Sure, I'd be glad to help you understand how to find the amplitude and period of the given function:
Given the function:
[tex]\[
y = 2 \cos \left(\frac{\pi}{4} \left(x - \frac{1}{2}\right)\right)
\][/tex]
### Amplitude
The amplitude of a cosine function in the form [tex]\( y = A \cos(Bx + C) \)[/tex] is the absolute value of the coefficient [tex]\( A \)[/tex].
In this case, the coefficient [tex]\( A \)[/tex] is 2. So, the amplitude is:
[tex]\[
\text{Amplitude} = 2
\][/tex]
### Period
The period of a cosine function in the form [tex]\( y = A \cos(Bx + C) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex].
The period [tex]\( P \)[/tex] is given by:
[tex]\[
P = \frac{2\pi}{B}
\][/tex]
Here, the coefficient [tex]\( B \)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex].
So, the period is:
[tex]\[
P = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \times \frac{4}{\pi} = 8
\][/tex]
### Summary
- The amplitude of the function is [tex]\( 2 \)[/tex].
- The period of the function is [tex]\( 8 \)[/tex].
Thus,
[tex]\[
\text{Amplitude} = 2
\][/tex]
[tex]\[
\text{Period} = 8
\][/tex]
