Learn on PengiBig Ideas Math, Algebra 1Chapter 8: Graphing Quadratic Functions

Lesson 4: Graphing f (x) = a(x − h)² + k

Property.

Section 1

Even and Odd Function Identification

Property

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.
A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain.
Most functions are neither even nor odd.

Examples

Section 2

Parameter 'a' Effects in Vertex Form

Property

In $f(x) = a(x - h)^2 + k$ , the parameter $a$ determines: - Direction: $a > 0$ opens upward, $a < 0$ opens downward - Width: $|a| > 1$ makes parabola narrower, $0 < |a| < 1$ makes parabola wider

Examples

Section 3

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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Even and Odd Function Identification

Property

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.
A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain.
Most functions are neither even nor odd.

Examples

Section 2

Parameter 'a' Effects in Vertex Form

Property

In $f(x) = a(x - h)^2 + k$ , the parameter $a$ determines: - Direction: $a > 0$ opens upward, $a < 0$ opens downward - Width: $|a| > 1$ makes parabola narrower, $0 < |a| < 1$ makes parabola wider

Examples

Section 3

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.

Overview

Lesson overview

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

Overview

Section 1

Even and Odd Function Identification

Property

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.
A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain.
Most functions are neither even nor odd.

Examples

Section 2

Parameter 'a' Effects in Vertex Form

Property

In $f(x) = a(x - h)^2 + k$ , the parameter $a$ determines: - Direction: $a > 0$ opens upward, $a < 0$ opens downward - Width: $|a| > 1$ makes parabola narrower, $0 < |a| < 1$ makes parabola wider

Examples

Section 3

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.