1. Stretching, Shrinking, and Reflecting
9-3: Transformations
Stretching, Shrinking, and Reflecting

2. y =|x|

3. y = - |x|
flip

4. y = 2|x|
stretch
stretch 2
stretch

5. y = - 2|x|
flip
stretch
stretch 2

6. y = |x|
shrink
shrink
1/2

7. y = |x|
flip
shrink
shrink
1/2

8. For y = c f(x) (multiply function by a constant)
If |c| > 1 (i.e. not a fraction)
graph is stretched vertically (opens more quickly/thinner)
y-value is multiplied by the constant
If |c| < 1 (i.e. a fraction)
graph is shrunk vertically (opens more slowly/wider)
y-value is multiplied by the constant
If c is negative, the graph is reflected across the x-axis. (y-values have opposite sign)

9. For y = f(cx) (multiply input value by a constant)
Suppose (6, 6) is a point on the graph of y = |x|. To get out a value of 6, we need to input 6.
Suppose we now have y = |2x|. What x-value must be input to get the same output of 6? We need to input 3, so the new ordered pair is (3, 6). Notice the input is changed by multiplying the x-value by the reciprocal of 2.
Suppose we now have y = |1/2x|. What x-value must be input to get the same output of 6? We need to input 12, so the new ordered pair is (12, 6). Notice the input is changed by multiplying the x-value by the reciprocal of 1/2.

10. Use the graph of f(x) to graph g(x) and h(x).
(0, 3)
(-4, 2)
(4, 1)
(1, -1)
(-2, -2)

11. stretch stretch 2 2 g(x) = 2[f(x)] (0, 6) (-4, 4) (0, 3) (4, 2)
(-4, 2)
stretch
stretch
(4, 1) 2 2
(1, -1)
(1, -2)
(-2, -2)
(-2, -4)
For each input, the output of g is twice the output of f, so the graph of g is stretched vertically by a factor of 2

12. 1/2 1/2 shrink shrink h(x) = f(2x) (0, 3) (0,3) (-4, 2) (-2, 2) (2, 1)
(4, 1)
shrink
shrink
(1, -1)
(1/2, -1)
(-2, -2)
(-1, -2)
For any given output, the input of h is one-half the input of f, so the graph of h is shrunk horizontally by a factor of ½.

13. For y = f(cx)
If |c| > 1
graph is shrunk horizontally
x-value is multiplied by the reciprocal of c.
If |c| < 1 (i.e. a fraction)
graph is stretched horizontally
x-value is multiplied by the reciprocal of c.
If c is negative, the graph is reflected across the y-axis.

14. Rule of Thumb
When the input is multiplied by a constant, there is a horizontal stretch or shrink.
When the output is multiplied by a constant, there is a vertical stretch or shrink.

15. Given y = f(x), sketch y = f(x)
flip
1/3
shrink
shrink
1/3

16. Given y = f(x), sketch y = - 3f(x)
flip
stretch 3
stretch 3