SINUSOIDAL FUNCTION

The general form of a sinusoidal function :

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

We can determine following graph properties for plotting one period of sine :

(i) Amplitude = |a| #Note if a < 0, there is a reflection over the x-axis.

(ii) Phase shift/horizontal shift by b #if -b, then right shift by b units. if +b, then left shift by b units.

This is an horizontal stretch or shrink of original sine.

(iii) Plotting Domain for one Period :

(iv) x-values for Plotting :

Doing this ensures that you are computing sine of easy values :

(v) Vertical shift : If +N, then we shift the the entire graph N units up. If -N, we shift the entire graph N units down.

Note :

We have the the patterns for the y-values of sin (x).

When there is NO reflection about x-axis :

0, 1, 0, -1, 0

When there is reflection about x-axis :

0, -1, 0, 1, 0

Problem 1 :

y = 4 sin [2(x - π)] + 3

Answer the following questions for the above sinusoidal function.

1. Determine the amplitude of the given function.

2. Is there reflection about the x-axis?

3. Is there a phase shift? If yes, what is this shift?

4. What is the period of the given function :

5. Determine the interval that you should plot over to ensure you plot an entire period.

6. Is there any vertical shift?

Solution :

Comparing

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

and

y = 4 sin [2(x - π)] + 3,

we get

a = 4, k =2, b = π and N = 3

Part 1 :

Amplitude = |a|

= |4|

Part 2 :

a = 4 > 0

NO reflection about the x-axis

Part 3 :

Yes, there is a phase shift or horizontal shift

x - b = x - π

π units to the right

Part 4 :

Part 5 :

Part 6 :

Yes, there is a vertical shift

+N = +3

Vertical shift by 3 units up

Problem 2 :

Solution :

Amplitude = 2

NO reflection about the x-axis

We get the following points from the above table.

Plot the above points on a xy-plane and sketch the graph of the given sinusoidal function.

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SINUSOIDAL FUNCTION

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