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

$y=\frac{1}{4}\sqrt{x}$

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:$y=\frac{1}{4}\sqrt{x}$

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

$$\begin{gathered} \text{For\hspace{0.17em}\hspace{0.17em}}x=0, \\ y=\frac{1}{4}\sqrt{0} \\ =\frac{1}{4}\left(0\right) \\ =0 \end{gathered}$$

$$\begin{gathered} \text{For\hspace{0.17em}\hspace{0.17em}}x=1, \\ y=\frac{1}{4}\sqrt{1} \\ =\frac{1}{4}\left(1\right) \\ =0.25 \end{gathered}$$

$$\begin{gathered} \text{For\hspace{0.17em}\hspace{0.17em}}x=4, \\ y=\frac{1}{4}\sqrt{4} \\ =\frac{1}{4}\left(2\right) \\ =0.5 \end{gathered}$$

$$\begin{gathered} \text{For\hspace{0.17em}\hspace{0.17em}}x=9, \\ y=\frac{1}{4}\sqrt{9} \\ =\frac{1}{4}\left(3\right) \\ =0.75 \end{gathered}$$

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

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

The parent function of$y=\frac{1}{4}\sqrt{x}$ is$y=\sqrt{x}$

The parent function is multiplied by ‘$\frac{1}{4}$’, a value less than 1 and greater than zero. So the graph is a vertical compression of $y=\sqrt{x}$.

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:$[0,\infty )$

Since the y-values are multiplied by $\frac{1}{4}$’ y takes all the positive real values including zero.

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

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