The question in your image asks for the amplitude of the function $y = -3 \sin(x)$, and also asks to identify the correct graph of this function from the given options.

1. Amplitude of the function:

The general form of the sine function is $y = A \sin(x)$, where $A$ represents the amplitude. The amplitude is the distance from the midline of the wave to the peak (or trough), which is always a positive number.

In the given function $y = -3 \sin(x)$, the coefficient $-3$ indicates that the amplitude is $|A| = 3$. The negative sign indicates a reflection over the x-axis, but it does not affect the amplitude.

Answer: The amplitude is 3.

2. Correct graph of the function:

The function $y = -3 \sin(x)$ is a sine function with:

• Amplitude of 3.
• A reflection over the x-axis (because of the negative sign).
• The period of a standard sine function, $2\pi$, remains unchanged since the coefficient of $x$ is 1.

Now, you can identify the correct graph:

• The graph should have peaks and troughs reaching 3 and -3, respectively.
• It should be reflected over the x-axis, meaning the first half of the sine wave should start by going downward.

Based on this, option B looks like the correct graph since it shows a sine wave with a reflection and an amplitude of 3.

Would you like more details on any part of this explanation? Here are some follow-up questions you can explore:

1. What effect does the negative sign in front of the sine function have on the graph?
2. How would the graph change if the coefficient of $x$ were something other than 1 (e.g., $y = 3 \sin(2x)$)?
3. What is the period of the function $y = \sin(x)$, and how does it relate to the graph?
4. How would you graph $y = 3 \cos(x)$ and how does it compare with $y = 3 \sin(x)$?
5. What is the formula for determining the amplitude and period of a general trigonometric function?

Tip: When graphing sine or cosine functions, always start by identifying the amplitude, period, and any reflections or phase shifts to guide your plot.