Question

write an equation of a sine graph with a phase shift right 3, a period of 5pi, a vertical translation down 6, and an amplitude of 3.

Answer 1

To write the equation of a sine graph with the specified transformations, we can start with the general form of the sine function:

\[
y = a \sin(b(x - c)) + d
\]

where:
- \(a\) is the amplitude,
- \(b\) determines the period,
- \(c\) is the phase shift,
- \(d\) is the vertical translation.

### Given:
- Amplitude (\(a\)): 3
- Period: \(5\pi\)
- Phase shift: right 3 (this means \(c = 3\))
- Vertical translation: down 6 (this means \(d = -6\))

### Step 1: Calculate \(b\) from the period
The period \(P\) of a sine function is given by the formula:

\[
P = \frac{2\pi}{b}
\]

Setting the period to \(5\pi\):

\[
5\pi = \frac{2\pi}{b}
\]

To solve for \(b\), we rearrange the equation:

\[
b = \frac{2\pi}{5\pi} = \frac{2}{5}
\]

### Step 2: Assemble the equation
Now we can plug in our values for \(a\), \(b\), \(c\), and \(d\):

- \(a = 3\)
- \(b = \frac{2}{5}\)
- \(c = 3\)
- \(d = -6\)

Thus, the equation of the sine graph is:

\[
y = 3 \sin\left(\frac{2}{5}(x - 3)\right) - 6
\]

### Final Equation
Putting it all together, we have:

\[
y = 3 \sin\left(\frac{2}{5}(x - 3)\right) - 6
\]

This is the equation of the sine graph with the specified transformations.