Question

Write a sine function that has an amplitude of 4, a midline of y, equals, 2y=2 and a period of one quarter
4
1
​
.

Answer 1

To create a sine function that meets the given criteria, we'll consider the general form of a sine function:

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

Where:
- \( A \) is the amplitude,
- \( B \) affects the period,
- \( C \) is the horizontal shift (phase shift),
- \( D \) is the vertical shift (midline).

### Given:
- Amplitude (\( A \)) = 4
- Midline (\( D \)) = 2 (since \( 2y = 2 \) implies \( D = 2 \))
- Period = \( \frac{1}{4} \)

### Finding \( B \):
The period of a sine function is given by:

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

Setting the period to \( \frac{1}{4} \):

\[
\frac{2\pi}{|B|} = \frac{1}{4}
\]

Solving for \( B \):

\[
|B| = 2\pi \cdot 4 = 8\pi
\]

We can take \( B = 8\pi \) for simplicity.

### Constructing the Function:
Now we can plug in all the values:

\[
y = 4 \sin(8\pi x) + 2
\]

This is the sine function with the specified amplitude, midline, and period.

### Final Function:
\[
y = 4 \sin(8\pi x) + 2
\]