Transformations of y = x 2 Parent Parabola

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1 Transformations of = 2 SUGGESTED LEARNING STRATEGIES: Marking the Tet, Interactive Word Wall, Create Representations, Quickwrite 1. Graph the parent quadratic function, f () = 2, on the coordinate grid below. Include the points that have -values -2, -1, 0, 1, and 2. M Notes ACTIVITY 3.6 A parent function is the simplest function of a particular tpe. For eample, the parent linear function is f () =. The parent absolute-value function is f () =. The points on the parent function graph that have -values -2, -1, 0, 1, and 2 are ke points that can be used when graphing an quadratic function as a transformation of the parent quadratic function. 2. Graph f () = 2 on the coordinate grid below. Then graph g () = 2-3 and h() = A transformation of a graph of a parent function is a change in the position, size or shape of the graph. 20 College Board. All rights reserved. 3. Identif and describe the transformations of the graph of f () = 2 that result in the graphs of g () and h(). Unit 3 Quadratic Functions and Comple Numbers 183

2 ACTIVITY 3.6 Transformations of = 2 M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Group Presentation 4. Graph f () = 2 on the coordinate grid below. Then graph g () = ( - 2) 2 and h() = ( + 3) 2. Translations are transformations that change the location of a graph but maintain the original shape of a graph. For this reason, the are known as rigid transformations.. Identif and describe the transformations of the graph of f () = 2 that result in the graphs of g () and h(). 6. Describe each function as a transformation of f () = 2. Then use that information to graph each function on the coordinate grid. a. a() = ( - 1) 2 b. w() = College Board. All rights reserved. 184 SpringBoard Mathematics with Meaning TM Algebra 2

3 Transformations of = 2 ACTIVITY 3.6 SUGGESTED LEARNING STRATEGIES: Create Representations, Group Presentation, Quickwrite c. d() = ( + 3) 2 - M Notes d. j() = ( - 1) Graph f () = 2 on the coordinate grid below. Then graph g () = 4 2 and h() = College Board. All rights reserved. 8. Identif and describe the transformations of the graph of f () = 2 that result in the graphs of g () and h(). Unlike a rigid transformation, a vertical stretch or vertical shrink will change the shape of the graph. Unit 3 Quadratic Functions and Comple Numbers 18

4 ACTIVITY 3.6 Transformations of = 2 M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Group Presentation, Quickwrite 9. Describe each function as a transformation of f () = 2. Then use the transformations to graph each function. a. f () = 2 2 b. f () = Graph f () = 2 on the coordinate grid below. Then graph g () = College Board. All rights reserved. Reflections over aes do not change the shape of the graph, so the are also rigid transformations. 11. Identif and describe the transformation of the graph of f () = 2 that results in the graph of g (). 186 SpringBoard Mathematics with Meaning TM Algebra 2

5 Transformations of = 2 ACTIVITY 3.6 SUGGESTED LEARNING STRATEGIES: Identif a Subtask, Create Representations, Group Presentation 12. Describe the transformation(s) of each function from the parent function. Then graph without the use of a graphing calculator. M Notes a. f () = -2 2 b. f () = College Board. All rights reserved. 13. Multiple transformations can be represented in the same function. Describe the transformations from the parent function. Then graph the function, using our knowledge of transformations onl. a. f () = -4 ( + 3) When graphing multiple transformations, ou ma find it easier to first perform reflections and stretches or shrinks, and then translate. Unit 3 Quadratic Functions and Comple Numbers 187

6 ACTIVITY 3.6 Transformations of = 2 M Notes SUGGESTED LEARNING STRATEGIES: Identif a Subtask, Create Representations, Group Presentation 13. () b. f () = 2 ( - 4) 2-3 c. f () = 2 ( + 1) 2-4 d. f () = - ( - 3) College Board. All rights reserved. 188 SpringBoard Mathematics with Meaning TM Algebra 2

7 Transformations of = 2 ACTIVITY 3.6 SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Marking the Tet, Create Representations, Quickwrite, Group Presentation A quadratic function in standard form, f () = a 2 + b + c, can be changed into verte form b completing the square. EXAMPLE 1 Write f () = in verte form. Step 1: Factor the leading coefficient f () = 3( 2-4) + 7 from the quadratic and linear terms. Step 2: Complete the square b taking f () = 3( ) + 7 half the linear coefficient [0.(-4) = -2], squaring it + 4 [(-2) 2 = 4], and then adding it inside the parentheses. Step 3: To maintain the value of f () = 3( ) - 3(4) + 7 the epression, multipl the leading coefficient [3] b the f () = 3( ) number added inside the parentheses [4]. Then subtract that product [12]. Step 4: Write the trinomial inside f () = 3 ( - 2) 2 - the parentheses as a perfect square. The function is in verte form. M Notes The verte form of a quadratic function is f() = a ( - h) 2 + k, where the verte of the function is (h, k). Notice that the transformations of f() = 2 are apparent when the function is in verte form. 20 College Board. All rights reserved. TRY THESE A Write each quadratic function in verte form. Write our answers in the M Notes space. Show our work. a. f () = b. g () = Write the each function in verte form. Then describe the transformation(s) from the parent function and graph without the use of a graphing calculator. a. f () = Unit 3 Quadratic Functions and Comple Numbers 189

8 ACTIVITY 3.6 Transformations of = 2 M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Group Presentation b. g () = CHECK YOUR UNDERSTANDING Write our answers on notebook paper or grid paper. Show our work. 1. Identif the transformations to the parent function f () = 2 evident in each function. a. g () = 12 2 b. h() = ( + 1) 2 c. m() = 2-47 d. n() = e. p() = 8π ( - 40) Write a quadratic function that represents the transformations described below. a. translated four units to left and 3 units up b. reflected over the -ais, stretched verticall b a factor of c. shrunk verticall b a factor of 1 4, translated 2 units right and ten units down 3. Graph each function using transformations. a. f () = ( + 3) 2 - b. f () = c. f () = -3 ( + 2) 2 + d. f () = 1 2 ( - 1) Write each quadratic function in verte form. a. f () = b. f () = MATHEMATICAL REFLECTION What are the advantages of recognizing a function as a transformation of a parent graph before graphing that function? 20 College Board. All rights reserved. 190 SpringBoard Mathematics with Meaning TM Algebra 2