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

3-7 Absolute Value Functions

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn how to graph and interpret absolute value functions using the general form f(x) = a|x − h| + k, identifying how the parameters a, h, and k produce translations, vertical stretches, and horizontal compressions of the parent function. Students practice describing and writing equations for transformed absolute value graphs, including identifying the vertex after horizontal and vertical shifts. The lesson is part of Unit 3: Linear and Nonlinear Functions and builds understanding of how operations on a function affect its graph.

Section 1

V-Shaped Graph Characteristics

Property

The absolute value function f(x)=∣x∣f(x) = |x| creates a V-shaped graph with vertex at (0,0)(0, 0), opening upward with two linear pieces: y=−xy = -x for x<0x < 0 and y=xy = x for x≥0x \geq 0.

Examples

Section 2

Absolute Value Function Vertex Form and Transformations

Property

The general form of a transformed absolute value function is g(x)=a∣x−h∣+kg(x) = a|x - h| + k, where the vertex is located at (h,k)(h, k). The parameter aa controls vertical stretch/compression and reflection, hh controls horizontal translation, and kk controls vertical translation.

Examples

Section 3

Vertex Identification for Absolute Value Functions

Property

The vertex of an absolute value function in the form f(x)=a∣x−h∣+kf(x) = a|x - h| + k is located at the point (h,k)(h, k). For functions not in vertex form, the vertex occurs where the expression inside the absolute value equals zero.

Examples

Section 4

Horizontal Translations of Piecewise Functions

Property

The graph of f(x)=∣x−h∣f(x) = |x - h| shifts the graph of f(x)=∣x∣f(x) = |x| horizontally hh units.

  • If h>0h > 0, shift the graph horizontally right hh units.
  • If h<0h < 0, shift the graph horizontally left ∣h∣|h| units.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

V-Shaped Graph Characteristics

Property

The absolute value function f(x)=∣x∣f(x) = |x| creates a V-shaped graph with vertex at (0,0)(0, 0), opening upward with two linear pieces: y=−xy = -x for x<0x < 0 and y=xy = x for x≥0x \geq 0.

Examples

Section 2

Absolute Value Function Vertex Form and Transformations

Property

The general form of a transformed absolute value function is g(x)=a∣x−h∣+kg(x) = a|x - h| + k, where the vertex is located at (h,k)(h, k). The parameter aa controls vertical stretch/compression and reflection, hh controls horizontal translation, and kk controls vertical translation.

Examples

Section 3

Vertex Identification for Absolute Value Functions

Property

The vertex of an absolute value function in the form f(x)=a∣x−h∣+kf(x) = a|x - h| + k is located at the point (h,k)(h, k). For functions not in vertex form, the vertex occurs where the expression inside the absolute value equals zero.

Examples

Section 4

Horizontal Translations of Piecewise Functions

Property

The graph of f(x)=∣x−h∣f(x) = |x - h| shifts the graph of f(x)=∣x∣f(x) = |x| horizontally hh units.

  • If h>0h > 0, shift the graph horizontally right hh units.
  • If h<0h < 0, shift the graph horizontally left ∣h∣|h| units.

Examples