Let us analyze the sinusoidal function from the graph:

Observations:

1. Amplitude: The maximum displacement from the midline is $2$. Thus, the amplitude is $A = 2$.

2. Vertical Shift: The midline of the function lies on $y = 0$, so there is no vertical shift ($D = 0$).

3. Period: The period is the horizontal length required for the wave to complete one full cycle. From the graph:

   • A full cycle occurs between $x = 0$ and $x = 2\pi$.
   • The period is $P = 2\pi$.

   The formula for the period of a sinusoidal function is: $P = \frac{2\pi}{B}$ Since $P = 2\pi$, we have $B = 1$.

4. Phase Shift: The graph starts at $y = 0$ and rises, which matches the behavior of a sine function. There is no phase shift ($C = 0$).

General Equation:

The general form of a sinusoidal function is: $y = A \sin(B(x - C)) + D$

Substituting the values:

• $A = 2$,
• $B = 1$,
• $C = 0$,
• $D = 0$,

The equation is: $y = 2 \sin(x)$

Would you like a step-by-step explanation or have further questions?

Follow-up Questions:

1. How do we identify the phase shift of a sinusoidal graph?
2. What is the difference between sine and cosine graphs in terms of their starting points?
3. How do we compute the period if $B \neq 1$?
4. How does changing the amplitude $A$ affect the graph visually?
5. What happens to the graph if a vertical shift ($D$) is added?

Tip:

For sinusoidal graphs, always identify the amplitude, period, phase shift, and vertical shift first—they are the building blocks of the function!