Question

if ( f(x)=sqrt{x} ), which equation describes the graphed function?
a. ( y = f(-x) - 3 )
b. ( y = -f(x) + 3 )
c. ( y = -f(x) - 3 )
d. ( y = f(-x) + 3 )

Explanation:

Step1: Analyze the original function \( f(x)=\sqrt{x} \)

The original function \( f(x)=\sqrt{x} \) has a domain \( x\geq0 \) and starts at the origin \((0,0)\), increasing as \( x \) increases.

Step2: Analyze the transformations

• Reflection: The graphed function is decreasing or has a negative - like behavior compared to the original \( f(x)=\sqrt{x} \). A reflection over the \( x \) - axis is given by \( y = - f(x) \), which changes the sign of the \( y \) - values of the original function.
• Vertical shift: The original function \( f(x)=\sqrt{x} \) has a \( y \) - intercept at \( (0,0) \). The graphed function has a \( y \) - intercept at \( (0, - 3) \)? Wait, no. Wait, let's re - examine. Wait, the original \( f(x)=\sqrt{x} \) at \( x = 0 \) is \( y=0 \). The graphed function: let's see the options. Let's check the transformations for each option.

Option A: \( y = f(-x)-3 \). \( f(-x)=\sqrt{-x} \), domain \( x\leq0 \). The graph of \( \sqrt{-x} \) is a reflection over the \( y \) - axis of \( \sqrt{x} \), and then shifted down 3 units. But the graphed function here: let's check the \( y \) - intercept. If \( x = 0 \), \( y=f(0)-3=0 - 3=-3 \). But the graph in the picture at \( x = 0 \) is \( y=-3 \)? Wait, no, the graph in the picture at \( x = 0 \) is \( y=-3 \)? Wait, the graph in the picture: when \( x = 0 \), the \( y \) - value is - 3? Wait, no, looking at the graph, the curve starts at \( (0,-3) \) and goes down? Wait, no, the original \( f(x)=\sqrt{x} \) is in the first quadrant. The graphed function: let's check the options.

Option B: \( y=-f(x)+3=- \sqrt{x}+3 \). At \( x = 0 \), \( y = 3 \), which does not match the graph's \( y \) - intercept (which seems to be - 3? Wait, no, the graph in the picture: the curve is in the fourth quadrant? Wait, no, the \( y \) - axis: the graph starts at \( (0,-3) \) and goes to the right, decreasing. Wait, maybe I made a mistake. Let's re - evaluate.

Wait, the original function \( f(x)=\sqrt{x} \): points like \( (0,0) \), \( (1,1) \), \( (4,2) \).

For option C: \( y=-f(x)-3=-\sqrt{x}-3 \). At \( x = 0 \), \( y=-0 - 3=-3 \). At \( x = 1 \), \( y=-\sqrt{1}-3=-1 - 3=-4 \). At \( x = 4 \), \( y=-\sqrt{4}-3=-2 - 3=-5 \). Let's check the graph: when \( x = 0 \), \( y=-3 \); when \( x = 1 \), \( y=-4 \); when \( x = 4 \), \( y=-5 \). This matches the general shape of the graph (starting at \( (0,-3) \) and decreasing as \( x \) increases).

For option A: \( y = f(-x)-3=\sqrt{-x}-3 \). The domain is \( x\leq0 \), but the graph in the picture has \( x\geq0 \) (since it is to the right of the \( y \) - axis), so A is out.

For option B: \( y=-f(x)+3=-\sqrt{x}+3 \). At \( x = 0 \), \( y = 3 \), which does not match the graph's \( y \) - intercept (which is - 3), so B is out.

For option D: \( y = f(-x)+3=\sqrt{-x}+3 \), domain \( x\leq0 \), and at \( x = 0 \), \( y = 3 \), which does not match, so D is out.

So the correct transformation is a reflection over the \( x \) - axis (\( y=-f(x) \)) and a vertical shift down by 3 units (\( y=-f(x)-3 \)).

Answer:

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