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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(3x). 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.

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.

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 2Π. Below are pictured the sine curve, along with the
following functions, each a horizontal stretch of the sine curve:
y = f (x) = sin(2x) and y = f (x) = sin().