Question

use transformations of ( f(x)=sqrt{x} ) to graph the following function.
( h(x)=sqrt{x + 5}-4 )
choose the correct graph below.

Explanation:

Step1: Recall transformation rules

For a function \(y = f(x + a)+b\), if \(a>0\), the graph of \(y = f(x)\) is shifted \(a\) units to the left. If \(b < 0\), the graph of \(y = f(x)\) is shifted \(|b|\) units down.
For \(h(x)=\sqrt{x + 5}-4\) and \(f(x)=\sqrt{x}\), here \(a = 5\) and \(b=-4\).

Step2: Analyze the vertex

The vertex of \(y=\sqrt{x}\) is \((0,0)\). After the transformation \(x\) - coordinate of the vertex: \(x+5 = 0\Rightarrow x=-5\), \(y\) - coordinate of the vertex: \(y=-4\). So the vertex of \(h(x)=\sqrt{x + 5}-4\) is \((-5,-4)\)

Answer:

Assuming that among the options, the graph with vertex \((-5,-4)\) is the correct one. Without seeing the actual visual details of the graphs in A, B, C, D (but based on vertex analysis), if we assume standard graphing, the graph which has its starting point (vertex) at \((-5,-4)\) is the correct one.