Learn on PengienVision, Algebra 1Chapter 5: Piecewise Functions

Lesson 1: The Absolute Value Function

In this Grade 11 enVision Algebra 1 lesson from Chapter 5: Piecewise Functions, students learn to analyze the absolute value function f(x) = |x| by identifying its key features, including the vertex, axis of symmetry, domain, and range. Students explore how multiplying the absolute value expression by positive or negative factors produces vertical stretches or reflections that affect the range while keeping the domain as all real numbers. The lesson also applies the function in real-world contexts, such as interpreting d(t) = 30|t − 1.5| to model distance over time.

Section 1

Absolute Value Function Properties

Property

FunctionDefinitionDomainRange
Absolute Value Functionf(x)=∣x∣f(x) = \lvert x \rvert(−∞,∞)(-\infty, \infty)[0,∞)[0, \infty)

Examples

Section 2

Vertex and Axis of Symmetry for Absolute Value Functions

Property

For the absolute value function f(x)=a∣x−h∣+kf(x) = a|x - h| + k, the vertex is at the point (h,k)(h, k) and the axis of symmetry is the vertical line x=hx = h.
For the parent function f(x)=∣x∣f(x) = |x|, the vertex is (0,0)(0, 0) and the axis of symmetry is x=0x = 0.

Examples

Section 3

Vertical Transformations of Absolute Value Functions

Property

The transformation f(x)=a∣x∣f(x) = a|x| vertically stretches or compresses the parent function f(x)=∣x∣f(x) = |x| by factor ∣a∣|a|.
When a>0a > 0, the graph opens upward with vertex at (0,0)(0,0). When a<0a < 0, the graph reflects across the x-axis and opens downward with vertex at (0,0)(0,0).

Examples

Section 4

Slope as rate of change

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

m=ΔyΔx=y2−y1x2−x1,x1≠x2m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}, x_1 \neq x_2

Examples

Find the slope between (−1,4)(-1, 4) and (3,−2)(3, -2): m=−2−43−(−1)=−64=−32m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Absolute Value Function Properties

Property

FunctionDefinitionDomainRange
Absolute Value Functionf(x)=∣x∣f(x) = \lvert x \rvert(−∞,∞)(-\infty, \infty)[0,∞)[0, \infty)

Examples

Section 2

Vertex and Axis of Symmetry for Absolute Value Functions

Property

For the absolute value function f(x)=a∣x−h∣+kf(x) = a|x - h| + k, the vertex is at the point (h,k)(h, k) and the axis of symmetry is the vertical line x=hx = h.
For the parent function f(x)=∣x∣f(x) = |x|, the vertex is (0,0)(0, 0) and the axis of symmetry is x=0x = 0.

Examples

Section 3

Vertical Transformations of Absolute Value Functions

Property

The transformation f(x)=a∣x∣f(x) = a|x| vertically stretches or compresses the parent function f(x)=∣x∣f(x) = |x| by factor ∣a∣|a|.
When a>0a > 0, the graph opens upward with vertex at (0,0)(0,0). When a<0a < 0, the graph reflects across the x-axis and opens downward with vertex at (0,0)(0,0).

Examples

Section 4

Slope as rate of change

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

m=ΔyΔx=y2−y1x2−x1,x1≠x2m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}, x_1 \neq x_2

Examples

Find the slope between (−1,4)(-1, 4) and (3,−2)(3, -2): m=−2−43−(−1)=−64=−32m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}.