Curriculum Dr. Kelly W. Edenfield Manager of School Partnerships - - PowerPoint PPT Presentation

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

• 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.

Webinar Goals

Possible Solution Strategy

f(x) = x2 + 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

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.

“New” Transformation Rules

Our Approach

• f(x) = 2x
• g(x) = 2x – 5 = f(x) – 5

Input Output for Parent Function, f(x) Output for Transformed Function, g(x)

• 3

0.125

• 4.875
• 1

0.5

• 4.5

1

• 4

1 2

• 3

3 8 3

Vertical Translations

• f(x) = 2x
• g(x) = 2x – 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

• g(x) = -0.5f(x)
• f(x) = 3x; g(x) = -0.5(3x) = -0.5f(x)

Input Output for Parent Function, f(x) Output for Transformed Function, g(x)

• 3

0.037

• 0.0185
• 1

0.333

• 0.1667

1

• 0.5

1 3

• 1.5

3 27

• 13.5

Vertical Dilations and Reflections

• f(x) = 3x
• g(x) = -0.5(3x) = -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) = x2
• g(x) = (x + 3)2 = f(x + 3)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 1 1 2 4

Horizontal Translations

• f(x) = x2
• g(x) = (x + 3)2 = f(x + 3)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 x’ + 3 = 0 1 1 2 4

Horizontal Translations

• f(x) = x2
• g(x) = (x + 3)2 = f(x + 3)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1

• 3

1 1 2 4

Horizontal Translations

• f(x) = x2
• g(x) = (x + 3)2 = f(x + 3)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

x’ + 3 = -2 4

• 1

x’ + 3 = -1 1 x’ + 3 = 0 1 x’ + 3 = 1 1 2 x‘ + 3 = 2 4

Horizontal Translations

• f(x) = x2
• g(x) = (x + 3)2 = f(x + 3)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

x’ + 3 = -2  x’ = -5 4

• 1

x’ + 3 = -1  x’ = -4 1 x‘ + 3 = 0  x’ = -3 1 x’ + 3 = 1  x’ = -2 1 2 x‘ + 3 = 2  x’ = -1 4

Horizontal Translations

• f(x) = x2; g(x) = (x + 3)2 = f(x + 3)
• In each case, we set x’ + 3 = x.
• So x’ = x – 3!

Original Input, x Transformed Input, x’ Output

• 2
• 5

4

• 1
• 4

1

• 3

1

• 2

1 2

• 1

4

Horizontal Translations

• f(x) = x2; 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) = x2
• g(x) = (x – 5)2 = f(x – 5)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 1 1 2 4

Another Example: Horizontal Translations

• f(x) = x2
• g(x) = (x – 5)2 = f(x – 5)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

x‘ – 5 = -2  x’ = 3 4

• 1

x‘ – 5 = -1  x’ = 4 1 x‘ – 5 = 0  x’ = 5 1 x‘ – 5 = 1  x’ = 6 1 2 x‘ – 5 = 2  x’ = 7 4

Another Example: Horizontal Translations

• f(x) = x2
• 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) = x2
• g(x) = (3x)2 = f(3x)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 1 1 2 4

Horizontal Dilations

• f(x) = x2
• g(x) = (3x)2 = f(3x)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 3x’ = 0 1 1 2 4

Horizontal Dilations

• f(x) = x2
• g(x) = (3x)2 = f(3x)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 1 1 2 4

Horizontal Dilations

• f(x) = x2
• g(x) = (3x)2 = f(3x)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

3x’ = -2 4

• 1

3x’ = -1 1 3x’ = 0 1 3x’ = 1 1 2 3x’ = 2 4

Horizontal Dilations

• f(x) = x2
• g(x) = (3x)2 = f(3x)
• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

3x’ = -2  x’ = -2/3 4

• 1

3x’ = -1  x’ = -1/3 1 3x’ = 0  x’ = 0 1 3x’ = 1  x’ = 1/3 1 2 3x’ = 2  x’ = 2/3 4

Horizontal Dilations

• f(x) = x2; 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 𝑙x, y).

• That is, y = f(kx) maps (x, y)  (1

𝑙x, y).

Another Example: Horizontal Dilations

• f(x) = x2; g(x) = (1

2x)2 = f(1 2x)

• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

4

• 1

1 1 1 2 4

Another Example: Horizontal Dilations

• f(x) = x2; g(x) = (

1 2x)2 = f( 1 2x)

• If we want the same output, what must be

the value of the “new” input?

Original Input, x Transformed Input, x’ Output

• 2

0.5x‘ = -2  x’ = -4 4

• 1

0.5x‘ = -1  x’ = -2 1 0.5x‘ = 0  x’ = 0 1 0.5x‘ = 1  x’ = 2 1 2 0.5x‘ = 2  x’ = 4 4

Another Example: Horizontal Dilations

• f(x) = x2
• 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).

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.

• 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.

Webinar Goals

kedenfield@carnegielearning.com 888.851.7094 ext 425 Follow me on Twitter @CLkedenfield pd@carnegielearning.com sales@carnegielearning.com www.carnegielearning.com

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