How do you graph y=sqrt(x-0.5), compare it to the parent graph and what is the domain and range?

May 3, 2017

See explanation

$x \in \mathbb{R} \mathmr{and} y \in \mathbb{R} \leftarrow \text{ what is called Real Numbers}$

Domain $\to \text{ input } \to x \ge 0.5 \to \left[0.5 , + \infty\right)$
Range $\to \text{ output } \to y \to \left(- \infty , + \infty\right)$

Explanation:

Just for identification purposes:

Identify the standardised graph of $y = x$ by the name ${G}_{1}$

Identify the graph of $y = x - 0.5$ by the name ${G}_{2}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{The affect of subtracting 0.5 from } x}$

Step 1:

Using ${G}_{1}$ look at the ordered pair point of $\left(x - 0.5 , y\right)$

Step 2:
Now go to ${G}_{2}$ and plot the y value for ${G}_{1}$ against the $x$ for ${G}_{2}$

Effectively it is 'shifting' $y = x$ to the right by 0.5
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To plot the root we have to remember that the square root of a value has a $\pm$ type answer

So we actually have $y = \pm \sqrt{x - 0.5}$

Thus we plot two graphs.$\text{ One for } y = + \sqrt{x - 0.5}$
$\text{ One for } y = - \sqrt{x - 0.5}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There is a set of numbers given the name 'Complex Numbers'

To avoid entering that 'realm' of mathematics we do not permit $x - 0.5 < 0$