Question

write a cosine function that has an amplitude of 3, a midline of y = 4 and a period of 2/3. answer f(x) =

Explanation:

Step1: Recall cosine function form

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

Step2: Determine A, C values

Given amplitude \( A = 3 \), midline \( C = 4 \) (since midline is \( y = 4 \)).

Step3: Calculate B value

Period is \( \frac{2}{3} \). Using period formula \( \frac{2\pi}{|B|} = \frac{2}{3} \). Solve for \( B \):
\( |B| = \frac{2\pi}{\frac{2}{3}} = 3\pi \), so \( B = 3\pi \) (we can take positive \( B \) for simplicity).

Step4: Form the function

Substitute \( A = 3 \), \( B = 3\pi \), \( C = 4 \) into the general form: \( f(x) = 3\cos(3\pi x) + 4 \).

Answer:

\( f(x) = 3\cos(3\pi x) + 4 \)