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 + 1} )
choose the correct graph of ( h(x) ) below.

Explanation:

Step1: Identify parent function

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

Step2: Apply horizontal shift

For $h(x)=-\sqrt{x+1}$, replace $x$ with $x+1$: shift $f(x)$ left 1 unit. New vertex at $(-1,0)$, domain $x\geq-1$.

Step3: Apply vertical reflection

Multiply by $-1$: reflect over x-axis. The function now decreases from $(-1,0)$ as $x$ increases.

Step4: Match to graph

Find graph starting at $(-1,0)$, decreasing, valid for $x\geq-1$.

Answer:

The correct graph is the first option (topmost graph, starting at $(-1,0)$ and decreasing downward to the right).