Write an equation for the function that is described by the given characteristics. A sine curve with a period of $4 \pi$, an amplitude of 8 , a left phase shift of $\frac { \pi } { 4 }$, and a vertical translation down 1 unit.
A)$y = 8 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 1$
B)$y = 1 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 8$
C) $y = 8 \sin \left( \frac { x } { 2 } + \pi \right) - 1$
D) $y = 8 \sin \left( x + \frac { \pi } { 2 } \right) - 1$
E) $y = 8 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) + 1$

Question

Write an equation for the function that is described by the given characteristics. A sine curve with a period of $4 \pi$, an amplitude of 8 , a left phase shift of $\frac { \pi } { 4 }$, and a vertical translation down 1 unit. 
A)$y = 8 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 1$
B)$y = 1 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 8$
C) $y = 8 \sin \left( \frac { x } { 2 } + \pi \right) - 1$
D) $y = 8 \sin \left( x + \frac { \pi } { 2 } \right) - 1$
E) $y = 8 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) + 1$

Quizplus · Accepted Answer

A sine curve with a period of $4\pi$ and a left phase shift of $\frac{\pi}{4}$ has an equation of the form $y=a\sin\left(\frac{x}{b}-c\right)$, where the amplitude is $|a|$, the period is $2\pi b$, and the phase shift is $c$. 
Plugging in the given values, we have:
$|a|=8$, so $a=8$ (since the sine curve is not reflected).
The period is $4\pi$, so $2\pi b = 4\pi$ and $b=2$.
The phase shift is $\frac{\pi}{4}$, so $c = \frac{\pi}{4}$.
Finally, we have a vertical translation down 1 unit, so we subtract 1 from the entire function, giving us the equation: 
$y=8\sin\left(\frac{x}{2}-\frac{\pi}{8}\right)-1$
Therefore, the correct choice is A.

Write an Equation for the Function That Is Described by the Given