HOW TO SKETCH THE GRAPH OF SINUSOIDAL FUNCTION

The general form of a sinusoidal function :

y = a sin [k(x - b)] + N

To sketch the graph of a sinuoidal function, first we have to sketch the graph of a basic sine function, which is

y = sin (x)

We have one pattern of y = sin (x) from x = 0 to x = 2π.

Then, we have to apply the following transformations one by one in the given order to get the graph of the given sinusoidal function.

vertical strech/compression (a)

horizontal stretch/compression (k)

reflection over x-axis (if a < 0)

horizontal shift (b)

vertical shift (N)

Example :

Sketch the graph of

Solution :

Sketch the graph of y = sin (x).

When x = 0,

y = sin (0)

y = 0

y = 1

When x = π,

y = sin (π)

y = 0

y = -1

When x = 2π,

y = sin (2π)

y = 0

sinusoidalfunction2.png

We get the following points from the above table.

Plot the above points on a xy-plane and sketch the graph of y = sin (x).

sinusoidalfunction3.png

Comparing

y = a sin [k(x - b)] + N

and

we get

a = 2  and  k = 2

Vertical Stretch/Compression :

a = 2 refers to vertical stretch by a factor of 2.

That is, at each point on the above graph, multiply the y-coordinate by 2.

sinusoidalfunction4.png

Horizontal Stretch/Compression :

k = 2 refers to horizontal compression by a factor of 2.

That is, at each point on the above graph, multiply the x-coordinate by ½.

sinusoidalfunction5.png

Horizontal Shift :

sinusoidalfunction6.png

The above is the graph of the given sinusoidal function. In the given sinusoidal function, since there is no value at the place of N, there is no vertical shift.

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