# How do you graph y=-2sqrt(x+1), compare it to the parent graph and what is the domain and range?

Jan 30, 2018

Range $\to y \in \left[0 , - \infty\right)$
Domain $\to x \in \left[- 1 , + \infty\right)$

#### Explanation:

For the solution not to enter the domain of complex numbers the content of the root must never be negative.

Thus the cut off is $x \ge - 1$

Deriving the 'cut off point'
The $x + 1$ 'shifts' ( translate ) the $x$ left. In that the x-intercept is:

$y = 0 = - 2 \sqrt{x + 1}$

$\sqrt{x + 1} = 0$

$x = - 1$

You can manipulate the given equation in such a way that you end up with a ul(color(red)("variant on ") +x=(-y)^2=(+y)^2. This has the form $\subset$ as it is a quadratice in $y$

However, the right hand side of $y = - 2 \sqrt{x + 1}$ will allways be negative so $y$ will always be negative. Thus you:
$\underline{\text{only have the bottom half of the } \subset}$

$\textcolor{b l u e}{\text{In summery you have:}}$

The parent graph of $y = {x}^{2}$ is rotated clockwise $\frac{\pi}{2}$ and then translated left by 1 on the x-axis. The multiplication by 2 makes it more narrow. The negative means you only have the lower part of the form $\subset$ ie $y \le 0$