9-3: Transformations

Stretching, Shrinking, and Reflecting

y =|x|

y = - |x|

y = 2|x|

y = - 2|x|

y = |x|1

2

y = |x|1

2

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)

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.

For y = f(cx)(multiply input value by a constant)

(-4, 2)

(-2, -2)

(0, 3)

(1, -1)

(4, 1)

Use the graph of f(x) to graph g(x) and h(x).

f(x)

g(x) = 2[f(x)]

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

(-4, 4)

(0, 6)

(-2, -4)

(1, -2)

(4, 2)(-4, 2)

(-2, -2)

(0, 3)

(1, -1)

(4, 1)

h(x) = f(2x)

(-2, 2)

(0,3)

(-1, -2)

(1/2, -1)

(2, 1)

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.

(-4, 2)

(-2, -2)

(0, 3)

(1, -1)

(4, 1)

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.

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.

Rule of Thumb

Given y = f(x), sketch y = f(x) 1

3

Given y = f(x), sketch y = - 3f(x)