1. Warm-up: Graph y =|x| y = x2 y =
HW: p (5 skip e, 7-11 all, 15, odd, 29, 33, 35, 56, 57, 62, 63, 64)

2. Objective: Shift, reflect, and stretch graphs
combinations of functions
composition of functions

3. Parent Functions:

4. Translations – In General
The transformation of any function f(x) looks like:
a represents the vertical stretch and orientation
h represents the horizontal shift
k represents the vertical shift.

5. Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = x3 + 4 .

6. Shifting Graphs
Graph the function.
down 2

7. Shifting Graphs
Graph the function.
left 3
down 4

8. Shifting Graphs
Graph the function.
right 2
down 5

9. Reflection About the x-Axis
Y=f(x)
Y=-f(x)

10. Reflect over the x-axis
Reflecting Graphs
Graph the function.
Reflect over the x-axis

11. Reflect over the x-axis
Reflecting Graphs
Graph the function.
left 1
Reflect over the x-axis
up 2

12. Reflection About the y-Axis
Y=f(-x)
Y=f(x)

13. Reflect over the y-axis
Reflecting Graphs
Graph the function.
Reflect over the y-axis

14. Vertical Stretching and Shrinking Graphs

15. Vertical Stretching and Shrinking Graphs
Y=f(x)
Y=2f(x)
Y=1/2f(x)

16. three times the y-value
Nonrigid Transformations
Graph the function.
three times the y-value

17. left 1 down 2 one fourth the y-value Nonrigid Transformations
Graph the function.
left 1
down 2
one fourth the y-value

18. Combine like terms & put in descending order
The sum f + g
Combine like terms & put in descending order

19. The difference f - g Distribute negative
CAUTION: Make sure you distribute the – to each term of the second function
Distribute negative

20. Should put in descending order.
The product f • g
FOIL
Should put in descending order.

21. Nothing more you could do here. (If you can reduce these you should).
The quotient f /g
Nothing more you could do here. (If you can reduce these you should).

22. FOIL first and then distribute the 2
The Composition Function
This is read “f composition g”
FOIL first and then distribute the 2

23. You could multiply this out but since it’s to the 3rd power we won’t
This is read “g composition f”
You could multiply this out but since it’s to the 3rd power we won’t

24. This is read “f composition f”

25. The DOMAIN of the Composition Function
The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.
The domain of g is x  1

26. Summary: Shift, reflect, and stretch graphs combinations of functions
composition of functions

27. HW: p (5 skip e, 7-11 all, 15, odd, 29, 33, 35, 56, 57, 62, 63, 64)