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.

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Vertical Stretches

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

The amplitude of the graph of

*any* periodic function is one-half the
absolute value of the sum of the maximum and minimum values of the function.

####
Horizontal Stretches

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