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 = 2sin(3x − 21) + 4

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

 

In this case the sine function is

y = 2sin(3x – 21) + 4

 

Comparing with the general form, the amplitude is

|A| = 2

 

Since B = 3, the period is

P = 2π/|B|

   = 2π/3

 

Observe that C = 21, so the phase shift is

C/B = 21/3

        = 7

 

Hence phase shift is 7 units to the right.

Next, observe that D = 4, so the graph shifts 4 units upward. Hence the midline is

y = 4

 

Graph is plotted using Maple as follows

 

STEP 1:

Open a new document in Maple

 

STEP 2:

Enter following Maple formula

plot{2sin(3x -21) + 4, x = 0..4π/3}

 

STEP 3:

Graph is obtained as shown.

 

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

Max(x, y) = (1.22, 6)

Min(x, y) = (0.2, 2)

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

Max(x, y) = (1.22, 6)

Min(x, y) = (0.2, 2)