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:

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 \(b = 4\pi\):

$$\text{Period} = \frac{2\pi}{4\pi} = \frac{1}{2}$$

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>