Question

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x > 0. Round answers to two decimal places if necessary.

y = 3sin(8(x + 4)) + 5

Answer 1

Periodic function:

A periodic function is a function that repeats its value over a fixed interval. The fixed interval is called as period of the function. Mathematically it is represented as f(x + P) = f(x).

Where P is the period of the function f(x)

 

Sine function:

A sine function is a trigonometric function representing ratio between opposite side to hypotenuse. It is an odd function with period 2π.

 

A general form of sine function is y = Asin(Bx – C) + D.

Here,

Amplitude is |A|

Period is P = 2π/|B|.

Phase shift is C/B and

Vertical shift is D.

 

In this case the sine function is f(x) = 3sin(8x + 32) + 5.

Comparing with the general form y = Asin(Bx – C) + D,

The amplitude is |A| = 3.

Since B = 8, the period is

P = 2π/|B|

P = π/4

 

Observe that C = -32, so the phase shift is

C/B = -32/8

       = -4

 

That is, phase shift is 4 units to the left.

Hence, horizontal shift is -4.

Next, observe that D = 5, so the graph shift 5 units upward.

Hence the midline is y = 5.

 

The graph of sine function f(x) = 3sin(8x + 32) + 5 is shown below:

 

From the plot we find that maximum and minimum is obtained at

Max(x, y) = (0.123, 8)

Min(x, y) = (0.516, 2)

From the plot we find that maximum and minimum is obtained at

Max(x, y) = (0.123, 8)

Min(x, y) = (0.516, 2)