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.