Find an equation for a sinusoidal function that has a period 2(pi), amplitude 2,and contains the point (pi,2)

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.

Discover how to find the equation of a sinusoidal function with a period of 2(pi), an amplitude of 2, and passing through the point (pi, 2). Learn the step-by-step solution with detailed explanations and equations. Suitable for advanced high school students in Grades 11-12.

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