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7.2: Introducing the Absolute Value Function

Difficulty Level: At Grade Created by: CK-12

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 \begin{align*}Y=\end{align*} 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: \begin{align*}43, 49, 47, 42, 54, 55, 58,\end{align*} \begin{align*}58, 61, 62, 49, 46\end{align*}.

Press STAT ENTER. Enter these \begin{align*}12\end{align*} data points into \begin{align*}L_1\end{align*}.

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

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

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

Press ENTER to execute. What do you notice about the numbers in \begin{align*}L_2\end{align*}? What is the highest difference? What is the smallest difference? When are the differences negative? Positive?

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

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

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

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

Press WINDOW. Set the window as shown.

Press GRAPH. Press TRACE to examine the relationships between the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinates of each point. When \begin{align*}x\end{align*} is positive, what happens to \begin{align*}y\end{align*}?

When \begin{align*}x\end{align*} is negative, what happens to \begin{align*}y\end{align*}? When will \begin{align*}y\end{align*} be negative? Why? When is \begin{align*}x\end{align*} negative?

(sample response: \begin{align*}x\end{align*} is negative whenever the temperature was lower than the mean; \begin{align*}y\end{align*} will not be negative because distances are positive)

Problem 2 - Compare Data Against Equations

Press \begin{align*}Y=\end{align*}. Enter the equation \begin{align*}y = x\end{align*} into \begin{align*}Y_1\end{align*}.

Press GRAPH. What is the relationship between \begin{align*}y = x\end{align*} and the scatter plot?

Return to \begin{align*}Y=\end{align*}. Enter the equation \begin{align*}y = -x\end{align*} into \begin{align*}Y_2\end{align*}.

Press GRAPH. What is the relationship between \begin{align*}y = -x\end{align*} and the scatter plot?

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

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

Complete the function as shown. Arrow left of \begin{align*}Y_3\end{align*}. Press ENTER to change the graph to a ‘thick line.’

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

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

Extension

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

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

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

Examine the table. Is the relationship between \begin{align*}Y_2\end{align*} and \begin{align*}Y_1\end{align*} what you were expecting? Why or why not? Where are the \begin{align*}Y\end{align*} values equal to zero?

Return to \begin{align*}Y=\end{align*}. Enter the equation \begin{align*}y = abs(x + 7)\end{align*} into \begin{align*}Y_2\end{align*} as shown.

Examine the graph. What seems to be the relationship between the graphs? How is this picture different from the graph with \begin{align*}y = abs(x) + 7\end{align*}?

Examine the table. Is the relationship between \begin{align*}Y_2\end{align*} and \begin{align*}Y_1\end{align*} what you were expecting? Why or why not? Where are the \begin{align*}Y\end{align*} values equal to zero?

Compare \begin{align*}y = abs(x) + 7\end{align*} to \begin{align*}y = abs(x + 7)\end{align*}. How are they similar? Different?

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TI.MAT.ENG.SE.1.Algebra-I.7.2