The image shows a trigonometric graph with the equation options below it. Let's analyze the graph and match it to the correct equation.

Key Observations:

1. The graph appears to be a sine or cosine function.
2. The graph starts at a peak of $y = 1$, descends to $y = -1$ at $x = \pi/2$, crosses zero at $x = \pi$, and completes one full period by $x = 2\pi$.
3. The amplitude of the graph is 1 (since it goes from $y = 1$ to $y = -1$).
4. The period of the graph is $2\pi$, but the phase shift moves the graph to the left.

Analysis:

• The starting point at the maximum value suggests that the function is either a cosine function or a sine function with a phase shift.
• The graph is shifted horizontally to the left by some amount, indicating the presence of a phase shift.
• The period of the sine or cosine function remains $2\pi$, which corresponds to the standard period.

Testing the Options:

1. Option A: $y = \sin\left(2x + \frac{\pi}{6}\right)$
   This equation represents a sine function with a horizontal phase shift. However, sine starts at zero, so this is likely not the right choice.

2. Option B: $y = \cos\left(2x + \frac{\pi}{6}\right)$
   This equation involves a cosine function with a phase shift to the left. Since the graph starts at its peak, this option is promising.

3. Option C: $y = \cos\left(2x - \frac{\pi}{3}\right)$
   This is also a cosine function but with a different phase shift. The sign and amount of phase shift will determine if this option fits.

4. Option D: $y = \sin\left(2x + \frac{\pi}{3}\right)$
   Since the graph looks more like a cosine function at first glance, this sine option seems less likely.

5. Option E: $y = \sin\left(2x - \frac{\pi}{3}\right)$
   This is another sine option, but it also seems unlikely due to the cosine-like behavior of the graph.

Conclusion:

Based on the cosine shape and phase shift, Option B $y = \cos\left(2x + \frac{\pi}{6}\right)$ looks like the correct equation for the given trigonometric graph.

Would you like further details on how the phase shift and period were determined?

Here are five related questions to deepen understanding:

1. How can we determine the phase shift from a trigonometric graph?
2. What is the difference between sine and cosine in terms of graph behavior?
3. How do you calculate the period of a sine or cosine function with a coefficient on $x$?
4. What does the amplitude of a trigonometric function represent, and how do we find it from the graph?
5. How does the horizontal shift affect the graph of sine or cosine functions?

Tip: When analyzing trigonometric graphs, always check where the function starts to identify whether it's a sine or cosine.