Question

graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. start with the graph of the basic function shown below. h(x) = √(x - 3) choose the correct graph below. the basic function graph is y = √x. options a, b, c, d are provided with their respective graphs.

Explanation:

Step1: Analyze the transformation rule

For the function \(y = f(x - h)\), it is a horizontal shift of the function \(y = f(x)\). If \(h>0\), the graph shifts \(h\) units to the right; if \(h < 0\), the graph shifts \(|h|\) units to the left.
For the basic function \(y=\sqrt{x}\) and the given function \(h(x)=\sqrt{x - 3}\), here \(h = 3>0\).

Step2: Determine the domain and key - point shift

The domain of \(y=\sqrt{x}\) is \(x\geq0\), and the domain of \(h(x)=\sqrt{x - 3}\) is \(x-3\geq0\), i.e., \(x\geq3\).
The key - point of \(y = \sqrt{x}\) is \((0,0)\). For \(h(x)=\sqrt{x - 3}\), when \(x=3\), \(h(3)=\sqrt{3 - 3}=0\). So the graph of \(y=\sqrt{x}\) is shifted 3 units to the right.

Answer:

Assuming that in the options, the graph which has the starting point at \((3,0)\) and has the same shape as \(y = \sqrt{x}\) (a curve increasing for \(x\geq3\)) is the correct one. If option C (assuming in the original multiple - choice setup where the shifted graph to the right by 3 units is labeled as C) has the starting point at \((3,0)\) and the correct increasing trend for \(x\geq3\), then the answer is C.