Parent functions and Transformations

Parent Functions and Transformations

1. Transformations of Parent Transformations 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=x
2

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
-5 555
45 2
5
4 2
5
Shrink (from 4)
Shifted right two units
(from -2)
Shifted down 5 units
(from -5)