Coordinate Algebra 5.4

Geometric Stretching, Shrinking, and Dilations

Stretching/ShrinkingHorizontal

• Affects the x-values• (2x, y) is a horizontal stretch• (x, y) is a horizontal shrink

Vertical• Affects the y-values• (x, 3y) is a vertical stretch• (x, y) is a vertical shrink)

Let’s Examine……..

CAT

C (-2, 0) A(1, -1) T(2, 3)

C ‘(-6, 0) A’(3, -1) T’(6, 3)

C’ A A’

T T’

C

Dilations

What is a Dilation?

• Dilation is a transformation that produces a figure similar to the original by proportionally shrinking or stretching the figure.

Dilated PowerPoint Slide

A dilation is a transformation that

produces an image that is the same shape as the original, but is a

different size.

What’s the difference?

• A dilation occurs when you stretch or shrink both the x and y values by the same scale factor

• Dilations preserve shape, whereas stretching and shrinking do not.

• Dilations create similar figures – Angle measures stay the same– Side lengths are proportional

Proportionally

• When a figure is dilated, it must be proportionally larger or smaller than the original.

Same shape, Different scale.

Let’s take a look…

We have a circle with a certain diameter.

Decreasing the size of the circle decreases the diameter.

And, of course, increasing the circle increases the diameter.So, we always have a circle with a certain diameter. We are just changing the size or scale.

Scale Factor and Center of Dilation

When we describe dilations we use the terms scale factor and center of dilation.

• Scale factor • Center of Dilation

Here we have Igor. He is 3 feet tall and the greatest width across his body is 2 feet.

He wishes he were 6 feet tall with a width of 4 feet.

He wishes he were larger by a scale factor of 2.

His center of dilation would be where the length and greatest width of his body intersect.

Scale Factor

• If the scale factor is larger than 1, the figure is enlarged.

• If the scale factor is between 1 and 0, the figure is reduced in size.

Scale factor > 1

0 < Scale Factor < 1

Are the following enlarged or reduced??

AC

DB

Scale factor of 0.75

Scale factor of 3

Scale factor of 1/5

Scale factor of 1.5

Example 1:• Quadrilateral ABCD has

vertices A(-2, -1), B(-2, 1), C(2, 1) and D(1, -1).

• Find the coordinates of the image for the dilation with a scale factor of 2 and center of dilation at the origin.

• Multiply all values by 2!• A’(-4, -2) B’(-4, 2) C;(4, 2)

and D’(2, -2)

A

B C

A’

B’ C’

D

D’

Example 2:• F(-3, -3), O(3, 3), R(0, -3)

Scale factor 1/3

• Multiple all values by 1/3 (same as dividing by 3!)

• F’(-1, -1) O’(1, 1) R’(0, -1)

F

O

R

F’

O’

R’

Finding a Scale Factor• The blue quadrilateral is a dilation image of

the red quadrilateral. Describe the dilation.J(0, 2) J’(0, 1)

K(6, 0) K’(3, 0)

L(6, -4) L’(3, -2)

M(-2,- 2) J’(-1, -1)

All values have been divided by 2. This means there is a scale factor of ½.

You have a reduction!

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Credits:

• Gallatin Gateway School• Texas A&M• Your fabulous 9th grade math teachers!