Question

select the correct answer.
if $f(x) = \sqrt{x}$, which equation describes the graphed function?
graph of a coordinate plane with a curve starting at (0, -3) and decreasing to the right
a. $y = f(-x) - 3$
b. $y = -f(x) + 3$
c. $y = -f(x) - 3$
d. $y = f(-x) + 3$

Explanation:

Step1: Analyze reflection and vertical shift

The original function is \( f(x)=\sqrt{x} \), which has domain \( x\geq0 \) and starts at (0,0), increasing. The graphed function is defined for \( x\leq0 \) (reflection over y - axis, so \( f(-x)=\sqrt{-x} \)) and shifted down. Let's check the y - intercept. The graph crosses the y - axis at (0, - 3).

Step2: Evaluate each option

• Option A: \( y = f(-x)-3=\sqrt{-x}-3 \). At \( x = 0 \), \( y=\sqrt{0}-3=-3 \), which matches the y - intercept. The domain is \( x\leq0 \) (since we have \( \sqrt{-x} \)), and the function is a reflection over y - axis and vertical shift down 3 units.
• Option B: \( y=-f(x)+3=-\sqrt{x}+3 \). Domain is \( x\geq0 \), but the graphed function is defined for \( x\leq0 \), so eliminate B.
• Option C: \( y=-f(x)-3=-\sqrt{x}-3 \). Domain is \( x\geq0 \), eliminate C.
• Option D: \( y = f(-x)+3=\sqrt{-x}+3 \). At \( x = 0 \), \( y = 3 \), which does not match the y - intercept (0, - 3), eliminate D.

Answer:

A. \( y = f(-x)-3 \)