QUESTION 7
Two trig functions namely, and , are shown below for
.
7.1
Determine;
7.1 .1 the value of and .
713 the range of .

Given Functions:

Assumptions:

7.1.1 Determine the Values of and

Determining in :

1. Amplitude Analysis:
   • The general form of the cosine function is , where:
     • is the amplitude.
     • affects the period.
     • is the phase shift.
     • is the vertical shift.
   • For :
     • Amplitude ( ): This is the coefficient multiplying the cosine function. It determines the peak value (maximum and minimum) of the function.
2. From the Graph:
   • Identify the Maximum and Minimum Values of from the graph.
   • The Amplitude is the absolute value of the maximum (or minimum) since cosine functions are symmetric.
   • Example: If the graph of reaches a maximum of 3 and a minimum of -3, then .

Determining in :

1. Period Analysis:
   • The general period of the sine function is given by:
   • For :
     • , so:
2. From the Graph:
   • Determine the Period of by measuring the distance between two consecutive peaks (or troughs).
   • Calculate using the period.
   • Example: If completes two full cycles within , then the period is , and:

7.1.3 Determine the Range of

1. Understanding Range:
   • The range of a function is the set of all possible output values (y-values).
2. For :
   • The cosine function oscillates between -1 and 1.
   • Multiplying by scales this range.
3. Example:
   • If , then:

Summary:

• is determined by the amplitude of the graph.
• is calculated based on the period of the graph.
• The range of is from to .