Transformations of Transformations of Parent Functions Parent Functions
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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)
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