1. Transformations

2. Key Vocabulary
Images
Pre-images
Dilations
Translations

3. The graphs of many functions are transformations of the graphs of very basic functions.
Example: The graph of y = x is the graph of y = x2 shifted upward three units. This is a vertical shift. x y -4 4 -8 8
y = x2 + 3
y = x2
The graph of y = –x2 is the reflection of the graph of y = x2 in the x-axis.
y = –x2

4. Vertical Shifts
If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units.
If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units. x y
f (x) + c
f (x) +c
f (x) – c -c

5. Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4.y -4 4 8
g(x) = |x| + 3
f (x) = |x|
h(x) = |x| – 4

6. Graphing Utility: Sketch the graphs given by–5 5 4 –4

7. Horizontal Shifts
If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. x y -c +c
If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units.
y = f (x + c)
y = f (x)
y = f (x – c)

8. Example: Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = (x + 4)3 .y -4 4
f (x) = x3
h(x) = (x + 4)3
g(x) = (x – 2)3

9. Graphing Utility: Sketch the graphs given by–5 6 7 –1

10. Example: Graph the function using the graph of .
First make a vertical shift 4 units downward.
Then a horizontal shift 5 units left. -4 x y 4 -4 y 4 x
(4, 2)
(0, 0)
(4, –2)
(–1, –2)
(0, – 4)
(– 5, –4)

11. The graph of a function may be a reflection of the graph of a basic function.x y
The graph of the function y = f (–x) is the graph of y = f (x) reflected in the y-axis.
y = f (–x)
y = f (x)
y = –f (x)
The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis.

12. Example: Graph y = –(x + 3)2 using the graph of y = x2.
First reflect the graph in the x-axis.
Then shift the graph three units to the left. x y
– 4 4 -4 x y 4
y = x2
(–3, 0)
y = – (x + 3)2
y = – x2

13. Vertical Stretching and Shrinking
If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c.
If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) shrunk vertically by c.
– 4 x y 4
y = x2
y = 2x2
Example: y = 2x2 is the graph of y = x2 stretched vertically by 2.
is the graph of y = x2 shrunk vertically by .

14. Dilation: Horizontal Stretching and Shrinking
If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by c.
If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by c.
y = |2x|
Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2.
- 4 x y 4
y = |x|
is the graph of y = |x| stretched horizontally by .

15. Graphing Utility: Sketch the graphs given by–5 5

16. Example: Multiple Transformations
Example: Graph using the graph of y = x3.
Graph y = x3 and do one transformation at a time.
- 4 4 8 x y
- 4 4 8 x y
Step 3:
Step 1: y = x3
Step 4:
Step 2: y = (x + 1)3
Example: Multiple Transformations