The formula for the period is [tex] \frac{2 \pi }{b} [/tex] and the formula for the phase shift is [tex] \frac{-c}{b} [/tex], the same b value as in the period. The general form of the sin function is [tex]y=Asin(bx+c)[/tex], where A is the amplitude. We are given the period as [tex] \frac{ \pi }{4} [/tex], so we will use the formula for the period to solve for b: [tex] \frac{ \pi }{4} = \frac{2 \pi }{b} [/tex]. Cross multiply to get [tex]b \pi =8 \pi [/tex] and b = 8. Now the phase shift. We will use the formula for phase shift along with the b value we found to find c. The phase shift (horizontal shift) is [tex] \frac{ \pi }{2} [/tex], so fitting that into our formula, [tex] \frac{ \pi }{2}= \frac{-c}{8} [/tex]. Cross multiply to get [tex]-2c=8 \pi [/tex] and [tex]c=-4 \pi [/tex]. Now let's put all this together. [tex]y=6sin(8x-4 \pi )[/tex]. We will factor out the 8 to get the final equation of [tex]y=6sin[8(x- \frac{ \pi }{2})] [/tex], choice C from above.
