1. Transformation of Functions

2. Three kinds of Transformations
Horizontal and Vertical Shifts
A function involving more than one transformation can be graphed by performing transformations in the following order:
Vertical shifting
Horizontal shifting
Reflecting
Stretching or shrinking
Expansions and Contractions
Reflections

3. How to recognize a vertical shift.
Basic function
Basic function
Transformed function
Transformed function
Recognize transformation
Recognize transformation
The inside part of the function
remains the same
The inside part of the function
remains the same
2 is THEN subtracted
15 is THEN subtracted
Original function
Original function

4. How to recognize a vertical shift.
Basic function
Transformed function
Recognize transformation
The inside part of the function
remains the same
3 is THEN added
Original function

5. The effect of the transformation on the graph
Replacing function with function – number SHIFTS the basic graph number units down
Replacing function with function + number SHIFTS the basic graph number units up

6. The graph of
Is like the graph of
SHIFTED 3 units up

7. The graph of
Is like the graph of
SHIFTED 2 units down

8. How to recognize a horizontal shift.
Basic function
Basic function
Transformed function
Transformed function
Recognize transformation
Recognize transformation
The inside part of the function
The inside part of the function
has been replaced by
has been replaced by

9. How to recognize a horizontal shift.
Basic function
Transformed function
Recognize transformation
The inside part of the function
has been replaced by

10. The effect of the transformation on the graph
Replacing x with x – number SHIFTS the basic graph number units to the right
Replacing x with x + number SHIFTS the basic graph number units to the left

11. The graph of
Is like the graph of
SHIFTED 2 units to the right

12. The graph of
Is like the graph of
SHIFTED 3 units to the left

13. How to recognize a vertical reflection.
Basic function
Transformed function
Recognize transformation
The inside part of the function remains the same
The function is then multiplied by -1
Original function
The effect of the transformation on the graph
Multiplying function by FLIPS the basic graph vertically

14. The graph of
Is like the graph of
FLIPPED vertically

15. How to recognize a horizontal reflection.
Basic function
Basic function
Transformed function
Transformed function
Recognize transformation
Recognize transformation
The inside part of the function
The inside part of the function
has been replaced by
has been replaced by
The effect of the transformation on the graph
Replacing x with -x FLIPS the basic graph horizontally

16. The graph of
Is like the graph of
FLIPPED horizontally

17. reflects about the x -axis
We know what the graph would look like if it was from our library of functions.
moves up 1
Graph using transformations
reflects about the x -axis
moves right 2

18. How to recognize a horizontal expansion or contraction
Basic function
Basic function
Transformed function
Transformed function
Recognize transformation
Recognize transformation
The inside part of the function
The inside part of the function
Has been replaced with
Has been replaced with

19. How to recognize a horizontal expansion or contraction
Basic function
Transformed function
Recognize transformation
The inside part of the function
Has been replaced with

20. The effect of the transformation on the graph
Replacing x with number*x
CONTRACTS
the basic graph horizontally if number is greater than 1.
Replacing x with number*x
EXPANDS
the basic graph horizontally if number is less than 1.

21. The graph of
Is like the graph of
CONTRACTED 3 times

22. The graph of
Is like the graph of
EXPANDED 3 times

23. How to recognize a vertical expansion or contraction
Basic function
Basic function
Transformed function
Transformed function
Recognize transformation
Recognize transformation
The inside part of the function
remains the same
The inside part of the function
remains the same
2 is THEN multiplied
4 is THEN multiplied
Original function
Original function

24. The effect of the transformation on the graph
Replacing function with number* function
EXPANDS
the basic graph vertically if number is greater than 1
Replacing function with number*function
CONTRACTS
the basic graph vertically if number is less than 1.

25. The graph of
Is like the graph of
EXPANDED 3 times vertically

26. The graph of
Is like the graph of
CONTRACTED 2 times vertically

27. g(x)
Write the equation of the given graph g(x). The original function was f(x) =x2
(a)
(b)
(c)
(d)

28. Example

29. Summary of Graph Transformations
Vertical Translation:
y = f(x) + k Shift graph of y = f (x) up k units.
y = f(x) – k Shift graph of y = f (x) down k units.
Horizontal Translation: y = f (x + h)
y = f (x + h) Shift graph of y = f (x) left h units.
y = f (x – h) Shift graph of y = f (x) right h units.
Reflection: y = –f (x) Reflect the graph of y = f (x) over the x axis.
Reflection: y = f (-x)
Reflect the graph of y = f(x) over the y axis.
Vertical Stretch and Shrink: y = Af (x)
A > 1: Stretch graph of y = f (x) vertically by multiplying each ordinate value by A.
0 < A < 1: Shrink graph of y = f (x) vertically by multiplying each ordinate value by A.
Horizontal Stretch and Shrink: y = Af (x)
A > 1: Shrink graph of y = f (x) horizontally by multiplying each ordinate value by 1/A.
0 < A < 1: Stretch graph of y = f (x) vertically by multiplying each ordinate value by 1/A.