# 7.2: Introducing the Absolute Value Function

*This activity is intended to supplement Algebra I, Chapter 6, Lesson 5.*

In this activity, you will examine data by comparing individual data points to the mean by finding the difference (positive or negative) and the distance from the mean, plot the distances versus the differences to examine the shape of the plot, investigate the absolute value function in the register to model the relationship between the distances and the differences, and extend the investigation of absolute value equations by examining tables and graphs.

## Problem 1 - Analyze the Data

The high temperatures in the first twelve days of February were: .

Press **STAT ENTER**. Enter these data points into .

Press **[MODE]** to return to the home screen. Press **[STAT]** to the ‘Math on Lists’ menu. Press 3 to select **3:mean**(.

This will paste the command onto the home screen. Press ) to complete the command to find the mean of . Press **ENTER** to execute.

Now that you know the mean of the temperatures, press **STAT ENTER** to return to the ‘statistics editor.’ Arrow to the top of as shown.

Press . This will command the calculator to subtract the mean of from each of the temperatures in .

Press **ENTER** to execute. What do you notice about the numbers in ? What is the highest difference? What is the smallest difference? When are the differences negative? Positive?

Move over to . Examine each entry in and determine is DISTANCE from the mean (how far away). Enter the distances in . What is the relationship between the distances and the differences from ? Why is this so?

Set up a scatter plot to compare the distances to the differences ( to ). Press . Press to select **1:Plot 1**.

Press **ENTER** to turn the plot **On**. Arrow down to the **Xlist**.

Press to use (the differences) as the list. Arrow down to the **Ylist**. Press to use (the distances) as the list.

Press **WINDOW**. Set the window as shown.

Press **GRAPH**. Press **TRACE** to examine the relationships between the and coordinates of each point. When is positive, what happens to ?

When is negative, what happens to ? When will be negative? Why? When is negative?

(sample response: is negative whenever the temperature was lower than the mean; will not be negative because distances are positive)

## Problem 2 - Compare Data Against Equations

Press . Enter the equation into .

Press **GRAPH**. What is the relationship between and the scatter plot?

Return to . Enter the equation into .

Press **GRAPH**. What is the relationship between and the scatter plot?

Press **[GRAPH]** to examine the tables for and . How are the values for and related? How are the values for and related? How are the values for and related? Where is each equal to zero?

Return to . Arrow down to . Press **MATH** to find the absolute value command **1:abs(**. Press **ENTER**. This will paste the command into .

Complete the function as shown. Arrow left of . Press **ENTER** to change the graph to a ‘thick line.’

Press **GRAPH**. What is the relationship between and the scatter plot? NOTE: In your textbook this function will be written as .

Press **[GRAPH]** to examine the tables. How are the values for related to and ? Where is equal to zero?

## Extension

Examine another absolute value equation. First, clear all earlier functions in and enter into . To clear functions, put cursor over the equals sign and press **[CLEAR]**.

Press **[GRAPH]** Examine the table. When are the values positive? When are they negative? When is zero?

Return to . Enter the equation into using **[MATH]** abs(Examine the graph. What seems to be the relationship between the graphs?

Examine the table. Is the relationship between and what you were expecting? Why or why not? Where are the values equal to zero?

Return to . Enter the equation into as shown.

Examine the graph. What seems to be the relationship between the graphs? How is this picture different from the graph with ?

Examine the table. Is the relationship between and what you were expecting? Why or why not? Where are the values equal to zero?

Compare to . How are they similar? Different?