# Lesson 14

Absolute Value Functions (Part 2)

### Lesson Narrative

Previously, students computed and plotted absolute errors of a set of data, recognizing that each absolute error is a distance from a target number. In this lesson, students use the idea of absolute value to generalize their observations.

Students define the absolute value function in terms of the distance of a number from 0 on the number line. They learn that it can be represented with an equation in the form of $$f(x) = |x|$$. Students also see that, because the graph of the absolute value function is composed of two linear pieces that form a V shape, the same function can also be defined as a piecewise function and described with this equation:

$$\displaystyle f(x)=\begin{cases} x,& x\geq 0 \\ \text{-}x,& x < 0 \end{cases}$$

Here, students also encounter functions defined by absolute value expressions that involve a constant term, and graphs of functions that have been translated horizontally or vertically. They also call these absolute value functions. To make sense of these functions and their graphs, students may relate expressions such as $$|x-5|$$ and $$|x+5|$$ to their earlier work on finding the absolute errors of guesses relative to a target number, say 5 or -5. They may recall that, in those activities, the V shape of the scatter plots shifted horizontally toward the target number.

Students look for and make use of structure (MP7) to relate the translation of a graph to the terms or values in the expression that defines the function. The distinctive V shape of the graph of an absolute value function is ideal for observing how adding or subtracting a constant term affects its graph.

In future units, students will see that the connections they notice here—between the parameters of a function and its graph—also extend to other kinds of functions.

### Learning Goals

Teacher Facing

• Analyze and describe (orally and in writing) the effects of adding a constant term to an expression defining an absolute value function.
• Define absolute value function in terms of the distance of the input from 0.
• Interpret an absolute value function described in words or in function notation, and create a table of values and a graph to represent the function.

### Student Facing

Let’s investigate distance as a function.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can describe the effects of adding a number to the expression that defines an absolute value function.
• I can explain the meaning of absolute value function in terms of distance.
• When given an absolute value function in words or in function notation, I can make sense of it, and can create a table of values and a graph to represent it.

Building Towards

### Glossary Entries

• absolute value

The absolute value of a number is its distance from 0 on the number line.

• domain

The domain of a function is the set of all of its possible input values.

• piecewise function

A piecewise function is a function defined using different expressions for different intervals in its domain.

• range

The range of a function is the set of all of its possible output values.