Describe the transformations required to obtain the graph of the function f(x) from the graph of the function g(x).
-$$f ( x ) = \sin \left( x - \frac { \pi } { 2 } \right) ; g ( x ) = \sin \left( x + \frac { \pi } { 3 } \right)$$
A) Translate to the left $\frac { 5 \pi } { 6 }$ units
B) Translate to the right $\frac { 5 \pi } { 6 }$ units
C) Translate to the right $\frac { \pi } { 6 }$ units
D) Translate to the left $\frac { \pi } { 2 }$ units

Question

Quizplus · Accepted Answer

To obtain the graph of $f(x) = \sin\left(x - \frac{\pi}{2}\right)$ from $g(x) = \sin\left(x + \frac{\pi}{3}\right)$, we need to shift $g(x)$ to the right by the difference in their phase shifts. The phase shift of $f(x)$ is $-\frac{\pi}{2}$ and that of $g(x)$ is $\frac{\pi}{3}$. The total shift is $-\left(-\frac{\pi}{2}\right) + \frac{\pi}{3} = \frac{\pi}{2} + \frac{\pi}{3} = \frac{5\pi}{6}$ units to the right.

Describe the Transformations Required to Obtain the Graph of the Function