To find the [tex]\( y \)[/tex]-intercept of the given sine function, we need to analyze the function's components and apply them correctly.
The general form of a sine function is:
[tex]\[ y = A \sin(B(x - C)) \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude.
- [tex]\( B \)[/tex] affects the period of the function.
- [tex]\( C \)[/tex] is the phase shift.
From the problem, we know:
- The amplitude ([tex]\( A \)[/tex]) is 3.
- The period is [tex]\( \pi \)[/tex].
- The phase shift ([tex]\( C \)[/tex]) is [tex]\(\frac{\pi}{4}\)[/tex].
### Step-by-Step Solution:
1. Determine [tex]\( B \)[/tex]:
The period ([tex]\( P \)[/tex]) of a sine function is related to [tex]\( B \)[/tex] by the equation:
[tex]\[ B = \frac{2\pi}{P} \][/tex]
Substituting the given period:
[tex]\[ B = \frac{2\pi}{\pi} = 2 \][/tex]
2. Identify the phase shift:
The phase shift ([tex]\( C \)[/tex]) is [tex]\(\frac{\pi}{4}\)[/tex].
3. Formulate the sine function:
With the given values of amplitude, [tex]\( B \)[/tex], and phase shift, the function becomes:
[tex]\[
y = 3 \sin(2(x - \frac{\pi}{4}))
\][/tex]
4. Determine the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
y = 3 \sin\left(2(0 - \frac{\pi}{4})\right)
\][/tex]
Simplify inside the sine function:
[tex]\[
y = 3 \sin\left(2 \left(- \frac{\pi}{4}\right)\right) = 3 \sin\left( -\frac{\pi}{2} \right)
\][/tex]
5. Evaluate the sine function:
[tex]\[
\sin\left(-\frac{\pi}{2}\right) = -1
\][/tex]
Therefore:
[tex]\[
y = 3 \times (-1) = -3
\][/tex]
### Conclusion:
Thus, the [tex]\( y \)[/tex]-intercept of the function is [tex]\(\boxed{-3}\)[/tex].
