9-3: transformations stretching, shrinking, and reflecting
TRANSCRIPT
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)
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