Trig Graphing Vocabulary

A sine wave, or sinusoid, is the graph of the sine function in trigonometry. In addition to mathematics, this function also occurs in other fields of study such as science and engineering. This wave pattern also occurs in nature as seen in ocean waves, sound waves and light waves. Even average daily temperatures for each day of the year resemble this wave. The term sinusoid was first use by Scotsman Stuart Kenny in 1789 while observing the growth and harvest of soybeans.

Let's see what vocabulary is needed to discuss sinusoids and other trigonometric graphs.

amplitude:

The amplitude of a sinusoidal function is one-half of the positive difference between the maximum and minimum values of a function.

Amplitude is the magnitude (height) of the oscillation (wave) of a sinusoidal function. Sometimes it is referred to as the "peak from center" of the graph.

The maximum of y = sin x is 1. The minimum is -1. The amplitude is 1/2(1 - (-1) ) = 1.

Note: while vertical shifts alter the maximum and minimum values of a function, they do not alter the amplitude. Also horizontal shifts (phase shifts) do not affect the amplitude.

period:

A periodic function is an oscillating (wave-like)function which repeats a pattern of y-values at regular intervals. One complete repetition of the pattern is called a cycle. The period of a function is the horizontal length of one complete cycle.

In functional notation we could say: The period is the smallest value of k in a function f for which there exists some constant k such that
f (t) = f (t + k) for every number t in the domain of f.

This sine curve, y = sin x, has a period of ,
the horizontal length of one complete cycle.

frequency:

The frequency of a trigonometric function is the number of cycles it completes in a given interval. This interval is generally

radians (or 360º) for the sine and cosine curves.

In terms of a formula:

This sine curve, y = sin x, completes 1 cycle in the interval from 0 to radians. Its frequency is 1 in the interval .

sinusoidal curve:

A sinusoidal curve is the graph of the sine function in trigonometry.

A sinusoid is the name given to any curve that can be written in the form

The example at the right is

The cosine wave is also said to be sinusoidal because of the relationship

which makes it the sine wave with a phase shift. The cosine function, which appears to have a "head start" on the sine function, is often said to lead the sine function (or the sine function lags the cosine function).

phase shift:

Phase shift is the measure of horizontal shifting. If the phase shift is positive, the horizontal shift is to the right. If the phase shift is negative, the horizontal shift is to the left.

From the sinusoidal equation,

the phase shift is obtained by determining the change being made to the x value. The phase shift is C.

Remember that the expression (x - C) from the equation will look like (for example):
   • (x - 2) where 2 is a positive value being
      subtracted, when the shift is to the right.
   • (x + 2) where 2 is a negative value being
      subtracted, when the shift is to the left.