Question

1. choose the correct graph of the function $y = -2\sqrt{x}$

Explanation:

Step1: Analyze the function's domain and sign

The function is \(y=-2\sqrt{x}\). The domain of \(\sqrt{x}\) is \(x\geq0\), and the coefficient \(- 2\) means \(y\leq0\) for all \(x\) in the domain. So the graph should be in the region where \(x\geq0\) and \(y\leq0\).

Step2: Check the first graph

The first graph has \(y\) - values positive for \(x\geq0\) (e.g., at \(x = 4\), \(y\) is positive), which does not match \(y=-2\sqrt{x}\) (since \(y\) should be negative here). So the first graph is incorrect.

Step3: Check the second graph

The second graph is in the region \(x\geq0\) and \(y\leq0\) (since it's below the x - axis for \(x\geq0\)). When \(x = 0\), \(y = 0\) (matches \(y=-2\sqrt{0}=0\)), and when \(x = 4\), \(y=-2\sqrt{4}=-4\) (the graph seems to pass through a point like \((4, - 4)\) - consistent with the function. So this graph is correct.

Answer:

The Middle Graph (assuming the second graph is the middle one here, as the first is top, second is middle - looking at the y - values: for \(y = - 2\sqrt{x}\), when \(x = 0\), \(y=0\); when \(x = 4\), \(y=-2\sqrt{4}=-4\). The first graph has positive y - values for \(x\geq0\), which is wrong. The second graph (middle) has \(y\) negative for \(x\geq0\) and passes through \((0,0)\) and \((4, - 4)\) - like points, so it's the correct one.