You've shared a photo of a multiple-choice question where you need to find a sine function based on given parameters. The function should have:

1. A positive amplitude of 5.
2. A horizontal shift of 3 units to the right.
3. A vertical displacement of -2.
4. A period of 6.

The general form of the sine function is $f(x) = A \sin(B(x - C)) + D$, where:

• $A$ is the amplitude.
• $B$ is the frequency factor, related to the period by $B = \frac{2\pi}{\text{period}}$.
• $C$ is the horizontal shift.
• $D$ is the vertical shift.

Given:

• Amplitude $A = 5$
• Horizontal shift $C = 3$
• Vertical displacement $D = -2$
• Period = 6, so $B = \frac{2\pi}{6} = \frac{\pi}{3}$

The correct form of the function should be: $f(x) = 5 \sin\left(\frac{\pi}{3}(x - 3)\right) - 2$

From the photo's options:

• $f(x) = 5 \sin\left(\frac{\pi}{3}(x - 3)\right) - 2$ aligns with the option: $f(x) = 5 \sin\left(\frac{\pi}{3}(x - 3)\right) - 2$

This matches the first option provided in the image.

Would you like to know more details on how to derive such equations or have any other questions on this topic? Here are a few related questions you might find helpful:

1. How do you determine the amplitude from a graph of a sine or cosine function?
2. What is the significance of the phase shift in trigonometric functions?
3. How does the frequency factor $B$ affect the graph of the sine function?
4. Could you provide an example of a sine function with a different period and explain how it affects the graph?
5. How do vertical shifts impact the y-intercepts of sine and cosine functions?

Tip: Always remember that the horizontal shift $C$ is subtracted inside the sine function if the shift is to the right, and added if the shift is to the left.