Question

begin by graphing the square root function, ( f(x)=sqrt{x} ). then, use transformations of this graph to graph the given function.
( g(x)=\frac{1}{3}sqrt{x - 3} )

Explanation:

Step1: Graph the parent function

The parent function is \(y = f(x)=\sqrt{x}\). Its domain is \(x\geq0\) and it passes through the points \((0,0)\) and \((1,1)\).

Step2: Horizontal translation

For the function \(y=\sqrt{x - 3}\), we use the transformation rule \(y = f(x - h)\) (horizontal translation). Here \(h = 3\), so the graph of \(y=\sqrt{x}\) is shifted 3 units to the right. The domain of \(y=\sqrt{x-3}\) is \(x\geq3\)

Step3: Vertical scaling

For the function \(g(x)=\frac{1}{3}\sqrt{x - 3}\), we use the transformation rule \(y=af(x)\). Here \(a=\frac{1}{3}\). Each \(y\) - value of the graph of \(y = \sqrt{x-3}\) is multiplied by \(\frac{1}{3}\). So, if \((x,y)\) is on the graph of \(y=\sqrt{x - 3}\), then \((x,\frac{1}{3}y)\) is on the graph of \(y=\frac{1}{3}\sqrt{x - 3}\)

Answer:

To graph \(g(x)=\frac{1}{3}\sqrt{x - 3}\), first graph \(y = \sqrt{x}\). Then shift the graph of \(y=\sqrt{x}\) 3 units to the right to get \(y=\sqrt{x - 3}\), and then vertically compress the graph of \(y=\sqrt{x - 3}\) by a factor of \(\frac{1}{3}\) to obtain the graph of \(g(x)=\frac{1}{3}\sqrt{x - 3}\)