Let's solve the problem step-by-step:
### Step 1: Understanding amplitude
The amplitude of a cosine function is the coefficient in front of the cosine. For a general function of the form [tex]\( y = A \cos(Bx) \)[/tex], the amplitude is [tex]\(|A|\)[/tex].
Given the amplitude is 2, we set [tex]\( |A| = 2 \)[/tex]. Therefore, [tex]\( A \)[/tex] can be either [tex]\( +2 \)[/tex] or [tex]\(-2\)[/tex].
### Step 2: Understanding the period
The period of the cosine function is determined by the coefficient [tex]\( B \)[/tex] inside the argument of the cosine function. For a general cosine function of the form [tex]\( y = A \cos(Bx) \)[/tex], the period [tex]\( T \)[/tex] is given by:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
We are given that the period is [tex]\( 6\pi \)[/tex]. So we set up the equation:
[tex]\[ \frac{2\pi}{B} = 6\pi \][/tex]
### Step 3: Solving for [tex]\( B \)[/tex]
To find [tex]\( B \)[/tex], we solve the equation:
[tex]\[ \frac{2\pi}{B} = 6\pi \][/tex]
Multiplying both sides by [tex]\( B \)[/tex] and then dividing by [tex]\(6\pi \)[/tex], we get:
[tex]\[ B = \frac{2\pi}{6\pi} = \frac{1}{3} \][/tex]
### Step 4: Forming the equation
Using the values we have found for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
1. Amplitude ([tex]\( A \)[/tex]) = 2
2. [tex]\( B = \frac{1}{3} \)[/tex]
The general form of the equation is:
[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]
### Step 5: Checking the options
Let's compare this with the given multiple-choice options:
1. [tex]\( y = -2 \cos \left( \frac{1}{6} x \right) \)[/tex]
2. [tex]\( y = -\frac{1}{2} \cos \left( \frac{1}{3} x \right) \)[/tex]
3. [tex]\( y = 2 \cos \left( \frac{1}{3} x \right) \)[/tex]
4. [tex]\( y = \frac{1}{2} \cos \left( \frac{1}{6} x \right) \)[/tex]
The correct option that matches our derived equation is:
[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]
Thus, the equation of the cosine function with amplitude 2 and period [tex]\( 6 \pi \)[/tex] is:
[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]
So, the correct answer is:
[tex]\[ y = 2 \cos \left( \frac{1}{3} x \right) \][/tex]
