This document discusses transformations of parent functions. It defines a parent function as the simplest form of a function, such as y=x, y=x^2, etc. Transformations include horizontal and vertical shifts which move the graph left, right, up or down; stretches which multiply the y-values making the graph skinnier; shrinks which reduce the y-values making the graph fatter; and reflections which flip the graph over the x-axis. Examples are provided to demonstrate how to graph different transformations of common parent functions. The document concludes by describing the transformation y=4x-2-5 as a shrink by a factor of 4, a right shift of 2 units, and a down shift of 5

1. Transformations of ParentTransformations of Parent
FunctionsFunctions

2. 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=x2

3. Transformations
Transformations are shifts up/down,
shifts left/right, or a change in size.
Ex:
Parent Transformations
y=x y=x+2
6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
y=x+2
6
4
2
-2
-4
-6
-5 5

4. Horizontal and Vertical Translations
If 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

5. 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

6. 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−3
6
4
2
-2
-4
-6
-5 5
y=x+2−1

7. Stretches
A 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=ax a>1

8. Shrinks
A 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=ax
0<a<1

9. ExamplesExamples
Graph the parent function:
Graph the transformations:
◦ Stretch:
 y’s are 3 times bigger
◦ Shrink:
 y’s are half as big
y=3x
y =
1
2
x
y=x
6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
stretch
parent
6
4
2
-2
-4
-6
-5 5
shrink

10. Reflections
A 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

11. Example
Graph 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

12. Describe this transformation
y=4x−2−5

13. Describe this transformation - Solution
y=4x−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)
Shrink (from 4)
Shifted right two units
(from -2)
Shifted down 5 units
(from -5)