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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 ‘$y$’.

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=0,\\ y=−\frac{1}{4}\sqrt{0}−1\\ =−\frac{1}{4}\left(0\right)−1\\ =−1\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=1,\\ y=−\frac{1}{4}\sqrt{1}−1\\ =−\frac{1}{4}\left(1\right)−1\\ =−0.25−1\\ =−1.25\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=2,\\ y=−\frac{1}{4}\sqrt{4}−1\\ =−\frac{1}{4}\left(2\right)−1\\ =−0.5−1\\ =−1.5\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=9,\\ y=−\frac{1}{4}\sqrt{9}−1\\ =−\frac{1}{4}\left(3\right)−1\\ =−0.75−1\\ =−1.75\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=16,\\ y=−\frac{1}{4}\sqrt{16}−1\\ =−\frac{1}{4}\left(4\right)−1\\ =−1−1\\ =−2\end{array}$

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, $xâ‰\yen 0,\text{ â¶Ä‰â¶Ä‰}⇒x∈[0,∞)$.

Therefore, domain:$[0,∞)$

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,$y≤−1,\text{ â¶Ä‰â¶Ä‰}⇒y∈(−∞,−1]$

Therefore, Range:$(−∞,−1]$