To determine the amplitude and period of the function [tex]\( y = \frac{1}{2} \cos \left( \frac{3}{4} x \right) \)[/tex], we will follow these steps:
1. Amplitude:
The amplitude of a cosine function [tex]\( y = a \cos(bx) \)[/tex] is the absolute value of the coefficient [tex]\( a \)[/tex]. In this case, the coefficient [tex]\( a \)[/tex] is [tex]\(\frac{1}{2}\)[/tex]. Therefore, the amplitude is:
[tex]$
\text{Amplitude} = \left| \frac{1}{2} \right| = \frac{1}{2}
$[/tex]
2. Period:
The period of the cosine function [tex]\( y = a \cos(bx) \)[/tex] is given by the formula:
[tex]$
\text{Period} = \frac{2\pi}{b}
$[/tex]
Here, [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the cosine function. For the given function, [tex]\( b = \frac{3}{4} \)[/tex]. Therefore, the period is:
[tex]$
\text{Period} = \frac{2\pi}{\frac{3}{4}} = 2\pi \times \frac{4}{3} = \frac{8\pi}{3}
$[/tex]
So, the exact values for the amplitude and period of the function [tex]\( y = \frac{1}{2} \cos \left( \frac{3}{4} x \right) \)[/tex] are:
- Amplitude: [tex]\( \frac{1}{2} \)[/tex]
- Period: [tex]\( \frac{8\pi}{3} \)[/tex]
