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.

f(x) = 3cos (6/5 x)

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)

 

Cosine function:

A cosine function is a trigonometric function representing ratio between adjacent sides to hypotenuse. It is an even function with period 2π.

 

A general form of cosine function is

y = Acos(Bx – C) + D

 

Consider the following cosine function:

f(x) = 3cos(6/5x)

 

Compare with the general form, the amplitude is

|A| = 3

 

Since B = 6/5, the period is

P = 2π/|B|

   = 5π/3

 

Since C and D are zero, there is no phase shift or vertical axis shift. Hence the graph decreases from the highest and decreases towards right.

 

Since there is not vertical shift, the midline is x-axis

y = 0

 

Use maple to draw the graph of cosine function as follows:

 

STEP 1:

Open a new document in Maple

 

STEP 2:

Enter following Maple formula

plot{3cos(6/5x), x = 0..10π/3}

 

STEP 3:

Obtain the graph as follows:

 

Determine the maximum and minimum as follows:

Max(x, y) = (0, 3)

Min(x, y) = (3.14, -3)

Determine the maximum and minimum as follows:

Max(x, y) = (0, 3)

Min(x, y) = (3.14, -3)