Operations on Functions
Stretches and Shrinks
Stretches and Shrinks
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, y
1)
is a point on the graph of
f (x)
,
(x, y
2)
is a point on the graph of
2f (x)
, and
(x, y
3)
is a point
on the graph of
f (x)
, then
y
2 = 2y
1
and
y
3 =
y
1
.
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)
.
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
(x
1, y)
is a point on the
graph of
f (x)
,
(x
2, y)
is a point on the graph of
f (
x)
, and
(x
3, y)
is a point on the graph of
f (2x)
, then
x
2 = 2x
1
and
x
3 =
x
1
. 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)
.
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.
