Graphing Absolute Value Functions. Objectives To graph an absolute value function To translate the graph of an absolute value function
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1 5-8 CC-0 CC-6 Graphing Absolute Value Functions Content Standards F.BF.3 Identif the effect on the graph of replacing f () b f () k, kf (), f (k), and f ( k) for specific values of k (both positive and negative); find the value of k given the graphs... Also F.IF.7.b Objectives To graph an absolute value To translate the graph of an absolute value Line 1 O 3 Line T C A DYNAMIC I V I E S I T Dnamic Activit Absolute Value with Linear Equations In the Solve It ou described how one line could be shifted to result in a second line. You can use a similar method to graph absolute value s. An absolute value has a V-shaped graph that opens up or down. The parent for the famil of absolute value s is. A translation is a shift of a graph horizontall, verticall, or both. The result is a graph of the same size and shape, but in a different position. Lesson Vocabular absolute value piecewise step translation Essential Understanding You can quickl graph absolute value equations b shifting the graph of. Problem 1 Describing Translations Below are the graphs of and. How are the graphs related? How can ou compare the graphs? Look for the characteristics that ou ve studied with other graphs, such as shape, size, or individual points. uu O uu 3 O 3 3 The graphs have the same shape. Notice each point on is units lower than the corresponding point on. The graph of is the graph of translated down units. 1 Common Core
2 1. a. How is the graph at the right related to the graph of? b. Reasoning What are the domain and range of each in Problem 1? O The graph of k is a translation of. Let k be a positive number. Then k translates the graph of up k units, while k translates the graph of down k units. Problem What is the graph of? Graphing a Vertical Translation absolute value The graph of the Identif the direction and amount of the translation. Translate the -intercept point and one point on each side of it. Draw the graph. Wh start with the graph of? Since is the parent of, ou can start with the graph of and shift it up. Start with the graph of. 6 O Draw the graph of b translating the graph of up units.. What is the graph of 7? The graphs below show what happens when ou graph and. 3 O O For a positive number h, h translates the graph of left h units, and h translates the graph of right h units. CC-6 Graphing Absolute Value Functions
3 How can ou check that the graph is correct? You can use the equation to check that points on the graph are solutions. Problem 3 Graphing a Horizontal Translation What is the graph of 5? Draw the graph of 5 b translating left 5 units O 6 3. What is the graph of 5? The absolute value is an eample of a piecewise. A piecewise is a that has different rules for different parts of its domain. For eample, when 0,. When 0,. Another eample of a piecewise is a step. A step is a that pairs ever number in an interval with a single value. The graph of a step can look like the steps of a staircase. Each piece of the graph is a horizontal segment that is missing its right endpoint, indicated b an open circle. O Problem Graphing a Step Function Transportation A school will charter buses so that the student bod can attend a football game. Each bus holds a maimum of 60 students. Make a graph that models the relationship between the number of students that go to the game b bus and the number of buses that are needed. You will need 0 buses for 0 students. As the number of students increases, the number of buses goes up b 1 ever time the number of students eceeds a multiple of 60. Draw a closed circle when the endpoints are part of the graph, and then draw an open point when the are not.. Make a graph that models the relationship between the number of students that go to the game b bus and the number of buses that are needed if each bus holds a maimum of 50 students. Number of Buses Number of Students Lesson Check Do ou know HOW? 1. How is the graph of 8 different from the graph of? How is it the same?. What is the equation for the translation of 9 units up? 3. What is the graph of 7? Do ou UNDERSTAND?. Compare and Contrast How are the graphs of and the same? How are the different? 5. Error Analsis A student is graphing the equation 10 and translates the graph of 10 units left. Describe the student s error. 3 Common Core
4 Practice and Problem-Solving Eercises A Practice Describe how each graph is related to the graph of. See Problem O O 1 O 5 Graph each b translating. See Problem Graph each b translating. See Problem Postage The table lists postage for letters weighing as much as 3 oz. You want to mail a letter that weighs.7 oz. Graph the step. How much will ou pa in postage? First-Class Postage See Problem. Weight 0 Weight 1 oz 1 oz Weight oz oz Weight 3 oz Price $. $.61 $.78 B Appl At the right is the graph of. Graph each b translating Write an equation for each translation of. O 6. units up 7..5 units left units down 9. units right 30. Writing Eplain how the relationship between and k is similar to the relationship between m and m b. CC-6 Graphing Absolute Value Functions
5 C Challenge 31. Think About a Plan What point(s) do the graphs of and have in common? How are these graphs related? Could a graph or a table help ou solve this problem? 3. What point(s) do the graphs of 7 and 3 have in common? Graph each translation of. Describe how the graph is related to the graph of a. Graph 3. b. The verte of an absolute value is the point at which its graph changes direction. What is the verte of the graph of 3? c. Reasoning What relationship do ou see between the verte and the equation? What is the verte of the graph of h k? 36. a. Graph b making a table of values. b. Translate to graph 3. c. Translate to graph ( 1). d. Translate to graph ( 1) Graph 7. SAT/ACT Standardized Test Prep 38. For f () 5 7, what value of gives f () 3? 39. What is the slope of the line at the right? 0. What is the value of f () 6 when 3? 1. What is the -intercept of the line? O Mied Review The data below follow a linear model. Write an equation of a trend line or use a graphing calculator to find an equation of the line of best fit. See Lesson Year 1 3 Ounces Price $5.30 $5.57 $5.8 $6.05 Calories Get Read! To prepare for Lesson 6-1, do Eercises 7. Graph each equation See Lesson Common Core
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