Question

Write a cosine function that has an amplitude of 4, a midline of y, equals, 2y=2 and a period of 22.

Answer 1

To write a cosine function with the specified characteristics, we can use the standard form of a cosine function:

\[ y = A \cos(B(x - C)) + D \]

where:
- \( A \) is the amplitude,
- \( B \) affects the period,
- \( C \) is the phase shift (which we can ignore here since a specific shift isn’t mentioned),
- \( D \) is the midline.

### Given Specifications:

1. **Amplitude (A)**: 4
2. **Midline (D)**: 2
3. **Period**: 22

### Step-by-Step Construction:

1. **Amplitude (A)**: The amplitude is 4, so \( A = 4 \).

2. **Midline (D)**: The midline is \( y = 2 \), so \( D = 2 \).

3. **Period**: The period of a cosine function is determined by the formula:

\[
\text{Period} = \frac{2\pi}{B}
\]

Setting this equal to 22:

\[
\frac{2\pi}{B} = 22
\]

To find \( B \), we solve the equation:

\[
2\pi = 22B \quad \Rightarrow \quad B = \frac{2\pi}{22} = \frac{\pi}{11}
\]

4. **Phase Shift (C)**: The phase shift is not specified, so we will set it to 0 for simplicity. Thus, \( C = 0 \).

### Final Function:

Putting it all together, we have:

\[
y = 4 \cos\left(\frac{\pi}{11} x\right) + 2
\]

This is the cosine function that has an amplitude of 4, a midline of \( y = 2 \), and a period of 22.