Graph the function. Compare to the parent graph. State the domain and range.

$\mathit{y}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\sqrt{\mathbf{x}}$

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: $y=-\frac{1}{3}\sqrt{x}$

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=0,\\ y=-\frac{1}{3}\sqrt{0}\\ =-\frac{1}{3}\left(0\right)\\ =0\end{array}$

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=4,\\ y=-\frac{1}{3}\sqrt{4}\\ =-\frac{1}{3}\left(2\right)\\ =-\frac{2}{3}\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ y=-\frac{1}{3}\sqrt{9}\\ =-\frac{1}{3}\left(3\right)\\ =-1\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=16,\\ y=-\frac{1}{3}\sqrt{16}\\ =-\frac{1}{3}\left(4\right)\\ =-\frac{4}{3}\end{array}$

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

The parent function of$y=-\frac{1}{3}\sqrt{x}$ is the simplest square root function.

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

Using graphing calculator, 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.

Coefficient of$\sqrt{x}$ is $-\frac{1}{3}$.

The absolute value of coefficient of $\sqrt{x}$is $\frac{1}{3}$.

As $\frac{1}{3}$is less than 1. The graph is a vertical compression of the parent graph $y=\sqrt{x}$.

Also as the coefficient $-\frac{1}{3}$of $\sqrt{x}$ is negative, the graph is a reflection across the $X$-axis.

Therefore, on comparison with the parent graph, the graph $y=-\frac{1}{3}\sqrt{x}$ a vertical compression and a reflection across $x$-axis.

Since ‘$x$’ is inside the root, the values inside the root must be positive.

Therefore, values of $x$is all positive real numbers including zero.

That is, $x\ge 0,\Rightarrow x\in [0,\infty )$.

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

In $y=-\frac{1}{3}\sqrt{x}$, the square root of$x$is multiplied by $-\frac{1}{3}$.

As square root is always positive, the least value it takes is zero.

Find the starting value of the function by substituting $x=0$in $y=-\frac{1}{3}\sqrt{x}$.

$\begin{array}{c}y=-\frac{1}{3}\left(\sqrt{0}\right)\\ =-\frac{1}{3}\left(0\right)\\ =0\end{array}$

In $y=-\frac{1}{3}\sqrt{x}$, coefficient of $\sqrt{x}$is $-\frac{1}{3}$, which is negative.

Therefore$y$ takes all the values of negative real numbers including zero.

That is,$y\le 0,\Rightarrow y\in (-\infty ,0]$

Therefore, Range:$(-\infty ,0]$