Let’s analyze the given function:
[tex]\[ y = -3 \cos \left(\frac{1}{2} x\right) \][/tex]
### 1. Amplitude
The amplitude of a cosine function [tex]\( y = A \cos(Bx) \)[/tex] is given by the absolute value of the coefficient in front of the cosine term.
In the function [tex]\( y = -3 \cos \left(\frac{1}{2} x \right) \)[/tex], the coefficient is [tex]\(-3\)[/tex].
Therefore, the amplitude is:
[tex]\[ \text{Amplitude} = | -3 | = 3 \][/tex]
### 2. Period
The period of a cosine function [tex]\( y = A \cos(Bx) \)[/tex] is calculated using the formula:
[tex]\[ \text{Period} = \frac{2\pi}{|B|} \][/tex]
where [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the cosine function.
In the function [tex]\( y = -3 \cos \left(\frac{1}{2} x \right) \)[/tex], the coefficient [tex]\( B \)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Plugging this value into the formula for the period, we get:
[tex]\[ \text{Period} = \frac{2\pi}{\left| \frac{1}{2} \right|} \][/tex]
Simplifying this expression:
[tex]\[ \text{Period} = \frac2\pi \cdot 2 = 2π \cdot 2 = 4π \cdot 2 = 8π \cdot 2 \approx 12.56637061435 \][/tex]
Therefore, the exact values of the amplitude and period are:
- Amplitude: [tex]\( 3 \)[/tex]
- Period: [tex]\( 12.566370614359172 \approx 2\pi \)[/tex]
