To find the equation of a sinusoidal function that meets the given conditions, let's start by using the general form of a sinusoidal function:

$y = A \sin(Bx + C) + D$

Where:

• $A$ is the amplitude.
• $B$ affects the period of the function.
• $C$ is the phase shift.
• $D$ is the vertical shift.

Step 1: Identify the given parameters

1. Amplitude ($A$): Given as 2, so $A = 2$.
2. Period: Given as $2\pi$. The period of a sinusoidal function is $\frac{2\pi}{|B|}$. Since the period is $2\pi$, we have: $\frac{2\pi}{|B|} = 2\pi$ Simplifying this gives $|B| = 1$. So $B = 1$ (assuming a positive value for simplicity).
3. Point: The function passes through the point $(\pi, 2)$.

Step 2: Substitute the known values into the equation

Now, we substitute the known values into the equation: $y = 2 \sin(x + C) + D$

Step 3: Use the point $(\pi, 2)$ to find $C$ and $D$

We know the function passes through $(\pi, 2)$. Substituting $x = \pi$ and $y = 2$ into the equation:

$2 = 2 \sin(\pi + C) + D$

This simplifies to:

$1 = \sin(\pi + C) + \frac{D}{2}$

Step 4: Analyze $\sin(\pi + C)$

• $\sin(\pi + C) = -\sin(C)$, so:

$1 = -\sin(C) + \frac{D}{2}$

At this point, we have a system of equations. If we assume no vertical shift $D = 0$ for simplicity, then:

$1 = -\sin(C)$

So $\sin(C) = -1$, meaning $C = -\frac{\pi}{2}$.

Step 5: Final equation

Substitute $C = -\frac{\pi}{2}$ and $D = 0$ back into the equation:

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

So, the equation of the sinusoidal function is:

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

This satisfies the given conditions: a period of $2\pi$, an amplitude of 2, and it contains the point $(\pi, 2)$.

Would you like more details or have any questions?

Relative Questions:

1. How does changing the amplitude affect the graph of a sinusoidal function?
2. What effect does the phase shift $C$ have on the graph of $y = A \sin(Bx + C) + D$?
3. How do you determine the vertical shift $D$ for a sinusoidal function?
4. What is the relationship between the period and the coefficient $B$ in a sinusoidal function?
5. How can we find the equation of a sinusoidal function given multiple points?

Tip:

When dealing with sinusoidal functions, always remember that the period is determined by $B$, and this influences how "stretched" or "compressed" the wave appears on the graph.