math American year 11

1. PARENT FUNCTIONS
and
TRANSFORMATIONS
Presented by:
D. Marwa
Elsherif

2.  A function is described as mathematical
machine that takes input number “x” and
computes output numbers f(x). We then
describe the input numbers “x” as being
the Domain of the function, while the
output numbers f(x) are the Range of
the functions.
•What is a function?

3. •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

5. •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

6. •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

7. •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

8. 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

9. •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  a x
a 1

10. •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  a x

0  a 1

11. •Examples
 Graph the parent function:
 Graph the transformations:
 Stretch:
y’s are 3 times bigger
 Shrink:
y’s are half as big

y  3 x

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

12. •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

13. •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

14. •Describe this transformation

y  4 x  2  5

15. •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)

16. Name That
Parent
Function!

17. WRITE THE EQUATION
OF THE PARENT
FUNCTION.

18. Name That Parent Function!

20. Name That Parent Function!

22. Name That Parent Function!

24. Name That Parent Function!

26. Name That Parent Function!

28. Name That Parent Function!

30. Name That Parent Function!

32. Name That Parent Function!

35. Name That Parent Function!

36. Constant Function

37. Name That Parent Function!

38. Square Root Function

39. Name That Parent Function!

40. Quadratic Function

41. Name That Parent Function!

42. Cube Function

43. Name That Parent Function!

44. Linear Function

45. Name That Parent Function!

46. Cube Root Function

47. • Presenter:
D. Marwa H. Elsherif