x+1)^2 - 4$ , the parabola opens downward ( $a=-1$ ), the vertex is $(-1, -4)$ , and the axis of symmetry is $x=-1$ .

Book overview

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

Continue this chapter

Chapter 2: Quadratic Functions and Equations

Back to enVision, Algebra 2

Lesson 1Current
Lesson 1: Vertex Form of a Quadratic Function
Learn Practice
Lesson 2
Lesson 2: Standard Form of a Quadratic Function
Learn Practice
Lesson 3
Lesson 3: Factored Form of a Quadratic Function
Learn Practice
Lesson 4
Lesson 4: Complex Numbers and Operations
Learn Practice
Lesson 5
Lesson 5: Completing the Square
Learn Practice
Lesson 6
Lesson 6: The Quadratic Formula
Learn Practice
Lesson 7
Lesson 7: Linear-Quadratic Systems
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

Vertex Form for a Quadratic Equation

Property

A quadratic equation $y = ax^2 + bx + c$ , $a \neq 0$ , can be written in the vertex form

y = a(x - x_v)^2 + y_v

where the vertex of the graph is $(x_v, y_v)$ .

Examples

The equation $y = -2(x - 1)^2 + 5$ is in vertex form. The vertex is at $(1, 5)$ , and because $a = -2$ is negative, the parabola opens downward.

To write $y = x^2 - 8x + 10$ in vertex form, find the vertex. $x_v = \frac{-(-8)}{2(1)} = 4$ . Then $y_v = 4^2 - 8(4) + 10 = -6$ . The vertex form is $y = (x - 4)^2 - 6$ .

Section 2

Graphing in Vertex Form Using Properties

Property

How To Graph a quadratic function in the form $f(x) = a(x - h)^2 + k$ using properties

Determine if the parabola opens upward ( $a > 0$ ) or downward ( $a < 0$ ).
Find the axis of symmetry, $x = h$ .
Find the vertex, $(h, k)$ .
Find the $y$ -intercept by calculating $f(0)$ .
Find the $x$ -intercepts by solving $f(x) = 0$ .
Graph the parabola using these key points.

Examples

For $f(x) = 3(x-2)^2 + 5$ , the parabola opens upward ( $a=3$ ), the vertex is $(2, 5)$ , and the axis of symmetry is $x=2$ .

For $f(x) = -(x+1)^2 - 4$ , the parabola opens downward ( $a=-1$ ), the vertex is $(-1, -4)$ , and the axis of symmetry is $x=-1$ .

Book overview

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

Continue this chapter

Chapter 2: Quadratic Functions and Equations

Back to enVision, Algebra 2

Lesson 1Current
Lesson 1: Vertex Form of a Quadratic Function
Learn Practice
Lesson 2
Lesson 2: Standard Form of a Quadratic Function
Learn Practice
Lesson 3
Lesson 3: Factored Form of a Quadratic Function
Learn Practice
Lesson 4
Lesson 4: Complex Numbers and Operations
Learn Practice
Lesson 5
Lesson 5: Completing the Square
Learn Practice
Lesson 6
Lesson 6: The Quadratic Formula
Learn Practice
Lesson 7
Lesson 7: Linear-Quadratic Systems
Learn Practice

Lesson 1: Vertex Form of a Quadratic Function

Property

Examples

Property

Examples

Chapter 2: Quadratic Functions and Equations

Lesson 1: Vertex Form of a Quadratic Function

Lesson 2: Standard Form of a Quadratic Function

Lesson 3: Factored Form of a Quadratic Function

Lesson 4: Complex Numbers and Operations

Lesson 5: Completing the Square

Lesson 6: The Quadratic Formula

Lesson 7: Linear-Quadratic Systems

Lesson overview

Property

Examples

Property

Examples

Chapter 2: Quadratic Functions and Equations

Lesson 1: Vertex Form of a Quadratic Function

Lesson 2: Standard Form of a Quadratic Function

Lesson 3: Factored Form of a Quadratic Function

Lesson 4: Complex Numbers and Operations

Lesson 5: Completing the Square

Lesson 6: The Quadratic Formula

Lesson 7: Linear-Quadratic Systems

Lesson 1: Vertex Form of a Quadratic Function

Lesson 2: Standard Form of a Quadratic Function

Lesson 3: Factored Form of a Quadratic Function

Lesson 4: Complex Numbers and Operations

Lesson 5: Completing the Square

Lesson 6: The Quadratic Formula

Lesson 7: Linear-Quadratic Systems