To derive the general equation of a sine function given its amplitude, period, and horizontal shift, let's proceed step-by-step:
1. Amplitude:
The amplitude [tex]\(A\)[/tex] is the coefficient in front of the sine function. For the given function, the amplitude is 2. So, [tex]\( A = 2 \)[/tex].
2. Period:
The period [tex]\(T\)[/tex] of a sine function is given by the formula [tex]\( T = \frac{2\pi}{B} \)[/tex], where [tex]\(B\)[/tex] is the frequency. We know that the period [tex]\(T\)[/tex] is given as [tex]\(\pi\)[/tex].
Using the period formula, we can solve for [tex]\(B\)[/tex]:
[tex]\[
\pi = \frac{2\pi}{B}
\][/tex]
Multiplying both sides by [tex]\(B\)[/tex]:
[tex]\[
\pi B = 2\pi
\][/tex]
Dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[
B = 2
\][/tex]
3. Horizontal Shift:
The horizontal shift [tex]\(C\)[/tex] is the value that shifts the sine function left or right. For the given function, the horizontal shift is [tex]\(\pi\)[/tex]. Therefore, [tex]\( C = \pi \)[/tex].
4. Putting it all together:
The general form of a sine function is [tex]\( y = A \sin(B(x - C)) \)[/tex].
- The amplitude [tex]\(A\)[/tex] is 2.
- The frequency [tex]\(B\)[/tex] is 2.
- The horizontal shift [tex]\(C\)[/tex] is [tex]\(\pi\)[/tex].
Therefore, substituting [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] into the general equation:
[tex]\[
y = 2 \sin(2(x - \pi))
\][/tex]
So the equation of the sine function is:
[tex]\[
y = 2 \sin(2(x - \pi))
\][/tex]
Among the given choices, the correct equation is:
[tex]\[
y = 2 \sin(2(x - \pi))
\][/tex]
