## 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:

- 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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