To write the equation of a cosine function with a given amplitude and period, it's important to recall the general form of the cosine function and how the amplitude and period are defined within it. The general form of a cosine function is:
[tex]\[ y = A \cos(Bx) \][/tex]
where
- [tex]\( A \)[/tex] represents the amplitude,
- [tex]\( B \)[/tex] affects the period of the function.
The amplitude [tex]\( A \)[/tex] is the maximum value of the function, which corresponds to the multiplier in front of the cosine function.
The period [tex]\( T \)[/tex] of the cosine function is given by the formula:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
Given the requirements:
1. Amplitude = 2
2. Period = 6Ï€
First, let's determine the amplitude.
- Here, the amplitude [tex]\( A \)[/tex] is directly given as 2. Therefore, we need the coefficient [tex]\( A \)[/tex] in front of the cosine function to be 2.
Next, let's determine the value of [tex]\( B \)[/tex] that will give us the desired period.
- The period [tex]\( T \)[/tex] is given by [tex]\( \frac{2\pi}{B} \)[/tex].
- We need this period to be [tex]\( 6\pi \)[/tex].
Setting up the equation for the period:
[tex]\[ 6\pi = \frac{2\pi}{B} \][/tex]
Solving for [tex]\( B \)[/tex]:
[tex]\[ B = \frac{2\pi}{6\pi} \][/tex]
[tex]\[ B = \frac{1}{3} \][/tex]
Thus, the cosine function that has an amplitude of 2 and a period of [tex]\( 6\pi \)[/tex] should have the form:
[tex]\[ y = 2 \cos\left(\frac{1}{3}x\right) \][/tex]
Reviewing the given options:
1. [tex]\( y = -2 \cos\left(\frac{1}{6} x\right) \)[/tex] - This has the correct amplitude but incorrect value of [tex]\( B \)[/tex].
2. [tex]\( y = -\frac{1}{2} \cos\left(\frac{1}{3} x\right) \)[/tex] - This has an incorrect amplitude, and while the period is correct, the overall sign is changed.
3. [tex]\( y = 2 \cos\left(\frac{1}{3} x\right) \)[/tex] - This has both the correct amplitude and period.
4. [tex]\( y = \frac{1}{2} \cos\left(\frac{1}{6} x\right) \)[/tex] - This has an incorrect amplitude and period.
Therefore, the correct equation is:
[tex]\[ y = 2 \cos\left(\frac{1}{3} x\right) \][/tex]
So, the correct choice is:
[tex]\[ y = 2 \cos\left(\frac{1}{3} x\right) \][/tex]
Thus, the answer is indeed:
[tex]\[ 3 \][/tex]
