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

$\mathit{y}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\sqrt{\mathbf{x}}\mathbf{-}\mathbf{1}$

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

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

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=1,\\ y=-\frac{1}{4}\sqrt{1}-1\\ =-\frac{1}{4}\left(1\right)-1\\ =-0.25-1\\ =-1.25\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=2,\\ y=-\frac{1}{4}\sqrt{4}-1\\ =-\frac{1}{4}\left(2\right)-1\\ =-0.5-1\\ =-1.5\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ y=-\frac{1}{4}\sqrt{9}-1\\ =-\frac{1}{4}\left(3\right)-1\\ =-0.75-1\\ =-1.75\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=16,\\ y=-\frac{1}{4}\sqrt{16}-1\\ =-\frac{1}{4}\left(4\right)-1\\ =-1-1\\ =-2\end{array}$

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

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

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

Also the parent function is subtracted by ‘$-1$’ the graph$y=-\frac{1}{4}\sqrt{x}-1$ is translated (shifted) downward by 1 units from the origin.

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 $y=-\frac{1}{4}\sqrt{x}-1$, the square root of x is multiplied by$-\frac{1}{4}$ and is subtracted by ‘1’.

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

Find the starting value of $y$by substituting $x=0$in $y=-\frac{1}{4}\sqrt{x}-1$

$\begin{array}{c}y=-\frac{1}{4}\left(0\right)-1\\ =0-1\\ =-1\end{array}$

In $y=-\frac{1}{4}\sqrt{x}-1$, coefficient of $\sqrt{x}$is $-\frac{1}{4}$, which is negative.

Therefore $y$takes all the values of real numbers which are less than or equal to ‘$-1$’.

That is,$y\le -1,\Rightarrow y\in (-\infty ,-1]$

Therefore, Range:$(-\infty ,-1]$