Question

question 9 of 10
describe the transformation of $f(x) = \sin x$ to $g(x) = \sin \left(x + \frac{\pi}{3}\
ight)$.

a. $f(x)$ is shifted $\frac{\pi}{3}$ units down.

b. $f(x)$ is shifted $\frac{\pi}{3}$ units to the right.

c. $f(x)$ is shifted $\frac{\pi}{3}$ units up.

d. $f(x)$ is shifted $\frac{\pi}{3}$ units to the left.

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( y = f(x + h) \), if \( h>0 \), the graph of \( f(x) \) is shifted left by \( h \) units; if \( h<0 \), it's shifted right by \( |h| \) units.

Step2: Analyze \( g(x)=\sin(x + \frac{\pi}{3}) \)

Compare \( g(x)=\sin(x + \frac{\pi}{3}) \) with \( y = f(x + h) \) where \( f(x)=\sin x \) and \( h=\frac{\pi}{3}>0 \). So, \( f(x) \) is shifted left by \( \frac{\pi}{3} \) units.

Answer:

D. \( f(x) \) is shifted \( \frac{\pi}{3} \) units to the left.