To write the equation of a sine or cosine function for the given graph, let's analyze the key properties of the function:

1. Amplitude

The amplitude is the distance from the midline (y = 0 in this case) to the maximum or minimum point. Here, the maximum value is $y = 4$ and the minimum value is $y = -4$. Therefore, the amplitude is: $A = 4$

2. Period

The period is the distance required for the function to complete one full cycle. Observing the graph, one full cycle starts at $x = -\frac{\pi}{2}$, reaches the next peak at $x = \frac{\pi}{2}$, and completes at $x = \frac{3\pi}{2}$. The length of this cycle is: $T = \frac{3\pi}{2} + \frac{\pi}{2} = 2\pi$ So, the period is $2\pi$.

3. Phase Shift

Since the graph passes through the origin $(0, 0)$ and follows the form of a sine wave, there’s no phase shift. For a sine wave, this implies it starts at zero when $x = 0$, which aligns with the sine function.

4. Vertical Shift

There’s no vertical shift, as the midline of the function is $y = 0$.

Equation Form

For a sine function with these characteristics, the general form is: $y = A \sin(Bx)$ Where:

• $A = 4$, as derived above.
• $B$ is determined by the period. Since the period $T = 2\pi$, and the relationship between $B$ and $T$ is $T = \frac{2\pi}{B}$, we get $B = 1$.

Final Equation

Thus, the equation of the function is: $y = 4 \sin(x)$

Alternatively, this graph could be described by a cosine function with a phase shift of $\frac{\pi}{2}$: $y = 4 \cos(x - \frac{\pi}{2})$

Either equation describes the graph accurately. Would you like any further details or have any questions?

Here are five related questions:

1. How would the equation change if the amplitude were halved?
2. What would the equation be if the function were vertically shifted up by 3 units?
3. How does a phase shift affect the appearance of the sine or cosine graph?
4. Can this graph be described with a tangent function? Why or why not?
5. What would be the effect on the graph if the period were $\pi$ instead of $2\pi$?

Tip: Remember, adjusting the amplitude or period directly affects the height and length of the cycles in sine and cosine graphs.