Question

find the period and amplitude of the function.

\\y = -4 \cos(4\pi x)\\

give the exact values, not decimal approximations.

period:
amplitude:

Explanation:

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<reasoning>

Identify the general form of the function

The given function is:
\[y = -4 \cos(4\pi x)\]
We compare this to the standard form of a cosine function:
\[y = a \cos(bx)\]
where \(a = -4\) and \(b = 4\pi\).

Calculate the amplitude

The amplitude is the absolute value of the coefficient \(a\):
\[\text{Amplitude} = |a| = |-4| = 4\]

Calculate the period

The period of a cosine function is given by the formula:
\[\text{Period} = \frac{2\pi}{|b|}\]
Substituting \(b = 4\pi\):
\[\text{Period} = \frac{2\pi}{4\pi} = \frac{1}{2}\]
</reasoning>

<answer>
Find the period and amplitude of the function.
\(y = -4 \cos(4\pi x)\)

Period: <blank>\(\frac{1}{2}\)</blank>

Amplitude: <blank>4</blank>
</answer>

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Answer:

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<reasoning>

Identify the general form of the function

The given function is:
\[y = -4 \cos(4\pi x)\]
We compare this to the standard form of a cosine function:
\[y = a \cos(bx)\]
where \(a = -4\) and \(b = 4\pi\).

Calculate the amplitude

The amplitude is the absolute value of the coefficient \(a\):
\[\text{Amplitude} = |a| = |-4| = 4\]

Calculate the period

The period of a cosine function is given by the formula:
\[\text{Period} = \frac{2\pi}{|b|}\]
Substituting \(b = 4\pi\):
\[\text{Period} = \frac{2\pi}{4\pi} = \frac{1}{2}\]
</reasoning>

<answer>
Find the period and amplitude of the function.
\(y = -4 \cos(4\pi x)\)

Period: <blank>\(\frac{1}{2}\)</blank>

Amplitude: <blank>4</blank>
</answer>

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