1. For each function, evaluate f(0), f(1/2), and f(-2)
Warm up 9/24
For each function, evaluate f(0), f(1/2), and f(-2)
f(x) = x2 – 4x
f(x) = -2x + 1
If f(x) = -3x, find f(2x) and f(x-1)
If f(x) = -2x + 3, find f(-2x) and f(2x-1)

2. Answers f(0)=0, f(1/2)=-1.75, f(-2)=12 =1, =0, =5 -6x , -3x + 3
=1, =0, =5
-6x , -3x + 3
4x + 3, -4x + 5

3. Lesson 2.7 - Parent Functions
The Parent Function is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent functions.

4. Parent Functions Family Rule Domain Range constant f(x) = c y = c
Linear
f(x) = x
quadratic
f(x) = x2
y ≥ 0

5. Parent Functions Family Rule Domain Range Cubic f(x) = x3 Square Root

6. Identify the parent function and describe the transformation1. 2. 3.
f(x) = x2
Up 4
f(x) = x2 + 4
f(x) = x
Down 3
f(x) = x-3

7. Transformations
A translation is type of transformation where a graph is moved horizontally and/or vertically.
What is a translation?

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

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

10. Given the graph f(x)=(x-h)+k.
The graph moves horizontally (h) units and vertically (k) units
So f(x) = (x-h) + k
Up/down
Left/right
Opposite of h

11. Example 1: Example 2: If the pre-image (original) is f(x) = 2x,
Describe the translation of the image of f(x) = 2(x – 3)+ 4.
h = _____ which means _____________
k = _____ which means______________
3 units to the right 3
4 units up 4
Example 2:
Pre-image f(x) = 3x
Image f(x) = 3(x+2) - 3
Describe the translation.
left 2, down 3

12. Example 3: Write the new equation.

13. Example 4: Given f(x) = -4x.
A. Find f(x+5).
-4(x+5)
-4x – 20
B. Find f(x-1)+6.
-4 (x-1)+6
-4x
-4x + 10

14. Example 5: The pre-image is the blue function defined as y =x
What would be the equation of the red function?
What would be the equation of the green function?
y = x + 3
y = (x – 3) – 1

15. VERTICAL STRETCHING OR COMPRESSING
The graph of y = a f (x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is
A vertical stretch away from the x-axis if
a > 1;
2. A vertical compression toward the x-axis if
0 < a < 1.
If a < 0, the graph of f is first reflected in the
x-axis, then vertically stretched or compressed.

16. VERTICAL STRETCHING OR COMPRESSING
The graph of y = a f (x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is
A vertical stretch away from the x-axis if
a > 1;
2. A vertical compression toward the x-axis if
0 < a < 1.
If a < 0, the graph of f is first reflected in the
x-axis, then vertically stretched or compressed.

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

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

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

20. REFLECTION IN THE x-AXIS
The graph of y = – f (x) is a reflection of the graph of y = f (x) in the x-axis.
If a point (x, y) is on the graph of y = f (x), then the point (x, –y) is on the graph of
y = – f (x).

21. REFLECTION IN THE x-AXIS

22. REFLECTION IN THE y-AXIS
The graph of y = f (–x) is a reflection of the graph of y = f (x) in the y-axis.
If a point (x, y) is on the graph of y = f (x), then the point (–x, y) is on the graph of
y = f (–x).

23. REFLECTION IN THE y-AXIS

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

25. The graph of
Is like the graph of
FLIPPED 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. 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

30. 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.