Transformations of Parent Transformations of Parent FunctionsFunctions

What is a parent function?A parent function is the simplest form of a function.

◦Examples inculde: (line with slope 1 passing through origin)

(a V-graph opening up with vertex at origin)

(a U-graph opening up with vertex at origin)

y x

y x

y x 2

TransformationsTransformations are shifts

up/down, shifts left/right, or a change in size.

Ex:Parent Transformations

y x

y x 26

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

y x 26

4

2

-2

-4

-6

-5 5

Horizontal and Vertical TranslationsIf the parent function is , then …

◦ shift up k units◦ shift down k units◦ shift right h units◦ shift left h units

y x

y x k

y x k

y x h

y x h

Example One

Graph:

◦Remember what the parent looks like and go from there…

◦…the parent shifted right 3 units!

y x 3

6

4

2

-2

-4

-6

-5 5

What would look like?

It would shift 2 units left and 1 unit down.

Example Two

Graph:

The parent shifted down 3 units.

y x 36

4

2

-2

-4

-6

-5 5

y x 2 1

StretchesA stretch multiples all the y-values by the same factor greater than one, stretching the graph vertically (making it skinnier than the parent!)

If the parent function is , then a stretch would be provided .

y x

y a x

a 1

ShrinksA shrink reduces all the y-values by the same factor less than one, compressing the graph vertically (making it fatter than the parent!)

If the parent function is , then a shrink would be when provided.

y x

y a x

0 a 1

ExamplesExamplesGraph the parent function:Graph the transformations:

◦Stretch: y’s are 3 times bigger

◦Shrink: y’s are half as big

y 3 x

y 12x

y x

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

stretchparent

6

4

2

-2

-4

-6

-5 5

shrink

ReflectionsA reflection over the x-axis changes the y-values to their opposites…(i.e. the parent flips!)

If the parent function is , then a reflection would be .

The reflection is over the x-axis ( )

y x

y x

ExampleGraph the parent function:Graph the reflection :

y x

6

4

2

-2

-4

-6

-5 5

y x

6

4

2

-2

-4

-6

-5 5

reflection

parent

Describe this transformation

y 4 x 2 5

Describe this transformation - Solution

y 4 x 2 5

6

4

2

-2

-4

-6

-5 5

Shrink (from 4)

Shifted right two units

(from -2)

Shifted down 5 units

(from -5)