Q. 12 CYU - Algebra 1

Question

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

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

Step-by-Step Solution

Answer

The domain of $y=-2\sqrt{x+1}$ is $\mathit{x}\mathbf{\in }\mathbf{[}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{\infty }\mathbf{)}$ and the range is $\mathit{y}\mathbf{\in }\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}$.

Step 1. State the concept of parent graph.

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.

Step 2. State the concept of domain and range.

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.

Step 3. Graph the function.

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

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

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

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=3,\\ y=-2\sqrt{3+1}\\ =-2\sqrt{4}\\ =-2\left(2\right)\\ =-4\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=8,\\ y=-2\sqrt{8+1}\\ =-2\sqrt{9}\\ =-2\left(3\right)\\ =-6\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=15,\\ y=-2\sqrt{15+1}\\ =-2\sqrt{16}\\ =-2\left(4\right)\\ =-8\end{array}$

Values of ‘$x$	Values of ‘$y$'	$\left(x,y\right)$
$-1$	0	$(-1,0)$
0	$-2$	$(0,-2)$
3	$-4$	$(3,-4)$
8	$-6$	$(8,-6)$
15	$-8$	$(15,-8)$

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

Step 4. Comparison with the parent graph.

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

That is, $y=\sqrt{x}$

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.

1 is added inside square root of 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 added inside the root, the graph is translated to the left by 1 units. 

Coefficient of $\sqrt{x}$ is ‘$-2$’.

The absolute value of coefficient of $\sqrt{x}$ 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 of $\sqrt{x}$ is negative, the graph is $y=-2\sqrt{x+1}$ a reflection across the $X$-axis. 

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

Step 5. State the domain and range.

Since the values inside the root must be positive.

$x+1\ge 0$

Adding ‘$-1$’ on both the 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: $(-\infty ,0]$