Question

write a cosine function that has a midline of $y = 5$, an amplitude of 4 and a period of $\frac{1}{3}$.

Explanation:

Step1: Recall the general form of a cosine function

The general form of a cosine function is \( y = A\cos(Bx)+C \), where \( A \) is the amplitude, \( B \) is related to the period by the formula \( \text{Period}=\frac{2\pi}{|B|} \), and \( C \) is the vertical shift (which determines the midline, \( y = C \)).

Step2: Determine the value of \( A \)

Given the amplitude is 4, so \( A = 4 \).

Step3: Determine the value of \( B \)

The period is given as \( \frac{1}{3} \). Using the period formula \( \text{Period}=\frac{2\pi}{|B|} \), we substitute the period value:
\( \frac{1}{3}=\frac{2\pi}{|B|} \)
To solve for \( |B| \), we can cross - multiply: \( |B|=2\pi\times3 = 6\pi \). Since we can take \( B>0 \) for a cosine function (the cosine function is even, so the sign of \( B \) doesn't affect the shape in terms of reflection over the y - axis for the basic form), \( B = 6\pi \).

Step4: Determine the value of \( C \)

The midline is \( y = 5 \), and the midline of the function \( y = A\cos(Bx)+C \) is \( y = C \), so \( C = 5 \).

Step5: Write the cosine function

Substituting \( A = 4 \), \( B = 6\pi \), and \( C = 5 \) into the general form, we get the function:
\( y = 4\cos(6\pi x)+5 \)

Answer:

\( y = 4\cos(6\pi x)+5 \)