Trigonometry: Graphs
Vertical and Horizontal Stretches
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.
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(
)
.
