Question

Describe the relationship between the graph of f and g Consider amplitude, period, and shift.

Answer 1

Step 1:

The graph of functions  and are in the form of .

where is amplitude, is the period and is the shift along -axis.

Consider the graph of the function .

Observe the graph.
The difference between the maximum height and the minimum height is twice of the amplitude of the function.

Amplitude of the function is .

Step 2:

Period of the function is .

The cosine function completes one half of the cycle between the times at maximum height and minimum height.

Then Period of the function is .

Step 3:

Phase shift along -axis is  the time where maximum height occurs.

The time at maximum height is .

Substitute the values , and in the function .

.

Comments

Contd..

Step 4:

Consider the graph of the function .

Observe the graph.

The difference between the maximum height and the minimum height is twice of the amplitude of the function.

Amplitude of the function is .

Step 5:

Period of the function is .

The cosine function completes one half of the cycle between the times at maximum height and minimum height.

Then Period of the function is .

Step 6:

Phase shift along -axis is  the time where maximum height occurs.

The time at maximum height is .

Substitute the values , and in the function .

.

Step 7:

Compare the graph of the functions and .

Amplitude of the function is and is .

Period and Phase shift of the functions and are same.

The amplitude of the graph of function is twice the amplitude of the graph of function .

Solution :

The amplitude of the graph of function is twice the amplitude of the graph of function .