Another common way that the graphs of trigonometric
functions are altered is by
stretching the graphs. Stretching a graph involves introducing a
coefficient into the function, whether that coefficient fronts the equation as
in
*y* = 3 sin(*x*)
or is acted upon by the trigonometric function, as in
*y* = sin(3*x*)
. Though both of the given examples result in stretches of the graph
of
*y* = sin(*x*)
, they are stretches of a certain sort. The first example
creates a vertical stretch, the second a horizontal stretch.

To stretch a graph vertically, place a coefficient in front of the function.
This coefficient is the amplitude of the function. For example, the
amplitude of
*y* = *f* (*x*) = sin(*x*)
is one. The amplitude of
*y* = *f* (*x*) = 3 sin(*x*)
is three. Compare the two graphs below.

Figure %: The sine curve is stretched vertically when multiplied by a coefficient

To horizontally stretch the sine function by a factor of
*c*
, the function must be
altered this way:
*y* = *f* (*x*) = sin(*cx*)
. Such an alteration changes the
period of the function. For
example, continuing to use sine as our representative trigonometric function,
the period of a sine function is
, where
*c*
is the coefficient of
the angle. Usually
*c* = 1
, so the period of the
sine function is
2*Π*
. Below are pictured the sine curve, along with the
following functions, each a horizontal stretch of the sine curve:
*y* = *f* (*x*) = sin(2*x*)
and
*y* = *f* (*x*) = sin()
.

Figure %: The sine function is stretched horizontally when the angle is
multiplied by a scalar