To determine the equation of the sine function from the graph, we need to identify several key characteristics:

1. Amplitude (A): The maximum displacement from the center line (y = 0). Here, the maximum y-value is 2, and the minimum y-value is -2. Therefore, the amplitude $A$ is 2.

2. Period (T): The length of one complete cycle. From the graph, one full cycle occurs from $x = 0$ to $x = 180$. Hence, the period $T$ is 180.

3. Frequency (f): The number of cycles per unit of time, which is the reciprocal of the period. Here, $f = \frac{1}{T} = \frac{1}{180}$.

4. Horizontal shift (phase shift): There appears to be no horizontal shift since the graph starts at the origin (0, 0).

5. Vertical shift (D): There appears to be no vertical shift since the graph oscillates around the x-axis (y = 0).

Given these observations, the general form of the sine function is: $y = A \sin(B(x - C)) + D$

Here:

• $A = 2$
• $B = \frac{2\pi}{T} = \frac{2\pi}{180} = \frac{\pi}{90}$
• $C = 0$ (no phase shift)
• $D = 0$ (no vertical shift)

So, the equation of the sine function is: $y = 2 \sin\left(\frac{\pi}{90} x\right)$

Would you like more details or have any questions?

Here are some additional questions you can ask:

1. How do you determine the amplitude of a sine function?
2. What is the general form of a trigonometric function?
3. How does the period of a function relate to its frequency?
4. What is the significance of phase shifts in trigonometric functions?
5. How can you identify vertical shifts in sine and cosine graphs?

Tip: When writing the equation of a sine or cosine function, always check the graph for amplitude, period, phase shift, and vertical shift.