The question asks for the equation of a sine or cosine function in the form $y = a \sin(bx)$ or $y = a \cos(bx)$, based on the given graph. Let's break down the graph to extract the required values for $a$ and $b$ for the function.

Step 1: Identify the amplitude (a)

The amplitude is the maximum distance from the middle of the wave to its peak or trough. In the graph:

• The maximum value appears to be 3, and the minimum value is -3.
• Thus, the amplitude $a = 3$.

Step 2: Identify the period (b)

The period is the distance required for the graph to complete one full cycle. The period can be found by looking at the distance along the x-axis for one complete wave.

• From the graph, one full cycle occurs between $x = 0$ and $x = 2\pi$.
• The formula for the period of a sine or cosine function is $\frac{2\pi}{b}$, and since the period is $2\pi$, we have: $\frac{2\pi}{b} = 2\pi \implies b = 1$

Step 3: Sine or Cosine?

Looking at the shape of the graph:

• The graph starts at 0, rises to a maximum, then drops to a minimum, and finally returns to 0. This behavior corresponds to the sine function.

Final Equation:

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

Would you like further details on this, or have any other questions?

Here are 5 related questions to deepen understanding:

1. How does the equation change if the graph shifts horizontally?
2. What happens to the graph if the amplitude $a$ is negative?
3. How do you determine the phase shift of a sine or cosine function?
4. What is the significance of the value $b$ in sine and cosine functions?
5. How would the equation look if the graph was a cosine function instead of sine?

Tip: Always verify the period and amplitude visually before deciding the equation type (sine or cosine).