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4.4 Sinusoidal Functions

What is a sinusoidal function?

A sinusoidal function is a function that is like a sine function in the sense that the function can be produced by shifting, stretching or compressing the sine function. If necessary you might like to review the graphing shortcuts.

Exercise 4.4.1

Sketch the graph of \(y = 2 \sin x\) from \(x = 0\) to \(x = 2\pi\).

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What to notice about the graph

Notice that this graph is twice as tall as the sine graph. The amplitude of a sinusoidal function is the distance from the midline of the graph to the highest point of the graph. In exercise 4.4.1, the amplitude is (2\).

Exercise 4.4.2

Sketch the graph of \(y = \sin ( x – \pi )\) from \(x = 0\) to \(x = 2\pi\). You’ll have to shift the graph of \(y = \sin x\) to the right by \(\pi\) units. In this case, \(\pi\) is called the phase shift of the function.

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Exercise 4.4.3

Sketch the graph of \(y = \sin ( 2x )\) from \(x = 0\) to \(x = 2\pi\). You’ll have to compress the graph of \(y = \sin x\) by one-half horizontally. This changes the period from \(2\pi\) to \(2\pi/2\). So this sinusoidal function has a period of \(\pi\). To draw the graph from \(x = 0\) to \(x = 2\pi\) you’ll have to draw two complete cycles of the graph.

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Exercise 4.4.4

Sketch the graph of \(y = 1 + \sin x\) from \(x = 0\) to \(x = 2\pi\).

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The general sinusoidal function

As you see, \(y = 1 + \sin x\) merely raises the graph of sine one unit. We say that this sinusoidal has a vertical shift of 1.

In general, all of these types of alterations may occur in a sinusoidal function. The general form of a sinusoidal is:

\[ f ( x ) = a \sin ( bx – c ) + d, \text{ for } b > 0 \]

The following formulas will be useful:

Two complete cycles of the graph may be drawn between \(x = c / b – 2\pi / b\) and \(x = c / b + 2\pi / b\).

But what if b is not positive?

Suppose \(f ( x ) = 3 \sin ( -2 x +\pi / 2 ) + 3\), for example. What do we do?

We replace \(\sin ( -2 x + π / 2 )\) with \(-\sin ( 2 x - π / 2 )\) ! We can do this because, remember, sine is an odd function. This means that \(\sin ( - x ) = - \sin ( x )\) for all \(x\) in the domain of the function.

Thus, \(f ( x ) = - 3 \sin ( 2 x - \pi / 2 ) + 3\).

Exercise 4.4.5

Sketch two complete cycles of the graph of \(f ( x ) = - 3 \sin ( 2 x - \pi / 2 ) + 3\).

Specify the amplitude, vertical shift, period and phase shift.

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