1. Introduction Absolute value functions can be used to model situations in which there is a peak in a phenomenon. One such example is a movie orchestra instructed to play at a higher sound level for a number of measures, then to play at a lower level, only to return back to the higher sound level. This represents a peak in the sound level. Sound engineers might graph the absolute value function for this situation in order to observe the sound levels at each measure and make adjustments quickly without having to perform cumbersome calculations. In this lesson, you will create and use various absolute value functions and their graphs to solve a variety of real-world problems. 1 6.2.3: Absolute Value Functions

2. Key Concepts An absolute value function is a function of the form where x is the independent variable and a, h, and k are real numbers. An absolute value function is a special type of piecewise function because the function values on either side of its maximum or minimum point are equal; for example, if, then f(3) = f(9), since and. In general terms, the maximum or minimum function value occurs at the point (h, k). 2 6.2.3: Absolute Value Functions

3. Key Concepts, continued For an absolute value function of the form, there is one extreme value for the range at y = k. The graph of the basic absolute value function, is comprised of lines that “open upward” with a right angle at the vertex, which is at the origin of (0, 0); the function also has symmetry about the y-axis. 3 6.2.3: Absolute Value Functions

4. Key Concepts, continued The graph of the basic negative absolute value function, is comprised of lines that “open downward” with a right angle at the vertex (the origin), and also has symmetry about the y-axis. 4 6.2.3: Absolute Value Functions

5. Key Concepts, continued 5 6.2.3: Absolute Value Functions

6. Key Concepts, continued The domain of an absolute value function is all real numbers, while the range is the set of non-negative real numbers: [0, ∞). Note that in the positive absolute value function,, the graph increases from y = 0 as the x-values approach negative infinity. That is, as the x-values decrease, the y-values increase. Additionally, increases from y = 0 as x approaches positive infinity—as the x-values increase, the y-values also increase. 6 6.2.3: Absolute Value Functions

7. Key Concepts, continued However, in the negative absolute value function,, the graph decreases from y = 0 as the x-values approach negative infinity; as the x-values decrease, the y-values also decrease. Additionally, decreases from y = 0 as x approaches positive infinity—as the x-values increase, the y-values decrease. If a in the general form is positive, the extreme value is the minimum of the range, and there is no maximum of the range. 7 6.2.3: Absolute Value Functions

8. Key Concepts, continued If a in the general form is negative, the extreme value is the maximum of the range, and there is no minimum of the range. Graphing calculators can be an efficient way to graph absolute value functions. Follow the directions specific to your model. 8 6.2.3: Absolute Value Functions

9. Key Concepts, continued On a TI-83/84: Step 1: Press [Y=]. At [Y1], press [2ND][0] to bring up the Catalog menu. Select “abs(”. Enter the expression contained within the absolute value bars, using [X, T, θ, n] for x. Enter the remaining part of the expression by moving the cursor to the left and/or right of the absolute value bars. Enter additional absolute value functions on separate lines. Step 2: Press [GRAPH]. Adjust the viewing window as needed. 9 6.2.3: Absolute Value Functions

10. Key Concepts, continued On a TI-Nspire: Step 1: Press the [home] key. Arrow down and over to the graphing icon and press [enter]. Step 2: Enter the expression at the bottom of the application, using x as the variable. Type “abs(x)” to enter an absolute value; for example, typing “abs(x+4)–5” is equivalent to the expression. Press [enter]. Adjust the viewing window as needed. 10 6.2.3: Absolute Value Functions

11. Common Errors/Misconceptions misinterpreting variables from a word problem not realizing that there are two equations for an absolute value function treating absolute value symbols like parentheses when solving for zeros 11 6.2.3: Absolute Value Functions

12. Guided Practice Example 2 Write the absolute value function represented in the graph. 12 6.2.3: Absolute Value Functions

13. Guided Practice: Example 2, continued 1.Determine the values of h and k from the graph. The standard form of an absolute value function is, where (h, k) is the vertex. Since (2.5, 3.5) is where the two parts of the graph meet, we know that this is the vertex; therefore, h = 2.5 and k = 3.5. 13 6.2.3: Absolute Value Functions

14. Guided Practice: Example 2, continued 2.Determine the value of a. Choose one of the given points to substitute into the general absolute value function, along with the values of h and k, to find a. A convenient point to use is (0, 0). 14 6.2.3: Absolute Value Functions Standard form for an absolute value function Substitute 2.5 for h and 3.5 for k. Substitute (0, 0) into the equation.

15. Guided Practice: Example 2, continued The value of a is –1.4. A negative value for a means the graph opens downward, which corresponds with the provided graph. 15 6.2.3: Absolute Value Functions Simplify. 0 = 2.5a + 3.5 Apply the absolute value. –3.5 = 2.5a Subtract 3.5 from both sides. a = –1.4 Divide and apply the Symmetric Property of Equality.

16. Guided Practice: Example 2, continued 3.Use the values for a, h, and k to write the equation of the absolute value function. The standard form of an absolute value function is. Substitute the values a = –1.4, h = 2.5, and k = 3.5 into this form. The standard form of the absolute value function shown in the graph is. 16 6.2.3: Absolute Value Functions ✔

17. Guided Practice: Example 2, continued 17 6.2.3: Absolute Value Functions

18. Guided Practice Example 4 Graph,, and using a graphing calculator. Compare the domains, intercepts, vertices, and symmetry. 18 6.2.3: Absolute Value Functions

19. Guided Practice: Example 4, continued 1.Use a graphing calculator to graph each function. Follow the directions appropriate to your model. The graphed functions should resemble the graph on the following slide. 19 6.2.3: Absolute Value Functions

20. Guided Practice: Example 4, continued 20 6.2.3: Absolute Value Functions

21. Guided Practice: Example 4, continued 2.Compare the domains of the functions. Look at the values of x for which the functions are defined. For each of the graphs, the lines are continuous as the values of x go from negative infinity to positive infinity. Therefore, the domain for all three graphs is all real numbers. 21 6.2.3: Absolute Value Functions

22. Guided Practice: Example 4, continued 3.Compare the x- and y-intercepts of each function. and both have x- and y-intercepts at (0, 0). has a y-intercept at (0, 5), but no x-intercept. 22 6.2.3: Absolute Value Functions

23. Guided Practice: Example 4, continued 4.Compare the vertices of each function. For these graphs, the vertex will be the lowest point on the absolute value function. and each have a vertex at (0, 0); has a vertex at (0, 5). 23 6.2.3: Absolute Value Functions

24. Guided Practice: Example 4, continued 5.Compare the lines of symmetry of each function. A line that goes through the vertex point of the absolute value graph is a line of symmetry. The line x = 0 (the y-axis) is the line of symmetry for all three functions. 24 6.2.3: Absolute Value Functions ✔

25. Guided Practice: Example 4, continued 25 6.2.3: Absolute Value Functions