TransformationsHorizontal/ Vertical Translation

Vertical Stretch/CompressionHorizontal Stretch/Compression

Reflection about x-axis/y-axis

Vertical TranslationFor any function y=f(x), adding any number k to

the function will cause the graph of the function to translate (shift) vertically k units.

Symbolically….y = f(x) + k

For each point on the graph, the y-value will change by k units while the x-value remains the same.

Symbolically…(x, y + k)

Vertical Translation y = f(x) + 24

2

-2

-4

-5

f x = x2

4

2

-2

-4

-5

f x = x2+2

4

2

-2

-4

-5

f x = x

4

2

-2

-4

-5

f x = x +2

4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = ex+2

Vertical Translation y = f(x) - 24

2

-2

-4

-5

f x = x2

4

2

-2

-4

-5

f x = x2-2

4

2

-2

-4

-5

f x = x

4

2

-2

-4

-5

f x = x -2

4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = ex-2

Horizontal TranslationFor any function y=f(x), replacing x with (x – h) in

the function will cause the graph of the function to translate (shift) horizontally h units.

Symbolically….y = f(x - h)

For each point on the graph, the x-value will change by h units while the y-value remains the same.

Symbolically…(x + h, y)

Horizontal Translation y = f(x-3)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = ex-34

2

-2

-4

-5

f x = x-34

2

-2

-4

-5

f x = x-3 2

Horizontal Translation y = f(x+3)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = x+3 2

4

2

-2

-4

-5

f x = x+3 4

2

-2

-4

-5

f x = ex+3

Vertical Stretch/CompressionFor any function y=f(x), multiplying any number a to

the function will cause the graph of the function to stretch/compress vertically by a factor of a.

Symbolically….y = a f(x)

For each point on the graph, the y-value will be multiplied by a while the x-value remains the same.

Symbolically…( x , a y )

Vertical Stretch y = 2 f(x)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = 2ex

4

2

-2

-4

-5

f x = 2x4

2

-2

-4

-5

f x = 2x2

Vertical Compression y = 1/2 f(x)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = 1

2 ex

4

2

-2

-4

-5

f x = 1

2 x4

2

-2

-4

-5

f x = 1

2 x2

Horizontal Stretch/CompressionFor any function y=f(x), replacing the x with bx in

the function will cause the graph of the function to stretch/compress horizontally by a factor of (1/b).

Symbolically….y = f( bx )

For each point on the graph, the x-value will be multiplied by (1/b) while the y-value remains the same.

Symbolically…( (1/b) x , y )

Horizontal Compression y = f ( 2x )4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = 2x 2

4

2

-2

-4

-5

f x = 2x4

2

-2

-4

-5

f x = e2x

Horizontal Stretch y = f((1/2)x)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = 1

2 x 4

2

-2

-4

-5

f x = e

1

2 x

4

2

-2

-4

-5

f x = 1

2 x 2

Reflection about x-axisFor any function y = f(x), if the function

expression is multiplied by -1, the graph of the function will be a reflection over the x-axis.

Symbolically…y = - f(x)

For each point on the graph, the y-value will be multiplied by -1.

Symbolically…( x, -1 y )

Reflection about x-axis y = - f(x)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = -ex

4

2

-2

-4

-5

f x = - x4

2

-2

-4

-5

f x = -x2

Reflection about y-axisFor any function y = f(x), if the x-value is

multiplied by -1, the graph of the function will be a reflection over the y-axis.

Symbolically…y = f(-x)

For each point on the graph, the x-value will be multiplied by -1.

Symbolically…(-1 x, y )

Reflection about y-axis y = f(-x)4

2

-2

-4

-5

f x = x24

2

-2

-4

-5

f x = x4

2

-2

-4

-5

f x = ex

4

2

-2

-4

-5

f x = e-x

4

2

-2

-4

-5

f x = -x4

2

-2

-4

-5

f x = -x 2

AssignmentWorksheet – “Move the Monster”