Lesson 3 Absolutely Valuable Solidify Understanding

Learning Focus

Relate piecewise functions to absolute value.

Identify features of an absolute value function.

What is the effect of taking the absolute value of a linear function?

How can we think of absolute value as a function?

Technology guidance for today’s lesson:

Compare Absolute Value and Piecewise Functions: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Michelle likes riding her bike to and from her favorite lake on Wednesdays. She created the following graph to represent the distance she is away from the lake while biking.

1.

Interpret the graph by writing three observations about Michelle’s bike ride.

2.

Write a piecewise function for this situation, with each linear function being in point-slope form using the point $(3, 0)$ . What do you notice?

3.

This particular piecewise function is called a linear absolute value function. What are some of the characteristics of linear absolute value functions that you see in this graph?

In this part of the task, you will solidify your understanding of piecewise functions and reason about absolute value functions.

Let: $f (x) = x$ and $g (x) = | f (x) |$ .

4.

Let: $f (x) = x$ and $g (x) = | f (x) |$

Complete the table of values from $[- 4, 4]$ for $f (x)$ and $g (x)$ . Explain how $g (x)$ affects the output values of $f (x)$ .

$x$	$f (x)$	$g (x)$
$- 4$
$- 3$
$- 2$
$- 1$
$0$
$1$
$2$
$3$
$4$

5.

Graph $f (x)$ .
On the same set of axes, using a different-colored pencil, graph $g (x) = | f (x) |$ .

6.

Explain the difference in the graphs of $f (x)$ and $g (x)$ . Explain why this difference occurs.

7.

Use the graph to write the piecewise function for $g (x)$ . Explain your process for creating this piecewise function.

8.

What similarities and differences do you observe in the piecewise form of $g (x)$ and the piecewise function you wrote for Michelle’s bike ride?

9.

The function is sometimes written in piecewise form and often written as $g (x) = | x |$ .

Answer each question below and explain your answer.

a.

What is the domain of $g (x) = | x |$ ?

b.

What is the range of $g (x) = | x |$ ?

c.

What is the minimum of $g (x) = | x |$ ?

d.

Is $g (x) = | x |$ continuous?

e.

For $x < 0$ , what is the rate of change of $g (x) = | x |$ ?

f.

For $x > 0$ , what is the rate of change of $g (x) = | x |$ ?

10.

Based on your analysis, how would you write the piecewise function that modeled Michelle’s bike ride in absolute value form?

Ready for More?

Besides Michelle’s bike ride, what is another context that could be modeled by an absolute value function?

Takeaways

Graph:

The absolute value function:

Vocabulary

absolute value
absolute value function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about the linear absolute value function. We learned that absolute value functions can be written as piecewise functions or using the operation because they have two distinct parts. We identified the domain and range and graphed the function.

Retrieval

Explain how the given function compares to the parent function, $f (x) = x^{2}$ , by describing the transformations that have occurred to the parent function to produce the given function.

1.

$g (x) = {(x - 7)}^{2} + 4$

2.

$h (x) = - {(x + 13)}^{2} - 9$

Evaluate the given expression using the variable provided.

3.

$- (b); b = - 74$

4.

$- | - (- n) |; n = 5$