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

$\mathit{y}\mathbf{=}\mathbf{-}\mathbf{2}\sqrt{\mathbf{x}\mathbf{-}\mathbf{1}}$

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 the parent graph.

Domain: The set of all possible values for which a given function is defined is called domain.

Range: The set of all possible values of the given function is called range.

The given function is: $y=-2\sqrt{x-1}$

To graph a function, find a few coordinates by substituting values of ‘$x$’ and find finding the respective values of ‘$y$’.

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=1\\ y=-2\sqrt{1-1}\\ =-2\sqrt{0}\\ =-2\left(0\right)\\ =0\end{array}$

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=5,\\ y=-2\sqrt{5-1}\\ =-2\sqrt{4}\\ =-2\left(2\right)\\ =-4\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=10,\\ y=-2\sqrt{10-1}\\ =-2\sqrt{9}\\ =-2\left(3\right)\\ =-6\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=17,\\ y=-2\sqrt{17-1}\\ =-2\sqrt{16}\\ =-2\left(4\right)\\ =-8\end{array}$

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

The parent function $y=-2\sqrt{x-1}$is the simplest square root function.

That is,$y=\sqrt{x}$

The graph of the parent function$y=\sqrt{x}$ is given below.

Note: Since the parent function is just used for comparison, it is graphed using a graphing calculator.

1 is subtracted inside the square root of the parent function $y=\sqrt{x}$and then is multiplied by ‘$-2$’. So the graph $y=-2\sqrt{x-1}$ is the translation and reflection of the parent graph $y=\sqrt{x}$.

Since 1 is subtracted inside the root, the graph is translated to the right by 1 unit.

The coefficient $\sqrt{x}$is ‘$-2$’.

The absolute value of the coefficient$\sqrt{x}$ is 2.

As 2 is greater than 1. The graph is a vertical stretch of the parent graph $y=\sqrt{x}$.

Also as the coefficient $\sqrt{x}$is negative, the graph is $y=-2\sqrt{x-1}$ a reflection across the $X$-axis.

Therefore, in comparison with the parent graph, the graph $y=-2\sqrt{x-1}$ is a reflection across the $x$-axis, a vertical stretch, and is translated to the right by 1 unit.

Since the values inside the root must be positive.

$x-1\ge 0$

Adding ‘1’ on both sides.

$\begin{array}{c}x-1+1\ge 0+1\\ x+0\ge 1\\ x\ge 1\end{array}$

Therefore, $x\ge 1\Rightarrow x\in [1,\infty )$

Therefore, domain: $[1,\infty )$

As the coefficient of the square root term ‘ $\sqrt{x-1}$‘ is ‘$-2$’ which is negative, $y$takes all the negative real values including zero.

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

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