Learn on PengiCalifornia Reveal Math, Algebra 1Unit 3: Linear and Nonlinear Functions

3-4 Transformations of Linear Functions

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn how to apply and identify transformations of linear functions, including vertical and horizontal translations, dilations, and reflections. Using the parent function f(x) = x as a starting point, students explore how adding or subtracting constants inside or outside the function shifts its graph on the coordinate plane. The lesson covers key vocabulary such as family of graphs, identity function, translation, dilation, and reflection within the context of Unit 3: Linear and Nonlinear Functions.

Section 1

Parent Function of Linear Functions and Introduction to Transformations

Property

The parent function of all linear functions is:

f(x) = x

Section 2

Vertical Translations: f(x) = x + k

Property

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

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

Examples

Section 3

Horizontal Translations: g(x) = f(x + h)

Property

The graph of $g(x) = f(x + h)$ shifts the graph of $f(x) = x$ horizontally $h$ units.

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

Examples

Section 4

Vertical Dilations of Linear Functions

Property

The coefficient $a$ in the function $f(x) = ax$ affects the graph of $f(x) = x$ by stretching or compressing it.

If $0 < |a| < 1$ , the graph of $f(x) = ax$ will be less steep than the graph of $f(x) = x$ .
If $|a| > 1$ , the graph of $f(x) = ax$ will be steeper than the graph of $f(x) = x$ .

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Parent Function of Linear Functions and Introduction to Transformations

Property

The parent function of all linear functions is:

f(x) = x

Section 2

Vertical Translations: f(x) = x + k

Property

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

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

Examples

Section 3

Horizontal Translations: g(x) = f(x + h)

Property

The graph of $g(x) = f(x + h)$ shifts the graph of $f(x) = x$ horizontally $h$ units.

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

Examples

Section 4

Vertical Dilations of Linear Functions

Property

The coefficient $a$ in the function $f(x) = ax$ affects the graph of $f(x) = x$ by stretching or compressing it.

If $0 < |a| < 1$ , the graph of $f(x) = ax$ will be less steep than the graph of $f(x) = x$ .
If $|a| > 1$ , the graph of $f(x) = ax$ will be steeper than the graph of $f(x) = x$ .

Examples

Overview

Lesson overview

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

Overview

Section 1

Parent Function of Linear Functions and Introduction to Transformations

Property

The parent function of all linear functions is:

f(x) = x

Section 2

Vertical Translations: f(x) = x + k

Property

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

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

Examples

Section 3

Horizontal Translations: g(x) = f(x + h)

Property

The graph of $g(x) = f(x + h)$ shifts the graph of $f(x) = x$ horizontally $h$ units.

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

Examples

Section 4

Vertical Dilations of Linear Functions

Property

The coefficient $a$ in the function $f(x) = ax$ affects the graph of $f(x) = x$ by stretching or compressing it.

If $0 < |a| < 1$ , the graph of $f(x) = ax$ will be less steep than the graph of $f(x) = x$ .
If $|a| > 1$ , the graph of $f(x) = ax$ will be steeper than the graph of $f(x) = x$ .