- Attributes and Transformations of Quadratic Functions TEKS FCUS TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening. TEKS ()(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Additional TEKS ()(E), ()(F), ()(G) VCABULARY Maimum value the greatest -value of a function Minimum value the least -value of a function Parabola the set of points in a plane that are the same distance from a fied point, the focus, as the are from a line, the directri Quadratic function a function that ou can write in the form f() = a + b + c Verte form The verte form of a quadratic function is f() = a( - h) + k, where a 0 and (h, k) are the coordinates of the verte of the function. Verte of a parabola the point where the function for the parabola reaches a maimum or a minimum value Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING The graph of an quadratic function is a transformation of the graph of the parent quadratic function, =. 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 Lesson - Attributes and Transformations of Quadratic Functions
Ke Concept Reflection, Stretch, and Compression Reflection, a and a Stretch, a Compression, 0 a If a 7 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 Ke Concept Horizontal h Move h units. (( h) Verte Maimum Value Translation of the Parabola Vertical Horizontal and Vertical k ( h) k k Move k units. kk k k h Move k units. Move h units. verte becomes (h, 0) verte becomes (0, k) verte becomes (h, k) PearsonTEXAS.com
Problem P TEKS Process Standard ()(D) Graphing a Function of the Form f() = a 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(). What is the graph of f () =? W Step Plot the verte (0, 0). Draw the ais of smmetr, = 0. S Step Find and plot two points on one side of the ais of smmetr. S 0 f() (0) 0 () () 8 Ais of Smmetr 0, f() f() f() (, ( 8 8) ((, 8)) (0, 0) (, ) (, 8) ((,,,, ) Verte (0, 0) Step Plot the corresponding points on the other side of the ais of smmetr. (, ) ((0, 0) (0 Step Sketch the curve. Problem bl 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() Ve Verte (0 (0, 0) Verte (0, ) B h() = ( ) Verte (, 0) Ais of Smmetr 0 g() f() Translate the graph of f down units to get the graph of g() = -. Lesson - h() h ( ( ) Verte (0, 0) Ais of Smmetr Translate the graph of f to the right units to get the graph of h() = ( - ). Attributes and Transformations of Quadratic Functions
Problem TEKS Process Standard ()(G) 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? Step Compare: = ( - ) - = a( - h) + k Step The verte is (h, k) = (, - ). Step The ais of smmetr is = h, or =. Step Since a 7 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 Ú-, since the minimum value of the function is -. 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. PearsonTEXAS.com
Problem Writing a Quadratic Function in Verte Form A Nature The picture shows the jump of a dolphin. The verte of the dolphin s jump is at the point (, 7). 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. 7 7 8 9 0 B A parabola passes through the point (, ), opens upward, and has an ais of smmetr at = and vertical stretch factor. What is the equation of the quadratic function? Since the parabola opens upward, ou know that the value of a is positive, so using the vertical stretch factor, a =. Because the ais of smmetr is =-, ou know that h =-. Using the verte form of the quadratic function, ou can obtain the equation f () = ( + ) + k. Since the graph passes through the point (-, -), ou can use this to find k. - = f (-) = (- + ) + k - = + k - = + k - = k The equation of the quadratic function is f () = ( + ) -. Lesson - Attributes and Transformations of Quadratic Functions
NLINE H M E W R K PRACTICE and APPLICATIN EXERCISES Scan page for a Virtual Nerd tutorial video. Graph each function. Describe how it was translated from f () =. For additional support when completing our homework, go to PearsonTEXAS.com.. f () = +. f () = ( - ). f () = -. f () = ( + ). f () = - 9. f () = ( + ) 7. f () = +. 8. f () = ( -.) Identif the verte, the ais of smmetr, the maimum or minimum value, and the domain and the range of each function. 9. =-.( + 0) 0. f () = 0.( -.). f () = ( +.). = 0.00( + ) -. f () =-( - ) -. = ( - ) + Write a quadratic function to model each graph... 8 7. Use a Problem-Solving Model ()(B) 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? 8. 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. Analze Mathematical Relationships ()(F) What do ou notice about the rate of change as the interval gets farther awa from the verte? e. Would our answer to part (d) change if the intervals were on the left side of the graph? Eplain. 9. 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. 8 PearsonTEXAS.com 7
Write the equation of each parabola in verte form. 0. verte (0, ), point (, -). verte (, ) -, point (, ). Create Representations to Communicate Mathematical Ideas ()(E) Write an equation of a parabola smmetric about =-0.. a. Select Tools to Solve Problems ()(C) 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? A X B Y A X Check that the aes of smmetr are correct. Write the equation of each parabola in verte form.. verte (-7, ), vertical stretch factor, opening upward. verte (, - ), vertical compression factor, opening downward. ais of smmetr =, vertical stretch factor 8, maimum - 7. ais of smmetr =., vertical compression factor 0., minimum -7. B Y 0 0 TEXAS Test Practice 8. ne parabola at the right has the equation = ( - ) +. Which equation represents the second parabola? A. =-(-) + C. = ( + ) - B. = (- - ) + D. =-( + ) - 9. Which sstem has the unique solution (, )? F. e = - H. e + = + = =- + G. e =- + - =- J. e + = - =- 0. What is the formula for the surface area of a right circular clinder, S = prh + pr, solved for h? A. h = S pr B. h = S pr C. h = S pr - r D. h = r - S pr 8 Lesson - Attributes and Transformations of Quadratic Functions