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LESSON 19.2 Name Class Date Transforming 19.2 Transforming Quadratic Quadratic Functions Functions Essential Question: How can you obtain the graph of g(x) = a(x - h) + k from the graph of 2 Resource f(x) = x 2? Common Core Math Standards Locker The student is expected to: F-BF.3 Explore Understanding Quadratic Functions of the Form g(x) = a(x - h) + k 2 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the Every quadratic function can be represented by an equation of the form g(x) = a(x - h) + k. The values of the 2 value of k given the graphs. Experiment with cases and illustrate an parameters a, h, and k determine how the graph of the function compares to the graph of the parent function, y = x 2. explanation of the effects on the graph using technology. Include Use the method shown to graph g(x) = 2(x - 3) + 1 by transforming the graph of ƒ(x) = x 2. 2 recognizing even and odd functions from their graphs and algebraic expressions for them. Also F-BF.1, F-BF.4, F-IF.4, F-IF.2 Graph ƒ(x) = x 2. y  10 Mathematical Practices 8 MP.8 Patterns 6 Language Objective 4 Students work in pairs or small groups to both give and listen to oral clues 2  Stretch the graph vertically by a factor of 2 to obtain the about graphs of quadratic functions. x graph of y = 2x 2. Graph y = 2x 2. -6 -4 -2 0 2 4 6 Notice that point (2, 4) moves to point (2, 8) . -2 ENGAGE  Translate the graph of y = 2x 2 right 3 units and up 1 unit -4 to obtain the graph of g(x) = 2(x - 3) + 1. Graph g(x) = 2 Essential Question: How can you © Houghton Mifflin Harcourt Publishing Company ( ) 2 x - 3 + 1. 2 obtain the graph of g(x ) = a(x - h) + k 2 Notice that point (2, 8) moves to point (5, 9) . from the graph of f(x) = x 2? (0, 0) while the vertex of the graph of g(x) =  The vertex of the graph of ƒ(x) = x 2 is Possible answer: Identify the vertex (h, k) and use (3, 1) . 2(x - 3) + 1 is 2 the sign of a to determine whether the graph opens up or down. Generate a few points on one side of the vertex and sketch the graph using those points and symmetry. PREVIEW: LESSON PERFORMANCE TASK Module 19 903 Lesson 2 View the Engage section online. Discuss the photo EDIT--Chan ges must DO NOT Key=NL-A;CA-A Correction be made throu gh “File info” Date Class and how the path of a ball used in sports can be Name forming Qu 19.2 Transtions adratic Func the graph of g(x) = a(x - h )2 + k from the graph of Resource Locker HARDCOVER PAGES 709716 modeled by a quadratic function. Then preview the obtain can you F-BF.4, ion: How 2 so F-BF.1, Essential Quest f(x) = x ? page CA2.Al starting on see the table standards, Form text of these of the Functions the full F-BF.3 For ic F-IF.4, F-IF.2 ing Quadrat Understand- h)2 + k values of the A1_MNLESE368187_U8M19L2.indd 903 Explore (x h) + k. The 3/25/14 3:29 AM 2 y=x . 2 g(x) = a form g(x) = a(x - function, Lesson Performance Task. on of the the parent graph of Turn to these pages to by an equati to the represented n compares of ƒ(x) = x2 . n can be of the functio the graph tic functio how the graph 2 1 by transforming Every quadra k determine ) = 2(x - 3) + y a, h, and g(x 10 parameters d shown to graph Use the metho 8 x) = x . find this lesson in the 2 Graph ƒ( 6  4 2 x hardcover student 2 to obtain the 4 6 of 0 2 -2 lly by a factor -6 -4 graph vertica = 2x2 . -2 Stretch the y  y = 2x . Graph (2, 8) . 2 graph of to point -4 4) moves point (2, edition. Notice that unit units and up 1 of y = 2x 2 3 right 2 g(x) = 1. Graph the graph (x - 3) + Translate g(x) = 2  the graph of to obtain 2 2(x - 3) + 1. (5, 9) . y g Compan to point 8) moves of g(x) = point (2, of the graph Notice that (0, 0) while the vertex x2 is of ƒ(x) = Publishin of the graph The vertex  (3, 1) . Harcour t + 1 is 2(x - 3) 2 n Mifflin © Houghto Lesson 2 903 3:02 AM 3/25/14 Module 19 903 L2.indd 7_U8M19 SE36818 A1_MNLE 903 Lesson 19.2 Reflect 1. Discussion Compare the minimum values of ƒ(x) = x 2 and g(x) = 2(x - 3) 2 + 1. How is the minimum value related to the vertex? EXPLORE The minimum value of f(x) = x 2 is 0 and the minimum value g(x) = 2(x - 3) +1 is 1. The 2 Understanding Quadratic Functions of the Form g(x ) = a(x - h) + k minimum value is the y-coordinate of the vertex. 2 2. Discussion What is the axis of symmetry of the function g(x) = 2(x - 3) 2 + 1? How is the axis of symmetry related to the vertex? The axis of symmetry of g(x) = 2(x - 3) + 1 is x = 3. The axis of symmetry always passes 2 INTEGRATE TECHNOLOGY through the vertex of the parabola. The x-coordinate of the vertex gives the equation of Students have the option of completing the activity the axis of symmetry of the parabola. either in the book or online. Explain 1 Understanding Vertical Translations A vertical translation of a parabola is a shift of the parabola up or down, with no change in the shape of the parabola. CONNECT VOCABULARY Vertical Translations of a Parabola This lesson discusses translation in terms of a The graph of the function ƒ(x) = x + k is the graph of ƒ(x) = x translated vertically. 2 2 transformation of a function graph. English learners If k > 0, the graph ƒ(x) = x 2 is translated k units up. may know about language translation. Discuss with If k < 0, the graph ƒ(x) = x 2 is translated ⎜k⎟ units down. students how the two meanings of translate are different. Example 1 Graph each quadratic function. Give the minimum or maximum value and the axis of symmetry.  g(x) = x 2 + 2 Make a table of values for the parent function f(x) = x 2 10 y EXPLAIN 1 and for g(x) = x 2 + 2. Graph the functions together. 8 Understanding Vertical Translations f(x) = x 2 g(x) = x 2 + 2 6 © Houghton Mifflin Harcourt Publishing Company x -3 9 11 4 -2 4 6 f(x) = x2 QUESTIONING STRATEGIES 2 -1 1 3 x How is the graph of g(x) = x 2 + 2 related to 0 0 2 -6 -4 -2 0 2 4 6 the graph of g(x) = x 2 – 5? Both are -2 1 1 3 translated graphs of the same parent function, 2 4 6 -4 f(x) = x 2, but g(x) = x 2 + 2 is translated 2 units up 3 9 11 and g(x) = x 2 - 5 is translated 5 units down. So, the graph of g(x) = x 2 + 2 is 7 units higher than the The function g(x) = x 2 + 2 has a minimum value of 2. graph of g(x) = x 2 - 5. The axis of symmetry of g(x) = x 2 + 2 is x = 0. Is the vertex of the graph of g(x) = x 2 + 2 the same as the vertex of the graph of g(x) = x 2 – 5? No; g(x) = x 2 + 2 has vertex (0, 2), Module 19 904 Lesson 2 and g(x) = x 2 - 5 has vertex (0, -5). PROFESSIONAL DEVELOPMENT A1_MNLESE368187_U8M19L2.indd 904 26/08/14 12:57 PM Math Background In this lesson, students graph the family of quadratic functions of the form g(x) = a(x - h) + k and compare those graphs to the graph of the parent 2 function f (x) = x 2. Some key understandings are: • The function f (x) = x 2 is the parent of the family of all quadratic functions. • To graph a quadratic function of the form g(x) = a(x - h) + k, identify the 2 vertex (h, k). Then determine whether the graph opens upward or downward. Then generate points on either side of the vertex and sketch the graph of the function. Transforming Quadratic Functions 904 g(x) = x - 5 2 B y 10 Make a table of values for the parent function ƒ(x) = x 2 and for g(x) = x 2 - 5. Graph the functions together. 8 6 x f(x) = x 2 g(x) = x 2 - 5 4 -3 9 4 2 -2 4 -1 x -6 -4 -2 0 2 4 6 -1 1 -4 -2 -4 0 0 -5 -6 1 1 -4 2 4 -1 3 9 4 The function g(x) = x 2 - 5 has a minimum value of -5. The axis of symmetry of g(x) = x 2 - 5 is x = 0 . Reflect 3. How do the values in the table for g(x) = x 2 + 2 compare with the values in the table for the parent function ƒ(x) = x 2? For each x in the table, g(x) is 2 greater than f(x). 4. How do the values in the table for g(x) = x 2 - 5 compare with the values in the table for the parent function ƒ(x) = x 2? © Houghton Mifflin Harcourt Publishing Company For each x in the table, g(x) is 5 less than f(x). Your Turn Graph each quadratic function. Give the minimum or maximum value and the axis of symmetry. 5. g(x) = x 2 + 4 y The function g(x) = x 2 + 4 has a minimum value of 4. 8 The axis of symmetry for g(x) = x + 4 is x = 0. 2 6 4 2 x -4 -2 0 2 4 Module 19 905 Lesson 2 COLLABORATIVE LEARNING A1_MNLESE368187_U8M19L2 905 28/03/14 2:31 AM Peer-to-Peer Activity Have students work in pairs. Have one student draw a graph of y = x 2 + k for some value of k. The second student then writes the equation for the graph. Students then compare their results and determine whether the equation is correct for the given graph. Have students take turns in the two roles. 905 Lesson 19.2 6. g(x) = x 2 - 7 y The function g(x) = x - 7 has a minimum value of -7. 2 The axis of symmetry for g(x) = x 2 - 7 is x = 0. 2 x EXPLAIN 2 -4 -2 0 2 4 -2 Understanding Horizontal -4 Translations -6 QUESTIONING STRATEGIES How is the graph of g(x) = (x – 1) related to 2 Explain 2 Understanding Horizontal Translations the graph of g(x) = (x + 2) ? Both are 2 A horizontal translation of a parabola is a shift of the parabola left or right, with no change in the shape of the translated graphs of the same parent function, parabola. f(x) = x 2 , but the graph of g(x) = (x - 1) is 2 Horizontal Translations of a Parabola translated 1 unit to the right and has vertex (1, 0), The graph of the function ƒ(x) = (x – h) is the graph of ƒ(x) = x 2 translated horizontally. while the graph of g(x) = (x + 2) is translated 2 2 If h > 0, the graph ƒ(x) = x 2 is translated h units right. 2 units to the left and has vertex (-2, 0). So, the If h < 0, the graph ƒ(x) = x 2 is translated ⎜h⎟ units left. graph of g(x) = (x - 1) is 3 units to the right of the 2 graph of g(x) = (x + 2) . 2 Example 2 Graph each quadratic function. Give the minimum or maximum value and the axis of symmetry. What is the vertex of the graph of g(x) = (x – h) ? (h, 0) 2 g(x) = (x - 1) 2  10 y Make a table of values for the parent function ƒ(x) = x 2 and for g(x) = (x - 1) . Graph the functions together. 8 2 6 f(x) = x 2 g(x) = (x -1) 2 x -3 © Houghton Mifflin Harcourt Publishing Company 9 16 4 -2 4 9 2 -1 1 4 x -6 -4 -2 0 2 4 6 0 0 1 -2 1 1 0 2 4 1 -4 3 9 4 The function g(x) = (x - 1) has a minimum value of 0. 2 The axis of symmetry of g(x) = (x - 1) is x = 1. 2 Module 19 906 Lesson 2 DIFFERENTIATE INSTRUCTION A1_MNLESE368187_U8M19L2.indd 906 26/08/14 12:57 PM Visual Cues Have students take a coordinate grid and label it “Vertex of g(x) = (x – h) + k.” 2 Have them place these points, labels, and functions into the four quadrants. (2, 3) h = 2, k = 3 g(x) = (x - 2) + 3 2 (-2, 3) h = -2, k = 3 g(x) = (x + 2) + 3 2 (–2, –3) h = –2, k = –3 g(x) = (x + 2) – 3 2 (2, –3) h = 2, k = –3 g(x) = (x – 2) – 3 2 Students can use this graph as a reminder of how the location of the vertex and the function are related. Transforming Quadratic Functions 906 g(x) = (x +1) 2 B 10 y AVOID COMMON ERRORS Make a table of values and graph the functions together. 8 Students may forget that they can use a pattern to f(x) = x 2 g(x) = (x + 1) 2 x write equations from graphs. Remind students that 6 -3 9 4 adding k to x 2 moves the graph up for k > 0 or down 4 for k < 0 and that subtracting h from x moves the -2 4 1 2 graph left for h < 0 or right for h > 0. This is true for -1 1 0 x -6 -4 -2 0 all nonzero values of k and h. 2 4 6 0 0 1 -2 1 1 4 -4 2 4 9 3 9 16 The function g(x) = (x +1) has a minimum value of 0 . 2 The axis of symmetry of g(x) = (x + 1) is x = -1 . 2 Reflect How do the values in the table for g(x) = (x - 1) compare with the values in the table for the parent 2 7. function ƒ(x) = x 2? For each x in the table, g(x) is the same as f(x - 1). How do the values in the table for g(x) = (x + 1) compare with the values in the table for the parent 2 8. function ƒ(x) = x 2? For each x in the table, g(x) is the same as f(x + 1). Your Turn © Houghton Mifflin Harcourt Publishing Company Graph each quadratic function. Give the minimum or maximum value and the axis of symmetry. g(x) = (x - 2) 10. g(x) = (x + 3) 2 2 9. Minimum: 0; axis of symmetry: x = 2 Minimum: 0; axis of symmetry: x = -3 y y 16 16 12 12 8 8 4 4 x x -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 Module 19 907 Lesson 2 LANGUAGE SUPPORT A1_MNLESE368187_U8M19L2 907 8/27/14 11:43 AM Communicate Math Have each student sketch a graph of a parabola on a card, and write a quadratic function in any form on another card. Then have them write clues about the graph and about the function. For example, “The parabola opens upward/downward. Its axis of symmetry is _____; the vertex is at the point _____. The function’s graph will open downward/upward.” Provide sentence stems if needed to help students begin their clues. Collect the graph and function cards in one pile, and the clue cards in another. Have other students match graph and function cards to fit the clues. 907 Lesson 19.2 Graphing g(x) = a(x - h) + k 2 Explain 3 The vertex form of a quadratic function is g(x) = a(x - h) + k, where the point (h, k) is the vertex. The axis of 2 symmetry of a quadratic function in this form is the vertical line x = h. EXPLAIN 3 To graph a quadratic function in the form g(x) = a(x - h) + k, first identify the vertex ( h, k ). Next, consider the Graphing g(x) = a(x - h) + k 2 2 sign of a to determine whether the graph opens upward or downward. If a is positive, the graph opens upward. If a is negative, the graph opens downward. Then generate two points on each side of the vertex. Using those points, sketch the graph of the function. Example 3 Graph each quadratic function. QUESTIONING STRATEGIES What can you tell about the graph of a g(x) = -3(x + 1) - 2 y x 2  0 function from its equation in the form -6 -4 -2 2 4 6 Identify the vertex. g(x) = a(x - h) + k? the location of the vertex 2 -2 The vertex is at (-1, -2). -4 and whether the graph opens upward or downward Make a table for the function. Find two points on each side of the vertex. -6 What are the domain and range for a -8 quadratic function whose graph opens x -3 -2 -1 0 1 -10 downward? The domain is all real numbers, and the g(x) -14 -5 -2 -5 -14 range is the set of all values less than or equal to the -12 Plot the points and draw a parabola through them. maximum value. -14 -16 AVOID COMMON ERRORS  g(x) = 2(x - 1) - 7 2 y Students may try to graph a quadratic function of the form g(x) = a(x - h) + k by using a value other 2 Identify the vertex. 15 The vertex is at (1, -7) . 12 than x = h. Remind them that they need to first identify and plot the vertex. Then they should Make a table for the function. Find two points on each side of the vertex. 9 © Houghton Mifflin Harcourt Publishing Company identify and plot other points and use the plotted 6 x -2 0 1 2 4 points to draw a parabola. 3 g(x) 11 -5 -7 -5 11 x 0 -3 3 INTEGRATE MATHEMATICAL Plot the points and draw a parabola through them. -3 PRACTICES -6 Focus on Modeling Reflect MP.4 Tell students that a transformed quadratic 11. How do you tell from the equation whether the vertex is a maximum value or a minimum value? function models the height of an object dropped If the value of a is positive, the vertex is a minimum value. If the value of a is negative, the from a given height, based upon the time since it was vertex is a maximum value. dropped. Sketch a quadratic function that models the situation, and draw students’ attention to the vertex Module 19 908 Lesson 2 (0, k) being the maximum point of the graph. Ask about the sign of a in the function g(x) = ax 2 + k, and note that the values to the left of the y-axis are not A1_MNLESE368187_U8M19L2.indd 908 26/08/14 11:56 PM considered. Transforming Quadratic Functions 908 Your Turn ELABORATE Graph each quadratic function. 12. g(x) = -(x - 2) + 4 13. g(x) = 2(x + 3) - 1 2 2 INTEGRATE MATHEMATICAL x -1 0 2 4 5 x -5 -4 -3 -2 -1 PRACTICES Focus on Technology g(x) -5 0 4 0 -5 g(x) 7 1 -1 1 7 MP.5 Give students a function in the form g(x) = a(x – h) + k. Have students use the 2 y y 6 8 whiteboard to identify and plot the vertex and then 4 6 identify and plot other points on the graph before 2 4 drawing the graph of the function. x 2 -6 -4 -2 0 2 4 6 x -2 0 -6 -4 -2 2 4 6 SUMMARIZE -4 -2 How do you graph a quadratic function of the -6 -4 form g(x ) = a(x – h) + k? First, identify and 2 plot the vertex. Then, identify and plot other points on the graph. Finally, draw the graph. Elaborate 14. How does the value of k in g(x) = x 2 + k affect the translation of ƒ(x) = x 2? If k > 0, the graph f(x) = x 2 is translated k units up. If k < 0, the graph f(x) = x 2 is translated k units down. 15. How does the value of h in g(x) = (x - h) affect the translation of ƒ(x) = x 2? 2 If h > 0, the graph f(x) = x 2 is translated h units right. © Houghton Mifflin Harcourt Publishing Company If h < 0, the graph f(x) = x 2 is translated h units left. 16. In g(x) = a(x - h) + k, what are the coordinates of the vertex? 2 (h, k) 17. Essential Question Check-In How can you use the values of a, h, and k, to obtain the graph of g(x) = a(x - h) + k from the graph ƒ(x) = x 2? 2 The graph of f(x) = x 2 is stretched or compressed by a factor of ⎜a⎟, and reflected across the x-axis if a is negative; it is translated h units horizontally and k units vertically. Module 19 909 Lesson 2 A1_MNLESE368187_U8M19L2.indd 909 3/25/14 3:29 AM 909 Lesson 19.2 Evaluate: Homework and Practice EVALUATE • Online Homework Graph each quadratic function by transforming the graph of ƒ(x) = x 2. Describe • Hints and Help the transformations. • Extra Practice g(x) = 2(x - 2) + 5 g(x) = 2(x + 3) - 6 2 2 1. 2. y y 16 8 14 6 ASSIGNMENT GUIDE 12 4 Concepts and Skills Practice 10 2 x Explore Exercises 1–4 8 -8 -6 -4 -2 0 2 4 6 Understanding Quadratic Functions of the Form g(x ) = a(x - h) + k 2 6 -2 4 -4 Example 1 Exercises 5–10 Understanding Vertical Translations 2 -6 x Example 2 Exercises 11–16 -8 -6 -4 -2 0 2 4 6 Understanding Horizontal Translations The parent function has been translated The parent function has been translated 3 units left and 6 units down. It has been Example 3 Exercises 17–27 2 units right and 5 units up. It has been Graphing g(x) = a(x - h) + k 2 stretched vertically by a factor of 2. stretched vertically by a factor of 2. 3. 1 (x - 3) - 4 g(x) = _ 2 4. g(x) = 3(x - 4) - 2 2 2 y y INTEGRATE MATHEMATICAL 8 10 PRACTICES © Houghton Mifflin Harcourt Publishing Company 6 8 Focus on Modeling 4 6 MP.4 Make sure that students understand where h, 2 4 k, and a come from. Give coordinates for a vertex and x 0 2 have students substitute the x- and y-values of the -8 -6 -4 -2 2 4 6 8 -2 x vertex into the equation of h and k, determine the -6 -4 -2 0 2 4 6 8 -4 -2 value of a, and then write the equation of the function. -6 -4 -8 The parent function has been translated The parent function has been translated 4 units right and 2 units down. It has been 3 units right and 4 units down. It has been stretched vertically by a factor of 3. vertically compressed by a factor of _1 2 . Module 19 910 Lesson 2 Exercise A1_MNLESE368187_U8M19L2.indd 910 Depth of Knowledge (D.O.K.) Mathematical Practices 26/08/14 12:57 PM 1–8 1 Recall of Information MP.6 Precision 9–12 2 Skills/Concepts MP.5 Using Tools 13–16 2 Skills/Concepts MP.2 Reasoning 17–20 2 Skills/Concepts MP.6 Precision 21–22 2 Skills/Concepts MP.2 Reasoning 23–25 3 Strategic Thinking MP.3 Logic 26–27 3 Strategic Thinking MP.2 Reasoning Transforming Quadratic Functions 910 Graph each quadratic function. VISUAL CUES 5. g(x) = x 2 - 2 6. g(x) = x 2 + 5 Have students create a design made of transformed y y parabolas and keep a record of each parabola’s 6 8 function. Encourage students to use their 4 6 imaginations to add colors and patterns to the design. 2 4 x 2 -4 -2 0 2 4 x INTEGRATE MATHEMATICAL -2 -4 -2 0 2 4 PRACTICES Focus on Technology 7. g(x) = x 2 - 6 8. g(x) = x 2 + 3 MP.5 Allow students to use graphing y y calculators to check their work. Some students 2 x 8 will be motivated to explore additional types of 0 -4 -2 2 4 6 transformations. -2 4 -4 2 -6 x -4 -2 0 -8 2 4 9. Graph g(x) = x 2 - 9. Give the minimum or maximum value and the axis of symmetry. y The function has a minimum value of -9. x The axis of symmetry is x = 0. 0 -4 -2 © Houghton Mifflin Harcourt Publishing Company 2 4 -2 -4 -6 -8 10. How is the graph of g(x) = x 2 + 12 related to the graph of ƒ(x) = x 2? The graph of g(x) = x 2 + 12 is the graph of f(x) = x 2 translated 12 units up. Module 19 911 Lesson 2 A1_MNLESE368187_U8M19L2.indd 911 3/25/14 3:23 AM 911 Lesson 19.2 Graph each quadratic function. Give the minimum or maximum value and the axis of symmetry. AVOID COMMON ERRORS 11. g(x) = (x - 3) 2 Some students may automatically say that the y function has a minimum when a parabola opens 36 downward, and a maximum when a parabola opens 30 upward, because of word association. Tell students to 24 visualize the graph before determining whether it has 18 a minimum or maximum. 12 6 x -8 -6 -4 -2 0 2 4 6 8 10 The function has a minimum value of 0. The axis of symmetry is x = 3. 12. g(x) = (x + 2) 2 y 28 24 20 16 12 © Houghton Mifflin Harcourt Publishing Company 8 4 x -8 -6 -4 -2 0 2 4 6 8 The function has a minimum value of 0. The axis of symmetry is x = -2. 13. How is the graph of g(x) = (x + 12) related to the graph of ƒ(x) = x 2? 2 The graph of g(x) = (x + 12) is the graph of f(x) = x 2 translated 12 units left. 2 14. How is the graph of g(x) = (x - 10) related to the graph of ƒ(x) = x 2? 2 The graph of g(x) = (x - 10) is the graph of f(x) = x 2 translated 10 units right. 2 Module 19 912 Lesson 2 A1_MNLESE368187_U8M19L2.indd 912 3/25/14 3:23 AM Transforming Quadratic Functions 912 15. Compare the given graph to the graph of the parent y KINESTHETIC EXPERIENCE function ƒ(x) = x 2. Describe how the parent function 8 must be translated to get the graph shown here. Display each function below, one at a time. Have 6 students discuss, in pairs, whether to lift their arms Translate the graph of the parent function 2 units 4 to the right. up in the shape of a U to signal the graph opens 2 upward, or move them downward in the shape of an x upside-down U, to signal that the graph opens -4 -2 0 2 4 6 -2 downward. Then have students demonstrate their decisions. -4 y = -3x 2 + 18 down 16. For the function g(x) = (x - 9) give the minimum or maximum value and the axis of symmetry. 2 y = 5x + 8 - __15 x 2 down The minimum value is 0. The axis of symmetry is x = 9. -2x 2 +y = -5 up Graph each quadratic function. Give the minimum or maximum value and the axis of 3x - y = -x 2 up symmetry. 17. g(x) = (x - 1) - 5 18. g(x) = -(x + 2) + 5 2 2 y y 6 6 4 4 2 2 x x -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -2 -2 -4 -4 © Houghton Mifflin Harcourt Publishing Company -6 -6 x -2 0 1 3 4 x -5 -3 -2 0 1 g(x) 4 -4 -5 -1 4 g(x) -4 4 5 1 -4 The function has a minimum value of -5. The function has a maximum value of 5. The axis of symmetry is x = 1. The axis of symmetry is x = -2. Module 19 913 Lesson 2 A1_MNLESE368187_U8M19L2.indd 913 26/08/14 12:57 PM 913 Lesson 19.2 1 (x + 1) - 7 19. g(x) = _ 2 1 (x + 3) 2 + 8 20. g(x) = -_ 4 3 PEERTOPEER ACTIVITY y y 6 10 Have students work in pairs. Have each student 4 8 change one or more of the parameters in f(x) = a(x – h) + k and graph the function. Then 2 2 6 x have students trade graphs and try to write the 4 -8 -6 -4 -2 0 2 4 6 function equation for the other student’s graph. Have -2 2 each student justify the function equation and discuss x -4 it with the partner. -12 -10 -8 -6 -4 -2 0 2 4 6 -6 -2 -8 -4 -6 x -8 -5 -1 3 5 x -7 -6 -3 0 3 g(x) 5.25 -3 -7 -3 2 g(x) 2.67 5 8 5 2.67 The function has a minimum value of -7. The function has a maximum value of 8. The axis of symmetry is x = -1. The axis of symmetry is x = -3. 21. Compare the given graph to the graph of the parent function ƒ(x) = x 2. y Describe how the parent function must be translated to get the graph 8 shown here. 6 Translate the graph of the parent function 3 units to the 4 right and 2 units up. 2 x © Houghton Mifflin Harcourt Publishing Company -2 0 2 4 6 -2 22. Multiple Representations Select the equation for the function represented by the graph of a parabola that is a translation of ƒ(x) = x 2. The graph has been translated 11 units to the left and 5 units down. a. g(x) = (x - 11) - 5 2 b. g(x) = (x + 11) - 5 2 c. g(x) = (x + 11) + 5 2 d. g(x) = (x - 11) + 5 2 e. g(x) = (x - 5) - 11 2 f. g(x) = (x - 5) + 11 2 g. g(x) = (x + 5) - 11 2 h. g(x) = (x + 5) + 11 2 Module 19 914 Lesson 2 A1_MNLESE368187_U8M19L2.indd 914 3/11/16 8:31 PM Transforming Quadratic Functions 914 JOURNAL H.O.T. Focus on Higher Order Thinking Critical Thinking Use a graphing calculator to compare the graphs of y = (2x) 2, In their journals, have students explain how to use y = (3x) , and y = (4x) with the graph of the parent function y = x2. Then compare 2 2 ( ) ( ) ( ) 2 2 2 the values of a, h, and k to obtain the graph of the graphs of y = __12 x , y = __13 x , and y = __14 x with the graph of the parent g(x) = a(x – h) + k from the graph of f(x) = x 2. 2 function y = x . 2 23. Explain how the parameter b horizontally stretches or compresses the graph of y = (bx) when b > 1. 2 When b > 1, the graph of y = (bx) is compressed horizontally by a factor 2 1 of __. b 24. Explain how the parameter b horizontally stretches or compresses the graph of y = (bx) when 0 < b < 1. 2 When 0 < b < 1, the graph of y = (bx) is stretched horizontally by a 2 1 factor of __. b 25. Explain the Error Nina is trying to write an equation for the function represented by the graph of a parabola that is a translation of ƒ(x) = x 2. The graph has been translated 4 units to the right and 2 units up. She writes the function as g(x) = (x + 4) + 2. Explain the error. 2 Nina should have subtracted 4 from x in the equation instead of adding it. 26. Multiple Representations A group of engineers drop an experimental tennis ball from a catwalk and let it fall to the ground. The tennis ball’s height above the ground (in feet) is given by a function of the form ƒ(t) = a(t - h) + k where t is the time (in 2 seconds) after the tennis ball was dropped. Use the graph to find the equation for ƒ(t). y 32 (0, 30) 24 16 (1, 14) © Houghton Mifflin Harcourt Publishing Company 8 t 0 1 2 3 4 f(t) = a(t - 0) + 30, or f(t) = at 2 + 30 2 14 = a(1) + 30 2 -16 = a The equation for the function is f(t) = -16t 2 + 30. 27. Make a Prediction For what values of a and c will the graph of ƒ(x) = ax 2 + c have one x-intercept? For any real value of a with a ≠ 0, the function will have one x-intercept when c = 0. Module 19 915 Lesson 2 A1_MNLESE368187_U8M19L2.indd 915 27/08/14 12:02 AM 915 Lesson 19.2 Lesson Performance Task INTEGRATE MATHEMATICAL PRACTICES The path a baseball takes after it has been hit is modeled by the graph. The baseball’s height above the ground is given by a function of the form Baseball’s Height Focus on Patterns y ƒ(t) = a(t - h) + k, where t is the time in seconds since the baseball 2 192 MP.8 Discuss with students which type of hit—a was hit. ground ball, a pop-up, or a line drive—would likely Height (ft) a. What is the baseball’s maximum height? At what time was 144 the baseball at its maximum height? make a baseball have the path shown in the graph. 96 Ask how knowing the maximum height of the ball b. When does the baseball hit the ground? 48 and the time it takes the ball to hit the ground help c. Find an equation for ƒ(t). x you write an equation to represent the path of the 0 2 4 6 8 d. A player hits a second baseball. The second baseball’s path is Time (s) baseball. modeled by the function g(t) = -16(t - 4) + 256. Which 2 baseball has a greater maximum height? Which baseball is The highest height of the ball, the vertex of the in the air for the longest? path, is the ordered pair (h, k), and the time it takes a. The vertex of the parabola is (3, 144). So, the baseball is at its the ball to hit the ground is the ordered pair (t, f(t)), maximum height of 144 feet after 3 seconds. so the h, k, t, and f(t) values can be substituted into b. The second x-intercept of the graph is (6, 0). So, the baseball hits the the standard form f(t) = a(t - h) + k to find the 2 ground after 6 seconds. c. The vertex of the parabola is (3, 144) and one intercept of the graph is value of a. (6, 0). Solve for a. f(t) = a(t - h) + k 2 0 = a(6 - 3) + 144 2 INTEGRATE MATHEMATICAL -144 = 9a PRACTICES -16 = a Focus on Critical Thinking So, f(t) = -16(t - 3) + 144. 2 MP.3 Point out that the model students write in the d. The vertex is (4,256) so the baseball was at its maximum height of form f(t) = a(t - h) + k to represent the path the 2 256 feet after 4 seconds. © Houghton Mifflin Harcourt Publishing Company g(t) = -16(t-4) + 256 2 baseball takes can be used to approximate the height 0 = -16(t-4) + 256 2 h in feet above the ground after t seconds because it -256 = -16(t-4) 2 does not account for air resistance, wind, or other 16 = (t-4) 2 √― real-world factors. 16 = t-4 ±4 + 4 = t 0, 8 = t The ball hits the ground after 8 seconds. So, the second baseball has a greater maximum height and it traveled longer in the air than the first. Module 19 916 Lesson 2 EXTENSION ACTIVITY A1_MNLESE368187_U8M19L2.indd 916 3/25/14 3:22 AM Have groups of students draw or tape a large, first-quadrant coordinate grid on the chalkboard. Have one student toss a tennis ball in front of the grid, making sure that the path of the ball stays within the grid’s borders, while another student videotapes the toss at a rate of about 15 frames per second. Then have students play back the video, marking points on the grid to show the path of the ball. Finally, have students use the model f(t) = a(t - h) + k to write a function 2 that models the path of the tennis ball. Students may discover that the angle at Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. which the ball is tossed affects the height and width of the curved path the ball 1 point: Student shows good understanding of the problem but does not fully follows. Have students save their data for Part 2 of the Extension Activity in the solve or explain his/her reasoning. following lesson. 0 points: Student does not demonstrate understanding of the problem. Transforming Quadratic Functions 916