More Than Meets the Eye: Seeing Structure in Graphical Transformations Across the Curriculum Dr. Kelly W. Edenfield Manager of School Partnerships Carnegie Learning
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
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.
Possible Solution Strategy f(x) = x 2 + 6x = x(x + 6) Zeros of the function are x = 0, -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.
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.
Are you convinced?
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.
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.
Transformation “Rules”
Transformation “Rules”
High School Standards • Functions to graph – Radical – Piecewise and analyze: – Logarithmic – Linear – Trigonometric – Exponential – Rational (+) – Quadratic – Higher-Order Polynomial • 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.
“New” Transformation Rules
Our Approach Input Output for Parent Output for Function, f(x) Transformed Function, g(x) -3 0.125 -4.875 -1 0.5 -4.5 0 1 -4 1 2 -3 3 8 3 • f(x) = 2 x • g(x) = 2 x – 5 = f(x) – 5
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).
More Vertical Transformations Input Output for Parent Output for Function, f(x) Transformed Function, g(x) -3 0.037 -0.0185 -1 0.333 -0.1667 0 1 -0.5 1 3 -1.5 3 27 -13.5 • g(x) = -0.5f(x) • f(x) = 3 x ; g(x) = -0.5(3 x ) = -0.5f(x)
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).
Horizontal Translations • f(x) = x 2 • g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Translations • f(x) = x 2 • g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 x’ + 3 = 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Translations • f(x) = x 2 • g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 -3 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Translations • f(x) = x 2 • g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x’ Output -2 x’ + 3 = -2 4 -1 x’ + 3 = -1 1 0 x’ + 3 = 0 0 1 x’ + 3 = 1 1 2 x‘ + 3 = 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Translations • f(x) = x 2 • g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x’ Output x’ + 3 = -2 x’ = -5 -2 4 x’ + 3 = -1 x’ = -4 -1 1 x‘ + 3 = 0 x’ = -3 0 0 x’ + 3 = 1 x’ = -2 1 1 x‘ + 3 = 2 x’ = -1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Translations • f(x) = x 2 ; g(x) = (x + 3) 2 = f(x + 3) Original Input, x Transformed Input, x’ Output -2 -5 4 -1 -4 1 0 -3 0 1 -2 1 2 -1 4 • In each case, we set x’ + 3 = x. • So x’ = x – 3!
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).
Another Example: Horizontal Translations • f(x) = x 2 • g(x) = (x – 5) 2 = f(x – 5) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Another Example: Horizontal Translations • f(x) = x 2 • g(x) = (x – 5) 2 = f(x – 5) Original Input, x Transformed Input, x’ Output x‘ – 5 = -2 x’ = 3 -2 4 x‘ – 5 = -1 x’ = 4 -1 1 x‘ – 5 = 0 x’ = 5 0 0 x‘ – 5 = 1 x’ = 6 1 1 x‘ – 5 = 2 x’ = 7 2 4 • If we want the same output, what must be the value of the “new” input?
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).
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.
Horizontal Dilations • f(x) = x 2 • g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Dilations • f(x) = x 2 • g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 3x’ = 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Dilations • f(x) = x 2 • g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x’ Output -2 4 -1 1 0 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Dilations • f(x) = x 2 • g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x’ Output -2 3x’ = -2 4 -1 3x’ = -1 1 0 3x’ = 0 0 1 3x’ = 1 1 2 3x’ = 2 4 • If we want the same output, what must be the value of the “new” input?
Horizontal Dilations • f(x) = x 2 • g(x) = (3x) 2 = f(3x) Original Input, x Transformed Input, x’ Output 3x’ = -2 x’ = -2/3 -2 4 3x’ = -1 x’ = -1/3 -1 1 3x’ = 0 x’ = 0 0 0 3x’ = 1 x’ = 1/3 1 1 3x’ = 2 x’ = 2/3 2 4 • If we want the same output, what must be the value of the “new” input?
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). 1 • If f(x, y) f(kx, y), then (x, y) ( 𝑙 x, y). • That is, y = f(kx) maps (x, y) ( 1 𝑙 x, y).
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 -2 4 -1 1 0 0 1 1 2 4 • If we want the same output, what must be the value of the “new” input?
Another Example: Horizontal Dilations 1 1 • f(x) = x 2 ; g(x) = ( 2 x) 2 = f( 2 x) Original Input, x Transformed Input, x’ Output 0.5x‘ = -2 x’ = -4 -2 4 0.5x‘ = -1 x’ = -2 -1 1 0.5x‘ = 0 x’ = 0 0 0 0.5x‘ = 1 x’ = 2 1 1 0.5x‘ = 2 x’ = 4 2 4 • If we want the same output, what must be the value of the “new” input?
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).
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