Graph the function, and compare to the parent graph. State the domain and range.

$\mathit{y}\mathbf{=}\mathbf{3}\sqrt{\mathbf{x}\mathbf{-}\mathbf{2}}$

Parent graph:The simplest form of the given function is called the parent function of that function and the graph of the parent function is called parent graph.

Domain: The set of all possible values for which given function defined is called domain.

Range: The set of all possible values of the given function is called range.

The given function is: $y=3\sqrt{x-2}$

In order to graph a function, find few co-ordinates by substituting values of ‘$x$’ and find finding the respective values of ‘$y$’.

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=2\\ y=3\sqrt{2-2}\\ =3\sqrt{0}\\ =3\left(0\right)\\ =0\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=3\\ y=3\sqrt{3-2}\\ =3\sqrt{1}\\ =3\left(1\right)\\ =3\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=6,\\ y=3\sqrt{6-2}\\ =3\sqrt{4}\\ =3\left(2\right)\\ =6\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=11,\\ y=3\sqrt{11-2}\\ =3\sqrt{9}\\ =3\left(3\right)\\ =9\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=18,\\ y=3\sqrt{18-2}\\ =3\sqrt{16}\\ =3\left(4\right)\\ =12\end{array}$

Plot these co-ordinates on a coordinate plane and join those points to get the required graph.

The parent function of$y=3\sqrt{x-2}$ is the simplest square root function.

That is,$y=\sqrt{x}$

The graph of parent function$y=\sqrt{x}$ is given below.

Note: Since the parent function is just used for comparison, it is graphed using graphing calculator.

2 is subtracted inside square root of parent function $y=\sqrt{x}$and then is multiplied by ‘3’. So the graph$y=3\sqrt{x-2}$ is the translation of the parent graph $y=\sqrt{x}$.

Since 2 is subtracted inside the root, the graph is translated to the right by 2 units.

Coefficient of$\sqrt{x-2}$ is ‘2’.

As 2 is greater than 1. The graph is a vertical stretch of the parent graph $y=\sqrt{x}$.

Therefore, on comparison with the parent graph, the graph $y=3\sqrt{x-2}$, a vertical stretch of$y=\sqrt{x}$ and is translated to the right by 2 units.

Since the values inside the root must be positive.

$x-2\ge 0$

Adding ‘2’ on both the sides.

$\begin{array}{c}x-2+2\ge 0+2\\ x+0\ge 2\\ x\ge 2\end{array}$

Therefore,$x\ge 2\Rightarrow x\in [2,\infty )$

Therefore, domain: $[2,\infty )$

As the coefficient of the square root term ‘$\sqrt{x-2}$ ‘is ‘3’ which is positive, y takes all the positive real values including zero.

Therefore,$y\ge 0,\Rightarrow y\in [0,\infty )$

Therefore, Range:$[0,\infty )$