We can also
stretch and
shrink the graph of a function. To stretch or
shrink the graph in the
y direction, multiply or divide the output by a
constant.
2f (x) is stretched in the
y direction by a factor of 2, and
f (x) is shrunk in the
y direction by a factor of 2 (or stretched
by a factor of
). Here are the graphs of
y = f (x),
y = 2f (x),
and
y = x. Note that if
(x, y1) is a point on the graph of
f (x),
(x, y2) is a point on the graph of
2f (x), and
(x, y3) is a point
on the graph of
f (x), then
y2 = 2y1 and
y3 = y1.
For example,
(3, 2) is on the graph of
f (x),
(3, 4) is on the graph of
2f (x), and
(3, 1) is on the graph of
f (x).
Graphs of f (x), 2f (x), and f (x)
To stretch or shrink the graph in the
x direction, divide or multiply the
input by a constant. As in translating, when we change the input, the function changes
to compensate. Thus, dividing the input by a constant stretches the function in
the
x direction, and multiplying the input by a constant shrinks the function
in the
x direction.
f (x) is stretched in the
x direction by a
factor of 2, and
f (2x) is shrunk in the
x direction by a factor of 2 (or
stretched by a factor of
frac12). Here is a graph of
y = f (x),
y = f (x), and
y = f (2x). Note that if
(x1, y) is a point on the
graph of
f (x),
(x2, y) is a point on the graph of
f (x), and
(x3, y) is a point on the graph of
f (2x), then
x2 = 2x1 and
x3 = x1. For example,
(- 2, 5) is on the graph of
f (x),
(- 4, 5) is
on the graph of
f (x), and
(- 1, 5) is on the graph of
f (2x).
Graphs of f (x), f (x), and f (2x)
We can understand the difference between altering inputs and altering outputs by
observing the following:
If g(x) = 3f (x): For any given input, the output iof g is three times the
output of f, so the graph is stretched vertically by a factor of 3.
If g(x) = f (3x): For any given output, the input of g is one-third the input
of f, so the graph is shrunk horizontally by a factor of 3.
