More Than Meets the Eye: Seeing Structure in Graphical Transformations Across the Curriculum

Transcription

1 More Than Meets the Eye: Seeing Structure in Graphical Transformations Across the Curriculum Dr. Kelly W. Edenfield Manager of School Partnerships Carnegie Learning

2 When I think of transformations I think of A. Translating, rotating, reflecting, and dilating geometric shapes B. Constructing similar and/or congruent shapes using classical tools C. Altering parent graphs on the coordinate grid D. More than 1 of the above

3 Webinar Goals Consider instructional implications of PARCC-like assessment items. Discuss the relationships across the Common Core transformation standards. Discuss ways to use mathematical structure (SMP7) to deepen students understanding of and fluency with transformations.

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5 Possible Solution Strategy f(x) = x 2 + 6x = x(x + 6) Zeros of the function are x = 0, -6.

6 Possible Solution 1. x-coordinate of the vertex depends on the value of k. True. By the rules, f(x + k) means to shift k units to the left. (If k is negative, then move right.) 2. x-coordinate of the vertex is negative for all values of k. False. k moves the graph left and right, so given extreme enough values of k (k -3), the vertex is non-negative.

7 Possible Solution 3. y-coordinate of the vertex is independent of the value of k. True. By the rules, we shifted horizontally but did not cause any vertical movement. So the y-coordinate of the vertex never changes.

8 Are you convinced?

9 Eighth Grade Standards Verify experimentally the properties of the rigid motions (reflections, rotations, translations). Understand congruence in relation to rigid motions and similarity in terms of all four transformations. Use coordinates to determine the effect of translations, rotations, reflections, and dilations on 2-D figures.

10 Eighth Grade Standards Verify experimentally the properties of the rigid motions (reflections, rotations, translations). Understand congruence in relation to rigid motions and similarity in terms of all four transformations. Use coordinates to determine the effect of translations, rotations, reflections, and dilations on 2-D figures.

11 Transformation Rules

12 Transformation Rules

13 High School Standards Functions to graph and analyze: Linear Exponential Quadratic Higher-Order Polynomial Radical Piecewise Logarithmic Trigonometric Rational (+) Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

14 New Transformation Rules

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16 Our Approach Input Output for Parent Function, f(x) Output for Transformed Function, g(x) f(x) = 2 x g(x) = 2 x 5 = f(x) 5

17 Vertical Translations f(x) = 2 x g(x) = 2 x 5 = f(x) 5 Parallel in reasoning to (x, y) (x, y ) = (x, y 5 ) If f(x, y) f(x, y + k), then (x, y) (x, y + k). That is, f(x) + k maps (x, y) (x, y + k).

18 More Vertical Transformations Input Output for Parent Function, f(x) Output for Transformed Function, g(x) g(x) = -0.5f(x) f(x) = 3 x ; g(x) = -0.5(3 x ) = -0.5f(x)

19 Vertical Dilations and Reflections f(x) = 3 x g(x) = -0.5(3 x ) = -0.5f(x) Parallel in reasoning to (x, y) (x, y ) = (x, -0.5y) If f(x, y) f(x, -ky), then (x, y) (x, -ky). That is, -kf(x) maps (x, y) (x, -ky).

20 f(x) = x 2 Horizontal Translations g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x Output If we want the same output, what must be the value of the new input?

21 f(x) = x 2 Horizontal Translations g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x Output x + 3 = If we want the same output, what must be the value of the new input?

22 f(x) = x 2 Horizontal Translations g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x Output If we want the same output, what must be the value of the new input?

23 f(x) = x 2 Horizontal Translations g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x Output -2 x + 3 = x + 3 = x + 3 = x + 3 = x + 3 = 2 4 If we want the same output, what must be the value of the new input?

24 f(x) = x 2 Horizontal Translations g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x Output -2 x + 3 = -2 x = x + 3 = -1 x = x + 3 = 0 x = x + 3 = 1 x = x + 3 = 2 x = -1 4 If we want the same output, what must be the value of the new input?

25 Horizontal Translations f(x) = x 2 ; g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x Output In each case, we set x + 3 = x. So x = x 3!

26 Horizontal Translations f(x) = x 2 ; g(x) = (x + 3) 2 = f(x + 3) Because x = x 3, we have our connection to (x, y) (x, y ) = (x 3, y). If f(x, y) f(x + k, y), then (x, y) (x k, y). That is, y = f(x + k) maps (x, y) (x k, y).

27 Another Example: Horizontal Translations f(x) = x 2 g(x) = (x 5) 2 = f(x 5) Original Input, x Transformed Input, x Output If we want the same output, what must be the value of the new input?

28 Another Example: Horizontal Translations f(x) = x 2 g(x) = (x 5) 2 = f(x 5) Original Input, x Transformed Input, x Output -2 x 5 = -2 x = x 5 = -1 x = x 5 = 0 x = x 5 = 1 x = x 5 = 2 x = 7 4 If we want the same output, what must be the value of the new input?

29 Another Example: Horizontal Translations f(x) = x 2 g(x) = (x 5) 2 = f(x 5) In each case, we set x 5 = x, so x = x + 5. Because x = x + 5, we have our connection to (x, y) (x, y ) = (x + 5, y).

30 Horizontal Translation Rule y = f(x + k) k > 0: (x, y) (x, y ) = (x k, y) Ask: If x + k = x, what is x? x' = x k. k < 0: (x, y) (x, y ) = (x + k, y) Ask: If x + k = x, what is x? x' = x k, but k is negative, so we simplify to x = x + k.

31 f(x) = x 2 Horizontal Dilations g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x Output If we want the same output, what must be the value of the new input?

32 f(x) = x 2 Horizontal Dilations g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x Output x = If we want the same output, what must be the value of the new input?

33 f(x) = x 2 Horizontal Dilations g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x Output If we want the same output, what must be the value of the new input?

34 f(x) = x 2 Horizontal Dilations g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x Output -2 3x = x = x = x = x = 2 4 If we want the same output, what must be the value of the new input?

35 f(x) = x 2 Horizontal Dilations g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x Output -2 3x = -2 x = -2/ x = -1 x = -1/ x = 0 x = x = 1 x = 1/ x = 2 x = 2/3 4 If we want the same output, what must be the value of the new input?

36 Horizontal Dilations f(x) = x 2 ; g(x) = (3x) 2 = f(3x) Because 3x = x x = (1/3)x, we have our connection to (x, y) (x, y ) = ((1/3)x, y). If f(x, y) f(kx, y), then (x, y) ( 1 x, y). k That is, y = f(kx) maps (x, y) ( 1 x, y). k

37 Another Example: Horizontal Dilations f(x) = x 2 ; g(x) = ( 1 2 x)2 = f( 1 2 x) Original Input, x Transformed Input, x Output If we want the same output, what must be the value of the new input?

38 Another Example: Horizontal Dilations f(x) = x 2 ; g(x) = ( 1 2 x)2 = f( 1 2 x) Original Input, x Transformed Input, x Output x = -2 x = x = -1 x = x = 0 x = x = 1 x = x = 2 x = 4 4 If we want the same output, what must be the value of the new input?

39 Another Example: Horizontal Dilations f(x) = x 2 g(x) = (0.5x) 2 = f(0.5x) In each case, we set 0.5x = x, so x = 2x. Because x = 2x, we have our connection to (x, y) (x, y ) = (2x, y) when f(x) f(0.5x).

40 Horizontal Dilations Rule y = f(kx) k > 1: (x, y) (x, y ) = ((1/k)x, y) Ask: If kx = x, what is x? x' = (1/k)x. 0 < k < 1: (x, y) (x, y ) = (kx, y) Ask: If kx = x, what is x? x' = (1/k)x, but k is already a fraction, so we simplify to x' = kx.

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43 Webinar Goals Consider instructional implications of PARCC-like assessment items. Discuss the relationships across the Common Core transformation standards. Discuss ways to use mathematical structure (SMP7) to deepen students understanding of and fluency with transformations.

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