Describe how the graph of the function f(x) = \sin \left(x - (\pi)/(6)\right) differs from the graph of \sin x.

Question

A. It shifts the graph of \sin x to the left by (\pi)/(6) units.
B. It shifts the graph of \sin x down by (\pi)/(6) units.
C. It shifts the graph of \sin x up by (\pi)/(6) units.
D. It shifts the graph of \sin x to the right by (\pi)/(6) units.

Answer 1

To analyze how the graph of the function \( f(x) = \sin \left(x - \frac{\pi}{6}\right) \) differs from the graph of \( \sin x \), we should look at the transformation applied to \( x \).

The function \( \sin(x - c) \) represents a horizontal shift of the sine function. Specifically:

• If \( c \) is positive, the graph shifts to the right by \( c \) units.
• If \( c \) is negative, the graph shifts to the left by \( |c| \) units.

In your case, you have \( c = \frac{\pi}{6} \) which is positive. Therefore, the graph of \( f(x) = \sin \left(x - \frac{\pi}{6}\right) \) is shifted to the right by \( \frac{\pi}{6} \) units.

Thus, the correct option is:

D. It shifts the graph of \(\sin x\) to the right by \(\frac{\pi}{6}\) units.

Answer 2

Are you sure

Answer 3

Yes, I am sure. Let's clarify again how the transformation works:

The function \( f(x) = \sin \left(x - \frac{\pi}{6}\right) \) can be analyzed by considering the argument of the sine function, which is \( x - \frac{\pi}{6} \).

In general, for a sine function of the form \( \sin(x - c) \):

• If \( c > 0 \), the graph shifts to the right by \( c \) units.
• If \( c < 0 \), the graph shifts to the left by \( -c \) units.

In your function, \( c = \frac{\pi}{6} \), which is positive, meaning the graph of \( \sin x \) will shift to the right by \( \frac{\pi}{6} \) units.

Therefore, the correct conclusion is that the graph of \( f(x) = \sin \left(x - \frac{\pi}{6}\right) \) is indeed shifted to the right by \( \frac{\pi}{6} \) units.

So, the answer is:

D. It shifts the graph of \(\sin x\) to the right by \(\frac{\pi}{6}\) units.