Among the variations on the graphs of the trigonometric functions are shifts--both horizontal and vertical. Such shifts are easily accounted for in the formula of a given function.

Take function *f*, where *f* (*x*) = sin(*x*). The graph of *y* = sin(*x*) is seen
below.

Figure %: The Graph of sine(x)

To shift such a graph vertically, one needs only to change the function to
*f* (*x*) = sin(*x*) + *c*, where *c* is some constant. Thus the
$y$-coordinate of the graph, which was previously
sin(*x*), is now sin(*x*) + 2. All values of *y* shift by two. The graph of
*y* = *f* (*x*) = sin(*x*) + 2 looks like this:

Figure %: By adding a constant to a function, like sine, the graph is shifted
vertically

To shift a graph horizontally, a constant must be added to the function within
parentheses--that is, the constant must be added to the
angle, not the whole function. Compare the
to the graph of
*y* = *f* (*x*) = sin(*x* + ).

Figure %: Horizontal shift

The graph of sine is shifted to the left by units. Does it look familiar? The graph is the same as the cosine graph. Sine and cosine are both periodic functions, and have the same domain and range. They are separated only by a phase shift: . This is the constant that must be added to create the necessary horizontal shift to make the graphs directly overlap each other. Because the cosine graph is only a phase shift of the sine graph, one or the other can be used to model motion, but both are not necessary. Most often sine is used.

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