Stretches and Shrinks Linear Functions

Stretches and Shrinks Linear Functions

Algebra 1

Mrs. LaGrange

Horizontal Shrink/Compression

A horizontal shrink is a transformation that occurs when we multiply all of the x-coordinates (inputs) by the same factor a, where a>1. This causes the graph to shrink toward the y-axis.

(Note that we use the parabola rather than the line for illustration as it is easier to see the shrink from the red graph to the blue graph.)

Horizontal Shrink/Compression:

*To create a horizontal shrink we multiply the x-value by a constant (a) greater than 1.

y = f(ax) where a>1. We say the graphy y is a horizontal shrink by a factor of 1/a.

*The y-intercept stays the same in a horizontal shrink.

The image shows f(x) = x+2 and f(2x). We say that the graph g(x) is a horizontal shrink by a factor of 1/2.

Horizontal Shrink/Compression

Note what happens to the x and y values after a horizontal shrink.

Note that the blue line is closer to the y-axis than the green line.

Horizontal Stretch

A horizontal stretch is a transformation where the graph stretches away from the y-axis.

In a stretch, 0 < a < 1.

*Note we illustrate with a parabola to see the stretch - f(x) is red and the stretch is blue. Notice that

a = 1/2.

_{Horizontal Stretch}

*To create a horizontal stretch we multiply the x-value by a constant (a) where 0 < a < 1.

y = f(ax) where 0<a<1. We say the graph y is a horizontal stretch by a factor of a.

*The y-intercept stays the same in a horizontal shrink.

*The image shows f(x) = x+2 and f((1/2)x). We say that the graph g(x) is a horizontal stretch by a factor of 2.

Vertical Shrink/Compression

A vertical shrink is a transformation that occurs when we multiply the y-coordinates (outputs) by the same factor a. When 0 < a <1, we have a vertical shrink.

y = a*f(x).

Vertical Stretch

A vertical stretch occurs when we multipy the y-coordinate (output) by a factor a where a>1. A vertical stretch takes the graph further away from the x-axis.

In the case of a vertical stretch or shrink the x-intercept stays the same.

Examples: Vertical and Horizontal Shrinks/Compressions

Multiple Choice

Describe the transformation:

 $g(x)=\frac{1}{5}f\left(x\right)$  

• Vertical stretch of 5

• Vertical compression of 1/5

• Horizontal stretch of 5

• Horizontal compression of 1/5

Multiple Choice

Describe the transformation:

 $g(x)=f\left(4x\right)$  

• Vertical stretch of 4

• Vertical compression of 1/4

• Horizontal stretch of 4

• Horizontal compression of 1/4

Examples: Horizontal and Vertical Stretches

Multiple Choice

Describe the transformation:

 $g(x)=f\left(\frac{1}{3}x\right)$  

• Vertical stretch of 3

• Vertical compression of 1/3

• Horizontal stretch of 3

• Horizontal compression of 1/3

Multiple Choice

Describe the transformation:

g(x) = 2f(x)

• Vertical stretch of 2

• Vertical compression of 1/2

• Horizontal stretch of 2

• Horizontal compression of 1/2

Combining Transformations

Sometimes we see that transformations are combined. Here is an example of the core concept.

Multiple Choice

Describe the transformation:

 $g(x)=6f\left(x\right)+1$  

• Vertical stretch of 6 ; move 1 to the right

• Vertical compression of 1/6 ; move 1 down

• Vertical stretch of 6 ; move 1 up

• Vertical compression of 1/6 ; move 1 to the left

Multiple Choice

Describe the transformation:

 $g(x)=9f\left(\frac{1}{2}x\right)$  

• Vertical stretch of 9 ; Horizontal stretch of 2

• Vertical compression of 1/9 ; Horizontal compression of 1/2

• Vertical stretch of 9 ; Horizontal compression of 1/2

• Vertical compression of 1/9 ; Horizontal stretch of 2

Multiple Choice

Describe the transformation:

 $g(x)=\frac{1}{2}f\left(7x\right)$  

• Vertical stretch of 2 ; Horizontal stretch of 7

• Vertical compression of 1/2 ; Horizontal compression of 1/7

• Vertical stretch of 2 ; Horizontal compression of 1/7

• Vertical compression of 1/2 ; Horizontal stretch of 7

Summary: