Question

begin by graphing the square root function, f(x)=\sqrt{x}. then, use transformations of this graph to graph the given function, h(x)=-\sqrt{x + 6} choose the correct graph of h(x) below.

Explanation:

Step1: Identify parent function

Parent function: $f(x)=\sqrt{x}$, domain $x\geq0$, range $y\geq0$, starts at $(0,0)$ and increases.

Step2: Analyze horizontal shift

For $h(x)=-\sqrt{x+6}$, replace $x$ with $x+6$: shift left 6 units. New vertex: $(0-6, 0)=(-6,0)$, domain $x\geq-6$.

Step3: Analyze reflection

Negative sign outside $\sqrt{\cdot}$: reflect over x-axis. Range becomes $y\leq0$, graph decreases from vertex.

Step4: Match to options

Find graph starting at $(-6,0)$, going downward/right, $y\leq0$.

Answer:

The correct graph is the top-most option (starting at (-6, 0), curving downward into the fourth quadrant, with y-values negative for all valid x ≥ -6).