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Given the function:
[tex]\[ y = 2 \cos \left( \frac{\pi}{4} \left(x - \frac{1}{2}\right) \right) \][/tex]

1. What is the amplitude of the function?
[tex]\[ 2 \][/tex]

2. What is the period of the function?
[tex]\[ \square \][/tex]
(Simplify your answer. Type an exact answer, using [tex]\(\pi\)[/tex] as needed. Use an integer or a simplified fraction.)



Answer :

GinnyAnswer
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]

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