Let's analyze the given graph and find the equation of the sinusoidal function:

Characteristics from the graph:

1. Amplitude: The maximum height from the centerline (y = 2) is 2 units, so the amplitude is 2.
2. Vertical Shift: The centerline is at $y = 2$, meaning the graph is shifted 2 units up.
3. Period: The graph completes one cycle from $0$ to $2\pi$, but since there are two full cycles between $0$ and $2\pi$, the period is $\pi$. This indicates a horizontal compression, so the angular frequency $b = 2$ (as the period $P = \frac{2\pi}{b}$).
4. Function Type: The graph starts at a peak at $x = 0$, which indicates it is a cosine function, not sine.

Equation Form:

The general equation of a cosine function is: $y = A \cos(Bx) + D$ where:

• $A = 2$ (amplitude),
• $B = 2$ (from the period),
• $D = 2$ (vertical shift).

Thus, the equation is: $y = 2 \cos(2x) + 2$

Answer:

The correct choice is B) $y = 2 \cos(2x)$.

Would you like a detailed explanation of any part of this process? Here are 5 related questions:

1. How do you determine the period of a sinusoidal function from a graph?
2. What are the key differences between sine and cosine graphs?
3. How does the vertical shift affect the graph of a sinusoidal function?
4. Can you explain how to derive the amplitude from a graph?
5. How does the phase shift affect sinusoidal functions?

Tip: The angular frequency $B$ in sinusoidal functions determines how many cycles fit into $2\pi$. The larger $B$, the more compressed the graph is.