Question

describe the transformation of f(x) to g(x)
(image of f(x) = sin x and g(x) graphs)
a. f(x) is shifted down 2 units to g(x).
b. f(x) is shifted down \\(\frac{\pi}{2}\\) units to g(x).
c. f(x) is shifted up 2 units to g(x).
d. f(x) is shifted up \\(\frac{\pi}{2}\\) units to g(x).

Explanation:

Step1: Analyze Vertical Shift

The graph of \( f(x)=\sin x \) (gray) and \( g(x) \) (red) are compared. The key is to check the vertical displacement. The \( y \)-intercept of \( f(x) \) is at \( (0,0) \), and for \( g(x) \), it's at \( (0, - 2) \) (or observing the entire graph, each point of \( g(x) \) is 2 units below \( f(x) \)).

Step2: Match with Options

Option A says shifted down 2 units, which matches the vertical shift (each point of \( f(x) \) moves 2 units down to get \( g(x) \)). Other options (shift down \( \frac{\pi}{2} \), up 2, up \( \frac{\pi}{2} \)) don't align with the visual shift.

Answer:

A. \( f(x) \) is shifted down 2 units to \( g(x) \)