To find the general equation of a sine function given the amplitude, period, and horizontal shift, we follow these steps:
1. Amplitude (A): The coefficient in front of the sine function represents the amplitude. Given that the amplitude is 6, this directly translates to the function being multiplied by 6.
2. Period (T): The period of a sine function is given by the formula [tex]\( T = \frac{2\pi}{B} \)[/tex], where [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the sine function. Given that the period is [tex]\( \frac{\pi}{4} \)[/tex], we can substitute this value into the formula and solve for [tex]\( B \)[/tex]:
[tex]\[
\frac{2\pi}{B} = \frac{\pi}{4}
\][/tex]
To find [tex]\( B \)[/tex], multiply both sides of the equation by [tex]\( B \)[/tex]:
[tex]\[
2\pi = B \cdot \frac{\pi}{4}
\][/tex]
Now, solve for [tex]\( B \)[/tex] by isolating it:
[tex]\[
B = \frac{2\pi}{\pi/4} = 8
\][/tex]
3. Horizontal Shift (C): The horizontal shift is given directly and is represented within the sine function as [tex]\( x - C \)[/tex]. Given that the horizontal shift is [tex]\( \frac{\pi}{2} \)[/tex], it appears in the equation as [tex]\( x - \frac{\pi}{2} \)[/tex].
Putting these values together into the general form of the sine function [tex]\( y = A \sin(B(x - C)) \)[/tex], we get:
[tex]\[
y = 6 \sin(8(x - \frac{\pi}{2}))
\][/tex]
Therefore, the correct equation for the sine function with the given parameters is:
[tex]\[ y = 6 \sin (8 (x - \frac{\pi}{2})) \][/tex]
So the correct answer is:
[tex]\[ y = 6 \sin \left(8 \left(x - \frac{\pi}{2}\right)\right) \][/tex]
