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Defining the Absolute Value Function • To investigate the concept of absolute value • To construct and interpret graphs of absolute-value functions. • To learn the piecewise definition of the absolute value function • To evaluate expressions containing absolute value. LESSON 7.5 Cal and Al both live 3.2 miles from school, but in opposite directions. If you assign the number 0 to the school, you can show that Cal and Al live in opposite directions from it by assigning +3.2 to AI's house and -3.2 to Cal's house. For both Cal and Al, the distance from school is 3.2 miles. Distance is never negative, but distance is often found by subtracting, which sometimes gives a negative result. For this reason, it is useful to have a function that turns opposite numbers into the same positive number. The absolute value of a number is its size, or magnitude, regardless of whether the number is positive or negative. A way to visualize the absolute value of a number is to picture its distance zero on a number line. A number and its opposite have the same magnitude, or absolute value, because they're the same distance from zero. For example, 3.2 and -3.2 are both 3.2 units from zero, so they both have an absolute value |3.2|=3.2 and |-3.2|=3.2 . Evaluate each expression. 5 5 2 17 3 6 6 8 4 4 0 Absolute value is useful for answering questions about distance, pulse rates, scores, and other data values that lie on opposite sides of a central point 5 mean. The difference of a data point from the mean of its data set is calla: deviation from the mean. In this investigation you will learn how the absolute-value function tells how much an item of data or a whole set of data deviates from the mean. Collect at least 10 pulse rates from your class. Record the data in a table and enter the numbers into list L1 on your calculator. Calculate the mean of the pulse rate using the graphing calculator. Find the difference between each data point and the mean of the data in list LI. Use L2 to find the difference between the values in L1 and the mean of the data. Record these numbers in a second row of your table. What do these numbers represent? Make a dot plot of the list L1 data and note the distance from each data point to the mean. Record your results in a third row of your table and enter them into list L3. How are these entries different from those in list L2? How are they alike? Next, plot points in the form (L2, L3). What numbers are in the domain and range of the graph? Use the trace function on your calculator and use the arrow keys to step through the data points. Which input numbers are unchanged as output numbers? Which input numbers are changed, and how? Does it make sense to connect these points with a continuous graph? Why or why not? How does this graph compare to the graph of Y1 = abs(x) on your calculator? Find the mean of the deviations stored in list L2. Compare it to the mean of the distances stored in list L3. Which do you think is a better measure of the spread of the data? In your own words, write the rule for the function you graphed in Step 8. What number is output as y when the input, x, is positive or equal to zero? What number is output when x is negative? How can you use operations to change these numbers? The absolute-value function is defined by two rules. if x 0 x x x if x 0 Solve each equation or inequality symbolically. x 7 12 x 2 7 12 x 2 7 12