Question

write a cosine function that has a midline of $y = 4$, an amplitude of 2 and a period of $\pi$.

Explanation:

Step1: Recall the general form of a cosine function

The general form of a cosine function is \( y = A\cos(Bx - C)+D \), where:

• \( |A| \) is the amplitude,
• \( \frac{2\pi}{|B|} \) is the period,
• \( \frac{C}{B} \) is the phase shift (we assume no phase shift here, so \( C = 0 \)),
• \( D \) is the vertical shift (which determines the midline \( y = D \)).

Step2: Determine the value of \( A \)

Given the amplitude is 2, so \( |A|=2 \). We can take \( A = 2 \) (we could also take \( A=- 2 \), but we'll use \( A = 2 \) for simplicity).

Step3: Determine the value of \( B \)

The period is given as \( \pi \). We know that the period \( T=\frac{2\pi}{|B|} \). Substituting \( T = \pi \) into the formula:
\( \pi=\frac{2\pi}{|B|} \)
To solve for \( |B| \), we can cross - multiply: \( |B|\times\pi=2\pi \), then divide both sides by \( \pi \) to get \( |B| = 2 \). We can take \( B = 2 \) (since we assume no phase shift and positive \( B \) for simplicity).

Step4: Determine the value of \( D \)

The midline is \( y = 4 \), and the midline of the cosine function \( y = A\cos(Bx - C)+D \) is \( y = D \). So \( D = 4 \).

Step5: Write the cosine function

Since we assume \( C = 0 \), \( A = 2 \), \( B = 2 \), and \( D = 4 \), substituting these values into the general form \( y=A\cos(Bx - C)+D \), we get:
\( y = 2\cos(2x)+4 \)

Answer:

\( y = 2\cos(2x)+4 \)