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

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

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

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

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

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=4,\\ y=-4\sqrt{4}\\ =-4\left(2\right)\\ =-8\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ y=-4\sqrt{9}\\ =-4\left(3\right)\\ =-12\end{array}$

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

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

The parent function of $y=-4\sqrt{x}$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 ‘$-4$’.

On comparing with the parent function, the graph $y=-4\sqrt{x}$is a vertical stretch of $y=\sqrt{x}$(since the absolute value of ‘$-4$’ is 4 which is greater than 1) and a reflection across the $X$-axis (since the coefficient of $\sqrt{x}$is ‘$-4$’ 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, $x\ge 0,\Rightarrow x\in [0,\infty )$.

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

In $y=-4\sqrt{x}$, the square root of $x$is multiplied by $-4$.

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 $y=-4\sqrt{x}$.

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

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

Therefore, $y$takes all the values in real numbers which are less than or equal to zero.

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

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