Path of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2.

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1 - Quadratic Functions and Transformations Content Standards F.BF. Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative) find the value of k given the graphs. Also A.CED., F.IF., F.IF. bjective To identif and graph quadratic functions Analze the path of the horse s jump. What additional information does it give ou? MATHEMATICAL PRACTICES In the computer game, Steeplechase, ou press the jump button and the horse makes the jump shown. The highest part of the jump must be directl above the fence or ou lose time. Where should this horse be when ou press jump? Eplain our reasoning Path of the Horse s Jump AMIC V I E S I T Dnamic Activit Quadratics in Verte Form Lesson L Vocabular V parabola quadratic function verte form ais of smmetr verte of the parabola minimum value maimum value In the Solve It, ou used the parabolic shape of the horse s jump. A parabola is the graph of a quadratic function, which ou can write in the form f () a b c, where a 0. Essential Understanding The graph of an quadratic function is a transformation of the graph of the parent quadratic function,. The verte form of a quadratic function is f () a( h) k, where a 0. The ais of smmetr is a line that divides the parabola into two mirror images. The equation of the ais of smmetr is h. The verte of the parabola is (h, k), the intersection of the parabola and its ais of smmetr. Ke Concept The Parent Quadratic Function The parent quadratic function is f (). Its graph is the parabola shown. The ais of smmetr is 0. The verte is (0, 0). Verte (0, 0) f() ( ) Ais of Smmetr 0 9 Chapter Quadratic Functions and Equations

2 How do ou choose points to plot? Choose the verte and two points on one side of the ais of smmetr that give integer values of f (). Problem What is the graph of f ()? Graphing a Function of the Form f () a Step Plot the verte (0, 0). Draw the ais of smmetr, 0. Step Find and plot two points on one side of the ais of smmetr. f(), f() 0 (0) 0 (0, 0) () (, ) () 8 (, 8) Step Plot the corresponding points on the other side of the ais of smmetr. Step Sketch the curve. Ais of Smmetr 0 f() ( ) (, 8) (, 8) (, ) (, ) Verte (0, 0) (0, 0) Got It?. a. What is the graph of f ()? b. Reasoning What can ou sa about the graph of the function f () a if a is a negative number? Eplain. Graphs of a and a are reflections of each other across the -ais. Increasing u a u stretches the graph verticall and narrows it horizontall. Decreasing u a u compresses the graph verticall and widens it horizontall. Ke Concept Reflection, Stretch, and Compression Reflection, a and a Stretch, a Compression, 0 a If a. 0, the parabola opens upward. The -coordinate of the verte is the minimum value of the function. If a, 0, the parabola opens downward. The -coordinate of the verte is the maimum value of the function. Minimum Value Verte Maimum Value Lesson - Quadratic Functions and Transformations 9

3 Problem Graphing Translations of f () Graph each function. How is each graph a translation of f ()? A g() How does g() differ from f()? For each value of, the value of g() is less than the value of f(). f() Verte (0, 0) Ais of Smmetr 0 f() g() h() h ( ( ) Ais of Smmetr Verte (0, 0) Translate the graph of f down units to get the graph of g(). Verte (0, ) Verte (, 0) B h() ( ) Translate the graph of f to the right units to get the graph of h() ( ). Got It?. Graph each function. How is it a translation of f ()? a. g () The verte form, f () a( h) k, gives ou information about the graph of f without drawing the graph. If a. 0, k is the minimum value of the function. If a, 0, k is the maimum value. b. h() ( ) Maimum Value f() 0. f() Minimum Value Problem Interpreting Verte Form How do ou use verte form? Compare ( ) to verte form a( h) k to find values for a, h, and k. For ( ), what are the verte, the ais of smmetr, the maimum or minimum value, the domain and the range? ( ) a( h) k Step Compare: Step The verte is (h, k) (, ). Step The ais of smmetr is h, or. Step Since a. 0, the parabola opens upward. k is the minimum value. Step Domain: All real numbers. There is no restriction on the value of. Range: All real numbers $, since the minimum value of the function is. Got It?. What are the verte, ais of smmetr, minimum or maimum, and domain and range of the function ( )? 9 Chapter Quadratic Functions and Equations

4 You can use the verte form of a quadratic function, f () a( h) k, to transform the graph of the parent function f (). Stretch or compress the graph of f () verticall b the factor u a u. If a, 0, reflect the graph across the -ais. Shift the graph u h u units horizontall and u k u units verticall. Ke Concept Translation of the Parabola Horizontal ZhZ Vertical k Move h units. (( h) Horizontal and Vertical ( h) k Move k units. ZkZ ZkkZ kz ZkZ ZhZ Move k units. Move h units. verte becomes (0, k) verte becomes (h, 0) verte becomes (h, k) Problem Using Verte Form A What is the graph of f () ( )? What do the values of a, h, and k tell ou about the graph? The graph is a stretched reflection of, shifted unit right and units up. Step Identif the constants a, h, and k. Because a 0, the parabola opens downward. Step Plot the verte (h, k) (, ) and draw the ais of smmetr. f() f() ( ) Step Plot two points. f() ( ). Plot (, ) and the smmetric point (0, ). Step Sketch the curve. B Multiple Choice What steps transform the graph of to ( )? Reflect across the -ais, stretch b the factor, translate unit to the right and units up. Stretch b the factor, translate unit to the right and units up. Reflect across the -ais, translate unit to the left and units up. Stretch b the factor, reflect across the -ais, translate unit to the left and units up. The correct choice is D. Got It?. What steps transform the graph of to ( )? Lesson - Quadratic Functions and Transformations 97

5 You can use the verte form of a quadratic function to model a real-world situation. Problem Writing a Quadratic Function in Verte Form Nature The picture shows the jump of a dolphin. What quadratic function models the path of the dolphin s jump? What is the verte? The verte is (, 7). h, k 7 Choose another point, (9, ), from the path. Substitute in the verte form. Solve for a. Substitute in the verte form. f() a( h) k a(9 ) 7 a 7 a a f() ( ) 7 models the path of the dolphin s jump Got It?. Suppose the path of the jump changes so that the ais of smmetr becomes and the height stas the same. If the path of the jump also passes through the point (, ), what quadratic function would model this path? Lesson Check Do ou know HW?. Graph the function f ().. Determine whether the function f () 0. ( ) 0 has a maimum or a minimum value.. Rewrite in verte form. Do ou UNDERSTAND? MATHEMATICAL PRACTICES. Vocabular When does the graph of a quadratic function have a minimum value?. Reasoning Is 0( ) a quadratic function? Eplain.. Compare and Contrast Describe the differences between the graphs of ( ) and ( ) Chapter Quadratic Functions and Equations

6 Practice and Problem-Solving Eercises MATHEMATICAL PRACTICES A Practice Graph each function. See Problem f () f () f () Graph each function. Describe how it was translated from f (). See Problem.. f (). f () ( ) 7. f () 8. f () ( ) 9. f () 9 0. f () ( ). f ().. f () (.) Identif the verte, the ais of smmetr, the maimum or minimum value, and the domain and the range of each function. See Problem...( 0). f () 0.(.). f () (.). 0.00( ) 7. f () ( ) 8. ( ) Graph each function. Identif the ais of smmetr. See Problem. 9. f () ( ) 0. ( ). f () ( ). ( 7) 8. ( ). f () ( 7) 0 Write a quadratic function to model each graph. See Problem B Appl 8. Think About a Plan A gardener is putting a wire fence along the edge of his garden to keep animals from eating his plants. If he has 0 meters of fence, what is the largest rectangular area he can enclose? To find the area of a rectangle, what two quantities do ou need? Choose one to be our variable and write the other in terms of this variable. How can a graph help ou solve this problem? What quadratic function represents the area of the garden? STEM 9. Manufacturing Th e equation for the cost in dollars of producing computer chips is C , where is the number of chips produced. Find the number of chips that minimizes the cost. What is the cost for that number of chips? Lesson - Quadratic Functions and Transformations 99

7 In Chapter, ou graphed absolute value functions as transformations of their parent function». Similarl, ou can graph a quadratic function as a transformation of the parent function. Graph the following pairs of functions on the same set of aes. Determine how the are similar and how the are different. 0. u u ; ( ). u u ; ( ). uu ;. u u; ( ) Describe how to transform the parent function to the graph of each function below. Graph both functions on the same aes.. ( ). ( ) You can find the rate of change for an interval between two points of a function b finding the slope between the points. Use the graph to find the -value for each -value. Then find the rate of change for each interval. a. (0, ) and (, ) b. (, ) and (, ) c. (, ) and (, ) d. Reasoning. What do ou notice about the rate of change as the interval gets further awa from the verte? e. Would our answer to part (d) change if the intervals were on the left side of the graph? Eplain. 8. Write a quadratic function to represent the areas of all rectangles with a perimeter of ft. Graph the function and describe the rectangle that has the largest area. Write the equation of each parabola in verte form. 9. verte (, ), point (, ) 0. verte (, ), point (, ). verte (0, ), point (, ). verte Q, R, point (, ). pen-ended Write an equation of a parabola smmetric about a. Technolog Determine the ais of smmetr for each parabola defined b the spreadsheet values at the right. b. How could ou use the spreadsheet columns to verif that the aes of smmetr are correct? c. What functions in verte form model the data? Check that the aes of smmetr are correct. A X B Y A X B Y Chapter Quadratic Functions and Equations

8 C Challenge Determine a and k so the given points are on the graph of the function.. (0, ), (, ); a( ) k. (, ), (0, ); a( ) k 7. (, ), (, 9); a( ) k 8. (, ), (, ); a( ) k 9. a. In the function a b c, c represents the -intercept. Find the value of the -intercept in the function a( h) k. b. Under what conditions does k represent the -intercept? Find the quadratic function a( h) whose graph passes through the given points. 0. (, ) and (, ). (, ) and (, ). (, ) and (7, ). (, ) and (, 0). (, 8) and (, 0). (, ) and (, 0) Standardized Test Prep SAT/ACT Short Response. ne parabola at the right has the equation ( ). Which equation represents the second parabola? ( ) ( ) ( ) ( ) 7. Which sstem has the unique solution (, )? e e e e 8. What is the formula for the surface area of a right circular clinder, S prh pr, solved for h? h S pr h S pr h S pr r h r S pr 9. An athletic club has feet of fencing to enclose a tennis court. What quadratic function can be used to find the maimum area of the tennis court? Find the maimum area, and the lengths of the sides of the resulting fence. Mied Review (LE_Ahead) s/b Solve placed each on sstem the standard of equations 0 Tet using laer a matri e 7. e 0 z 7. z z See Lesson -. Get Read! To prepare for Lesson -, do Eercises 7 7. Find the verte of the graph of each function. See Lesson u u 7. u u 7. u u Lesson - Quadratic Functions and Transformations 0