## 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:

• Amplitude $$= | a |$$
• Period $$= 2\pi/ b$$
• Phase shift $$= c / b$$
• Vertical shift $$= d$$

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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