Q29. - Algebra 1

Question

Graph the function. Compare to the parent graph. State the domain and range.

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

Step-by-Step Solution

Answer

The domain of $y=\sqrt{x-4}$ is $\mathit{x}\mathbf{\in }\mathbf{[}\mathbf{4}\mathbf{,}\mathbf{\infty }\mathbf{)}$ and the range is $\mathit{y}\mathbf{\in }\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$.

Step 1. State the concept of the 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 the parent graph.

Step 2. State the concept of domain and range.

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.

Step 3. Graph the function.

The given function is: $y=\sqrt{x-4}$

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

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

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

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=13,\\ y=\sqrt{13-4}\\ =\sqrt{9}\\ =3\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=20,\\ y=\sqrt{20-4}\\ =\sqrt{16}\\ =4\end{array}$

Values of ‘$x$	Values of ‘$y$'	$\left(x,y\right)$
4	0	(4,0)
5	1	(5,1)
8	2	(8,2)
13	3	(13,3)
20	4	(20,4)

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

Step 4. Comparison with the parent graph.

The parent function $y=\sqrt{x-4}$ 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 the graphing calculator. 

The value 4 is subtracted inside the square root of the parent function $y=\sqrt{x}$. So the graph $y=\sqrt{x-4}$ is the translation of the parent graph $y=\sqrt{x}$.

Since 4 is subtracted inside the root, the graph is translated to the right by 4 units.

Step 5. State the domain and range.

Since the values inside the root must be positive. 

$x-4\ge 0$

Adding ‘4’ on both sides.

$\begin{array}{c}x-4+4\ge 0+4\\ x+0\ge 4\\ x\ge 4\end{array}$

$x\ge 4\Rightarrow x\in [4,\infty )$

Therefore, domain: $[4,\infty )$

The $y$ values are $\sqrt{x-4}$. As square roots are positive, $y$ takes all the positive real values including zero. 

Therefore, $y\ge 0,\Rightarrow y\in [0,\infty )$

Therefore, Range: $[0,\infty )$