The problem provided asks to determine the equation of the graph, which is of the form $y = a \sin(bx)$ or $y = a \cos(bx)$, where $b > 0$.

Let's break it down based on the graph:

1. Amplitude (a):

   • The maximum value of the graph is 2, and the minimum value is -2. This suggests that the amplitude $a$ is 2.

2. Period:

   • The graph completes one full cycle from $x = 0$ to $x = \frac{2\pi}{3}$, which means the period $T$ is $\frac{2\pi}{3}$.
   • The period of a sinusoidal function is given by $T = \frac{2\pi}{b}$, so we can set this equal to $\frac{2\pi}{3}$ and solve for $b$: $\frac{2\pi}{b} = \frac{2\pi}{3} \implies b = 3$

3. Sine or Cosine:

   • At $x = 0$, the graph is at its maximum value (2), which is characteristic of a cosine graph. A sine graph typically starts at 0. Hence, the function is of the form $y = a \cos(bx)$.

Putting it all together:

The equation of the graph is: $y = 2 \cos(3x)$

Would you like any more details, or do you have any questions?

Relative Questions:

1. How do we determine the amplitude of a trigonometric graph?
2. What is the difference between the sine and cosine graphs in terms of their starting points?
3. How do you find the period of a trigonometric function from the graph?
4. What role does the value of $b$ play in a trigonometric equation?
5. How can you recognize a phase shift in a trigonometric graph?

Tip: The amplitude affects the vertical stretch of the graph, while the period affects its horizontal compression/stretch.