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
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).
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.
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.
Horizontal Shift :
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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