From the graph provided, we need to determine the equation of the sinusoidal function. The general forms of the sine and cosine functions are:

$y = A \sin(\omega x) \quad \text{or} \quad y = A \cos(\omega x)$

Where:

• $A$ is the amplitude (the maximum vertical distance from the axis),
• $\omega$ is the angular frequency (which relates to the period),
• $x$ is the variable.

Step 1: Determine the amplitude $A$

From the graph, the peak is at $y = 4$ and the trough is at $y = -4$. Therefore, the amplitude is: $A = 4$

Step 2: Determine the period and frequency $\omega$

The period is the horizontal length of one complete cycle. From the graph, one full cycle happens between $0$ and $4\pi$, so the period $T = 4\pi$. The angular frequency $\omega$ is related to the period by the formula: $T = \frac{2\pi}{\omega}$ Solving for $\omega$: $\omega = \frac{2\pi}{T} = \frac{2\pi}{4\pi} = \frac{1}{2}$

Step 3: Determine if it is sine or cosine

The graph starts at $y = 0$ when $x = 0$, which is a characteristic of a sine function. Thus, we use the sine function form.

Final Equation

Now we can plug the values into the equation: $y = 4 \sin\left(\frac{1}{2}x\right)$

This is the equation that represents the graph.

Would you like more details or have any further questions?

Related questions:

1. How would the equation change if the graph represented a cosine function?
2. What is the phase shift for the sine or cosine function if the graph doesn't start at the origin?
3. How can we adjust the equation if the amplitude was 5 instead of 4?
4. What effect does changing the angular frequency $\omega$ have on the graph?
5. How do we determine the vertical shift if the graph is not centered on the x-axis?

Tip:

Always remember, the period of a sine or cosine function is $T = \frac{2\pi}{\omega}$, which helps you find $\omega$ quickly.