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 4, a left phase shift of $\frac { \pi } { 4 }$ , and a vertical translation down 5 units.

A) $y = 4 \sin \left( x + \frac { \pi } { 8 } \right) - 5$
B) $y = 4 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) + 5$
C) $y = 4 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 5$
D) $y = 4 \sin \left( \frac { x } { 2 } + \pi \right) - 5$
E) $y = 5 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 4$

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 4, a left phase shift of $\frac { \pi } { 4 }$ , and a vertical translation down 5 units.
 
A) $y = 4 \sin \left( x + \frac { \pi } { 8 } \right) - 5$
B) $y = 4 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) + 5$
C) $y = 4 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 5$
D) $y = 4 \sin \left( \frac { x } { 2 } + \pi \right) - 5$
E) $y = 5 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 4$

Quizplus · Accepted Answer

The general form of a sine function is $y = A \sin(B(x - C)) + D$, where $A$ is the amplitude, $B$ determines the period by $2\pi/B$, $C$ is the phase shift, and $D$ is the vertical translation. Given an amplitude of 4, a period of $4\pi$, a phase shift to the left of $\pi/4$, and a vertical translation down 5 units, we adjust the formula accordingly. The period adjustment gives $B = 2\pi / (4\pi) = 1/2$, and the phase shift to the left of $\pi/4$ means we add $\pi/4$ inside the sine function, but since $B = 1/2$, it becomes $1/2 \cdot (\pi/4) = \pi/8$. Therefore, the correct equation is $y = 4 \sin \left( \frac { x } { 2 } + \frac { \pi } { 8 } \right) - 5$.

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