The graph provided shows a periodic function resembling a sine or cosine wave. To derive its equation, let's identify key features:

1. Amplitude:

The amplitude is the distance from the centerline (y = 0) to the maximum value. The maximum value is 2 and the minimum value is -2. Thus, the amplitude is:

$A = 2$

2. Period:

The period is the length of one full cycle. From the graph, one full cycle starts at $-\frac{\pi}{4}$ and ends at $\frac{5\pi}{4}$, making the total period:

$\text{Period} = \frac{5\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{6\pi}{4} = \frac{3\pi}{2}$

The standard period of sine and cosine functions is $2\pi$. To match the period of $\frac{3\pi}{2}$, we use the formula for period adjustment:

$\text{New Period} = \frac{2\pi}{B} = \frac{3\pi}{2} \implies B = \frac{4}{3}$

3. Phase Shift:

The graph starts at $-\frac{\pi}{4}$, which suggests a phase shift to the right by $\frac{\pi}{4}$.

4. Equation Form:

Since the graph resembles a sine wave that starts at a point below the x-axis (suggesting a reflection or negative sine wave), we use the sine form with a negative amplitude:

$y = A \sin(Bx - C)$

Using all the identified parameters:

• Amplitude $A = -2$
• Period $B = \frac{4}{3}$
• Phase shift $C = \frac{\pi}{4}$

Thus, the equation is:

$y = -2 \sin\left(\frac{4}{3}x - \frac{\pi}{4}\right)$

Do you want further clarification on any step? Also, here are five follow-up questions for deeper understanding:

1. What is the difference between sine and cosine functions in terms of graph starting points?
2. How does the amplitude affect the graph's appearance?
3. What is the relationship between the period and the value of $B$?
4. How can you determine whether to use sine or cosine based on a graph?
5. Why is a phase shift necessary when the graph starts away from the origin?

Tip: For any sine or cosine function, always check amplitude, period, and phase shift first to write the equation correctly.