1. Transformations of functions

2. Vocabulary Transformation: moves the graph up or down, left or right,
stretches or shrinks it, or flips it.

3. Vocabulary Translation: a transformation that shifts the graph
horizontally or vertically

4. Your turn: 1. What is the difference between the two graphs?
Graph the following on your calculator:
Without deleting the previous equation, graph:
1. What is the difference between the two graphs?

5. Your turn: 2. What is the difference between the two graphs?
Keep the following in your calculator:
Change the second equation to:
2. What is the difference between the two graphs?

6. Your turn: 3. What does adding or subtraction “k” do to the function?
We call this a vertical translation (or “shift”)

7. Your turn: 4. What is the difference between the two graphs?
Enter the following on your calculator:
“2nd” “0” (catalog) then “enter”
Without deleting the previous equation, enter the following (close the parentheses before you add ‘5’)
4. What is the difference between the two graphs?

8. Your turn: 5. What is the difference between the two graphs?
Keep the following in your calculator:
Change the second equation to:
5. What is the difference between the two graphs?

9. Your turn: 6. What does adding or subtraction “k” do to the function?
We call this a vertical translation (or “shift”)

10. Does this work for every function?
Let’s try another function that we’ll learn about later.
Graph the following:
Without deleting the previous equation, graph the following
(make sure to close the parentheses before you add 5)

11. Vocabulary
Parent Function: the simplest function in a family of functions.
Parent of the linear function family:
Parent of the parabola function family:
Parent of the absolute value function family:

12. Vocabulary
Parent Function: the simplest function in a family of functions.
These are the only three we have learned so far.
We will learn several more this year. In the next chapter
will will learn the following:

13. Your turn: 7. What does adding a number to the parent function
do to the graph of the parent function?

14. Your turn: 8. What is the difference between the
Enter the following on your calculator:
8. What is the difference between the
graphs of these two equations?
Replace the previous with the following:
9. What is the difference between the
graphs of these two equations?

15. Your turn: 10. Enter the following on your calculator:
What is the difference between the
graphs of these two equations?
11. Replace the previous with the following:
What is the difference between the
graphs of these two equations?

16. Your turn:
12. What does replacing ‘x’ with ‘x ± h’ do to the parent function?
13. Without graphing, describe how the parent of the parabola
function family been transformed by replacing
‘x’ with ‘(x + 2)’ and then adding ‘3’ to the function?

17. Your turn:
14. Enter the following on your calculator (make sure you put
a ‘2’ into the “y-editor” before you enter the “abs( “ function).
What is the difference between the
graphs of these two equations?
15. Replace the previous with the following:
What is the difference between the
graphs of these two equations?

18. Your turn:
16. Enter the following on your calculator (make sure you put
a ‘2’ into the “y-editor” before you enter the “abs( “ function).
What is the difference between the
graphs of these two equations?
17. Replace the previous with the following:
What is the difference between the
graphs of these two equations?

19. Your turn:
18. What “transformation” occurs to the parent function when
you multiply the parent function by a number greater than ‘1’?
19. What “transformation” occurs to the parent function when
you multiply the parent function by a number less than ‘1’?

20. Summary Vertical shift Horizontal shift Vertical stretch
What does adding or subtraction “k” do to the parent function?
Vertical shift
What does adding or subtraction “h” do to the parent function?
Horizontal shift
What does multiplying by ‘a’ do to the parent function?
Vertical stretch

21. The “Transformation Equation”
shift
stretch
shift
Vertical stretch
Horizontal shift
Vertical shift

22. Without graphing predict how the graph of the following
function is going to be different from the parent function.
Vertically stretched by a factor of 2,
shifted left 4 and up 5
Vertically stretched by a factor of 2/3,
shifted right 1 and down 2
Vertically stretched by a factor of 0.25,
shifted left 1 and down 6
Vertically stretched by a factor of 37,
shifted right 20
shifted down 1/2

23. Your turn:
Describe the transformation of the parent function.
20.
21.
22.

24. Your turn:
Without using your calculator, draw both the parent function
and an approximate graph of the following (on the same plot)
23.
24.

25. Your turn: 25. What is the difference between them?
Graph the following two equations on your calculator:
25. What is the difference between them?
Graph the following two equations on your calculator:
26. What is the difference between them?

26. Reflections across the ‘x’ axis
Changing ‘y’ to ‘– y’
reflects the point
across the x-axis.
(2, 3)
y = f(x)  y = -f(x)
Multiplying the function
by (-1)
reflects the function
across the x-axis.
(2, -3)

27. Your turn: Vocabulary:
26. What does multiplying the parent function by (-1) do
to the parent function?
Vocabulary:
Reflection: a mirror image of the graph across a boundary line.

28. Reflections across the ‘x’ and ‘y’ axis
Changing ‘x’ to ‘– x’
reflects the point
across the y-axis.
(-2, 3)
(2, 3)
y = f(x)  y = f(-x)
Replacing and input value
by (-1) times the imput value
reflects the function
across the y-axis.

29. Your turn:
27. What does replacing ‘x’ in the parent function with (-x) do
to the parent function?
(Linear function)
For the other 2 functions we’ve learned (square function and
absolute value function) a reflection across the y-axis looks
just like the orginal function.

30. Vocabulary Reflection: makes “mirror-image” of the graph
across some line.

31. Vocabulary rise run Absolute Value Function: A function of the form:
Slope
(h, k)
rise
run
Vertex (h, k)

32. Comparing the General Equation to a specific example.
Slope = ?
vertex = ?
Slope = ?
vertex = ?

33. Your Turn:
Find the slope and vertex of each graph.
28.
29.
30.

34. More Transformations f(x): (0, 0) (2, 3) y = f(x + 3) – 4 (0, 0) x - 3
shift
stretch
shift
f(x): (0, 0) (2, 3)
y = f(x + 3) – 4
(0, 0)
x - 3
(-3, 0)
(-3, 0)
y – 4
(-3, -4)
(2, 3)
2 - 3
(-1, 0)
(-1, 3)
3 – 4
(-1, -1)
f(x + 3) – 4: (-3, -4) (-1, -1)

35. More Transformations f(x): (2, 4) y = 5 * f(x + 3) – 4 (2, 4) x - 3
shift
stretch
shift
f(x): (2, 4)
y = 5 * f(x + 3) – 4
(2, 4)
Left 3
x - 3
(-1, 4)
Vertical stretch
factor of 5
(-1, 4)
5 * y
(-1, 20)
(-1, 8)
y – 4
(-1, 16)
Down 4
5 *f(x + 3) – 4: (-1, 16)

36. Your Turn: Graph: f(x+2) – 1
31. Given: f(x) is made up of a segment with
end points (2, 3) and (-4, 7)
Graph: f(x+2) – 1

37. Your Turn: 32. Graph without using your calculator:

38. Effect of ‘h’ and ‘k’ on the Absolute Value Function
h = 0, k = 0 1
h = 1, k = 1

39. Effect of ‘a’ on the Absolute Value Function
Sign change
causes a “Reflection”
As the value of ‘a’ changes, the shape of the graph changes.

40. Effect of ‘h’ and ‘k’ on the Absolute Value Function
(0, 0)
(0, 0)
(2, 1)
(-3, -2)