The question asks for the equation of the sine wave shown in the image. Let us analyze the graph and solve it step by step.

Observations from the Graph:

1. Amplitude: The sine wave has a maximum value of 2 and a minimum value of -2. Hence, the amplitude is 2.

2. Period: The wave completes one full cycle between $0^\circ$ and $180^\circ$. This corresponds to a period of $\frac{\pi}{3}$ radians (or 180 degrees).

   • The formula for period is $T = \frac{2\pi}{b}$, where $b$ is the frequency coefficient in $y = a \sin(bx)$. From this, $b = \frac{2\pi}{T} = \frac{2\pi}{\pi/3} = 6$.

3. Phase Shift: The sine wave starts from 0 at $x = 0$, so there is no phase shift.

The general form of the sine equation is: $y = a \sin(bx + c) + d$

• $a = 2$ (amplitude),
• $b = \frac{1}{3}\pi$ (frequency factor).

Matching Options:

Among the options given, the correct choice is: $\boxed{y = 2 \sin\left(\frac{1}{3}\pi x\right)}$

Let me know if you'd like further clarifications!

Related Questions:

1. What is the amplitude of a sine wave, and how is it calculated?
2. How does the frequency of a sine wave affect its graph?
3. How is the period of a sine function calculated from its equation?
4. Can a sine function have a vertical shift? How would it appear on the graph?
5. How do you determine the phase shift of a sine or cosine wave?

Tip:

Always compare the key features of the graph (amplitude, period, phase shift, and vertical shift) to the standard sine equation $y = a \sin(bx + c) + d$ for accurate matching.