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

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