A sine function has an amplitude of 3, a period of π, and a phase shift of  π/4.  What is the y-intercept of the function?

Given the sine function with the following characteristics:

- Amplitude: \( A = 3 \)
- Period: \( T = \pi \)
- Phase shift: \( \phi = \frac{\pi}{4} \)

The general form of a sine function is:

\[
y(x) = A \sin(Bx - C) + D
\]

Where:
- \( A \) is the amplitude,
- \( B \) is related to the period by \( B = \frac{2\pi}{T} \),
- \( C \) is related to the phase shift by \( C = B \cdot \text{(phase shift)} \),
- \( D \) is the vertical shift (which is 0 in this case since it's not mentioned).

### Step 1: Determine \( B \)
Given the period \( T = \pi \):

\[
B = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2
\]

### Step 2: Determine \( C \)
Using the phase shift \( \phi = \frac{\pi}{4} \):

\[
C = B \cdot \phi = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}
\]

### Step 3: Write the equation of the sine function
The equation becomes:

\[
y(x) = 3 \sin(2x - \frac{\pi}{2})
\]

### Step 4: Find the y-intercept
The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:

\[
y(0) = 3 \sin\left(2(0) - \frac{\pi}{2}\right) = 3 \sin\left(-\frac{\pi}{2}\right)
\]

Since \( \sin\left(-\frac{\pi}{2}\right) = -1 \), we have:

\[
y(0) = 3(-1) = -3
\]

### Final Answer:
The y-intercept of the function is \(-3\).

Would you like more details on any specific step or have any questions? 

Here are five related questions to deepen your understanding:
1. How would the y-intercept change if the amplitude were doubled?
2. What effect would a vertical shift have on the y-intercept?
3. How does the phase shift influence the overall shape of the sine function?
4. If the period were doubled, what would be the new value of \( B \) in the sine function?
5. How can the phase shift be determined if only the function and the period are known?

**Tip:** Understanding how the period and phase shift affect the function's graph can help you quickly identify important features like intercepts and maximum points.

Learn how to find the y-intercept of a sine function with an amplitude of 3, a period of π, and a phase shift of π/4. This problem explores trigonometric concepts such as amplitude, period, and phase shift in sine functions. Suitable for high school students studying trigonometry.

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