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7.2: Introducing the Absolute Value Function

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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 Y= 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: 43, 49, 47, 42, 54, 55, 58, 58, 61, 62, 49, 46.

Press STAT ENTER. Enter these 12 data points into L_1.

Press 2^{nd} [MODE] to return to the home screen. Press 2^{nd} [STAT] \rightarrow \rightarrow to the ‘Math on Lists’ menu. Press 3 to select 3:mean(.

This will paste the command onto the home screen. Press 2^{nd}\ [1] ) to complete the command to find the mean of L_1. 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 L_2 as shown.

Press 2^{nd}\ [1] - 52. This will command the calculator to subtract the mean of 52 from each of the temperatures in L_1.

Press ENTER to execute. What do you notice about the numbers in L_2? What is the highest difference? What is the smallest difference? When are the differences negative? Positive?

Move over to L_3. Examine each entry in L_1 and determine is DISTANCE from the mean (how far away). Enter the distances in L_3. What is the relationship between the distances and the differences from L_2? Why is this so?

Set up a scatter plot to compare the distances to the differences (L_3 to L_2). Press 2^{nd} \ Y=. Press 1 to select 1:Plot 1.

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

Press 2^{nd}\ [2] to use L_2 (the differences) as the x list. Arrow down to the Ylist. Press 2^{nd}\ [3] to use L_3 (the distances) as the y list.

Press WINDOW. Set the window as shown.

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

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

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

Problem 2 - Compare Data Against Equations

Press Y=. Enter the equation y = x into Y_1.

Press GRAPH. What is the relationship between y = x and the scatter plot?

Return to Y=. Enter the equation y = -x into Y_2.

Press GRAPH. What is the relationship between y = -x and the scatter plot?

Press 2^{nd} [GRAPH] to examine the tables for Y_1 and Y_2. How are the values for X and Y_1 related? How are the values for X and Y_2 related? How are the values for Y_1 and Y_2 related? Where is each Y equal to zero?

Return to Y=. Arrow down to Y_3. Press MATH \rightarrow to find the absolute value command 1:abs(. Press ENTER. This will paste the command into Y_3.

Complete the function as shown. Arrow left of Y_3. Press ENTER to change the graph to a ‘thick line.’

Press GRAPH. What is the relationship between y = abs(x) and the scatter plot? NOTE: In your textbook this function will be written as y = \left | x \right |.

Press 2^{nd} [GRAPH] to examine the tables. How are the values for Y_3 related to Y_1 and Y_2? Where is Y equal to zero?

Extension

Examine another absolute value equation. First, clear all earlier functions in Y= and enter y = x+7 into Y1. To clear functions, put cursor over the equals sign and press [CLEAR].

Press 2^{nd} [GRAPH] Examine the table. When are the Y_1 values positive? When are they negative? When is Y_1 zero?

Return to Y=. Enter the equation y = abs(x) + 7 into Y_2 using [MATH] \rightarrow abs(Examine the graph. What seems to be the relationship between the graphs?

Examine the table. Is the relationship between Y_2 and Y_1 what you were expecting? Why or why not? Where are the Y values equal to zero?

Return to Y=. Enter the equation y = abs(x + 7) into Y_2 as shown.

Examine the graph. What seems to be the relationship between the graphs? How is this picture different from the graph with y = abs(x) + 7?

Examine the table. Is the relationship between Y_2 and Y_1 what you were expecting? Why or why not? Where are the Y values equal to zero?

Compare y = abs(x) + 7 to y = abs(x + 7). How are they similar? Different?

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