Write an equation for the function that is described by the given characteristics.
A cosine curve with a period of 4$\pi$, an amplitude of 7, a right phase shift of $\frac { \pi } { 2 }$ , and a vertical translation up 2 units.

A) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
B) $y = 2 + 7 \cos \left( \frac { x } { 2 } + \pi \right)$
C) $y = 2 - 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
D) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \pi \right)$
E) $y = 2 + 7 \cos ( x - \pi )$

Question

Write an equation for the function that is described by the given characteristics.  
A cosine curve with a period of 4$\pi$, an amplitude of 7, a right phase shift of $\frac { \pi } { 2 }$ , and a vertical translation up 2 units.
 
A) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
B) $y = 2 + 7 \cos \left( \frac { x } { 2 } + \pi \right)$
C) $y = 2 - 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$
D) $y = 2 + 7 \cos \left( \frac { x } { 2 } - \pi \right)$
E) $y = 2 + 7 \cos ( x - \pi )$

Quizplus · Accepted Answer

The general form of a cosine function is $y = A \cos(B(x - C)) + D$, where $A$ is the amplitude, $\frac{2\pi}{B}$ is the period, $C$ is the phase shift, and $D$ is the vertical translation. Given an amplitude of 7, a period of $4\pi$, a phase shift to the right of $\frac{\pi}{2}$, and a vertical translation up 2 units, we can determine the values as follows: $A = 7$, $B = \frac{2\pi}{4\pi} = \frac{1}{2}$, $C = \frac{\pi}{2}$, and $D = 2$. The phase shift in the cosine function is represented as a subtraction inside the function, so a right shift of $\frac{\pi}{2}$ translates to subtracting $\frac{\pi}{2}$ inside the function. However, because $B = \frac{1}{2}$, the actual value to subtract becomes $\frac{\pi}{2} \times \frac{1}{2} = \frac{\pi}{4}$. Therefore, the correct equation is $y = 2 + 7 \cos \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right)$.

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