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

$\mathit{y}\mathbf{=}\mathbf{-}\mathbf{2}\sqrt{\mathbf{x}}$

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: $y=-2\sqrt{x}$

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

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=0,\\ y=-2\sqrt{0}\\ =-2\left(0\right)\\ =0\end{array}$

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

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

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ y=-2\sqrt{9}\\ =-2\left(3\right)\\ =-6\end{array}$

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=16,\\ y=-2\sqrt{16}\\ =-2\left(4\right)\\ =-8\end{array}$

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

The parent function of $y=-2\sqrt{x}$is the simple 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 ‘$-2$’.

On comparing with the parent function, the graph$y=-2\sqrt{x}$ is a vertical stretch of$y=\sqrt{x}$ (since the absolute value of ‘$-2$’ is 2 which is greater than 1) and a reflection across the $X$-axis (since the coefficient of$\sqrt{x}$ is ‘$-2$’ 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, $x\ge 0,\Rightarrow x\in [0,\infty )$.

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

In $y=-2\sqrt{x}$, the square root of $x$is multiplied by $-2$.

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 $y=-2\sqrt{x}$.

$\begin{array}{l}y=-2\sqrt{0}\\ =-2\left(0\right)\\ =0\end{array}$