1. Unit 3 Functions (Linear and Exponentials)
Parent Functions and Transformations

2. Transformation of Functions
Recognize graphs of common functions
Use shifts to graph functions
Use reflections to graph functions
Graph functions w/ sequence of transformations

3. The following basic graphs will be used extensively in this section
The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

4. The identity function f(x) = x

5. The exponential function

6. The quadratic function

7. The square root function

8. The absolute value function

9. The cubic function

10. The rational function

11. Transformations happen in 3 forms: (1) translations (2) reflections (3) stretching.

12. Vertical Translation OUTSIDE IS TRUE! Vertical Translation
the graph of y = f(x) + k is the graph of y = f(x) shifted up k units;
the graph of y = f(x)  k is the graph of y = f(x) shifted down k units.

13. Horizontal Translation
INSIDE LIES!
Horizontal Translation
the graph of y = f(x  h) is the graph of y = f(x) shifted right h units;
the graph of y = f(x + h) is the graph of y = f(x) shifted left h units.

14. The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

15. Recognizing the shift from the equation, examples of shifting the function f(x) =
Vertical shift of 3 units up
Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

16. Use the basic graph to sketch the following:

17. Combining a vertical & horizontal shift
Example of function that is shifted down 4 units and right 6 units from the original function.

18. Use the basic graph to sketch the following:

19. Reflection about the x-Axis
The graph of y = - f (x) is the graph of y = f (x) reflected about the x-axis.
Reflection about the y-Axis
The graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis.

20. Stretching and Shrinking Graphs
Let f be a function and c a positive real number.
If a > 1, the graph of y = a f (x) is the graph of y = f (x) vertically stretched by multiplying each of its y-coordinates by a.
If 0 < a < 1, the graph of y = a f (x) is the graph of y = f (x) vertically shrunk by multiplying each of its y-coordinates by a.
g(x) = 2x2
f (x) = x2 10 9 8
h(x) =1/2x2 7 6 5 4 3 2 1 -4 -3 -2 -1 1 2 3 4

21. The big picture…

22. Sequence of Transformations
A function involving more than one transformation can be graphed by performing transformations in the following order.
1. Horizontal shifting
2. Vertical stretching or shrinking
3. Reflecting
4. Vertical shifting

23. Example
Use the graph of f(x) = x3 to graph g(x) = -2(x+3)3 - 4

24. Example Explain the difference in the graphs Vertical Shift Up 3 Units
Horizontal Shift Left 3 Units
Vertical Shift Up 3 Units

25. A combination If the parent function is Describe the graph of
The parent would be horizontally shifted right 3 units and vertically shifted up 6 units

26. If the parent function is
What do we know about
The graph would be vertically shifted down 5 units and vertically stretched two times as much.

27. What can we tell about this graph?
It would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.

28. Transformations of Exponential Functions

29. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each.

30. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each.

31. Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.
State the domain, range and the horizontal asymptote for each.

32. Graph using transformations and determine the domain, range and horizontal asymptote.B) A) D) C)