Question

write a cosine function that has an amplitude of 4, a midline of y = 5 and a period of \\(\frac{1}{4}\\).
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Explanation:

Step1: Recall cosine function form

The general form of a cosine function is \( f(x) = A\cos(Bx) + k \), where \( |A| \) is the amplitude, \( \frac{2\pi}{|B|} \) is the period, and \( k \) is the midline.

Step2: Determine \( A \)

Given amplitude is 4, so \( A = 4 \) (or -4, but we'll use 4 for simplicity).

Step3: Determine \( B \)

Period is \( \frac{1}{4} \). Using the period formula \( \text{Period} = \frac{2\pi}{|B|} \), we solve for \( B \):
\( \frac{1}{4} = \frac{2\pi}{|B|} \)
Multiply both sides by \( |B| \): \( \frac{|B|}{4} = 2\pi \)
Multiply both sides by 4: \( |B| = 8\pi \), so \( B = 8\pi \) (or -8\pi, we'll use 8\pi).

Step4: Determine \( k \)

Midline is \( y = 5 \), so \( k = 5 \).

Step5: Construct the function

Substitute \( A = 4 \), \( B = 8\pi \), and \( k = 5 \) into the general form:
\( f(x) = 4\cos(8\pi x) + 5 \)

Answer:

\( f(x) = 4\cos(8\pi x) + 5 \)