Question

which of the following is the graph of $y = -4\sqrt{x}$?

Explanation:

Step1: Analyze the domain

The function is \( y = -4\sqrt{x} \). The square root function \( \sqrt{x} \) is defined for \( x \geq 0 \), so the domain of \( y \) is \( x \geq 0 \). This means the graph should only exist for non - negative x - values.

Step2: Analyze the sign of y - values

For \( x\geq0 \), \( \sqrt{x}\geq0 \), and we have a coefficient of - 4. So \( y=-4\sqrt{x}\leq0 \) for all \( x \) in the domain. This means the graph should be in the region where \( y\leq0 \) (below or on the x - axis) for \( x\geq0 \).

Step3: Analyze the shape and key points

When \( x = 0 \), \( y=-4\sqrt{0}=0 \), so the graph passes through the origin \((0,0)\). When \( x = 1 \), \( y=-4\sqrt{1}=-4 \); when \( x = 4 \), \( y=-4\sqrt{4}=-8 \). The parent function \( y = \sqrt{x} \) has a curve that starts at the origin and increases slowly. The function \( y=-4\sqrt{x} \) is a vertical stretch (by a factor of 4) and a reflection over the x - axis of \( y = \sqrt{x} \), so it should start at the origin and decrease (since it's reflected) as \( x \) increases, with \( y \) values being negative.

Looking at the given graphs, the first graph has positive y - values for \( x\geq0 \), which does not match. The second graph (assuming the lower graph is the one with \( y\leq0 \) for \( x\geq0 \) and passing through the origin with the correct shape) should be the graph of \( y=-4\sqrt{x} \). Since the first graph is in the upper half - plane (positive y) for \( x\geq0 \) and the second graph (the lower one) is in the lower half - plane (negative y) for \( x\geq0 \) (and considering the domain and the shape), the correct graph is the lower one (the second graph shown, probably the one at the bottom of the image).

Answer:

The lower graph (the second graph in the given image)