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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: $y=−\frac{1}{2}\sqrt{x}$

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

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=0,\\ y=−\frac{1}{2}\sqrt{0}\\ =−\frac{1}{2}\left(0\right)\\ =0\end{array}$

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

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

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=9,\\ y=−\frac{1}{2}\sqrt{9}\\ =−\frac{1}{2}\left(3\right)\\ =−1.5\end{array}$

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

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

The parent function of$y=−\frac{1}{2}\sqrt{x}$ 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.

The parent function is multiplied by $−\frac{1}{2}$.

On comparing with the parent function, the graph $y=−\frac{1}{2}\sqrt{x}$is a vertical compression of $y=\sqrt{x}$(since the absolute value of $−\frac{1}{2}$is $\frac{1}{2}$ which is less than 1 and greater than zero) and a reflection across the $X$-axis (since the coefficient of $\sqrt{x}$is ‘$-\frac{1}{2}$’ which is negative).

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 $y=−\frac{1}{2}\sqrt{x}$, the square root of x is multiplied by $−\frac{1}{2}$.

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

Find the starting value of $y$by substituting $x=0$in $y=−\frac{1}{2}\sqrt{x}$.

$\begin{array}{c}y=−\frac{1}{2}\left(\sqrt{0}\right)\\ =−\frac{1}{2}\left(0\right)\\ =0\end{array}$

In $y=−\frac{1}{2}\sqrt{x}$, coefficient of $\sqrt{x}$is $−\frac{1}{2}$, which is negative.

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

That is,$y≤0,\text{ â¶Ä‰â¶Ä‰}⇒y∈(−∞,0]$

Therefore, Range:$(−∞,0]$