# How do I rotate a parabola from opening up to opening horizontally?

Aug 22, 2014

You can swap $x$ and $y$ to make it open to the right; this is the common one and is known as the inverse relation. Or you can swap $- x$ with $y$ to make it open to the left.

Let's do an example for the inverse relation of $y = {x}^{2} + 6 x + 8$. We start by swapping $x$ with $y$:

$x = {y}^{2} + 6 y + 8$
$x = \left({y}^{2} + 6 y\right) + 8$ start completing the square for $y$
$x = \left({y}^{2} + 6 y + 9\right) + 8 - 9$
$x = {\left(y + 3\right)}^{2} - 1$
$x + 1 = {\left(y + 3\right)}^{2}$
$\pm \sqrt{x + 1} = y + 3$
$- 3 \pm \sqrt{x + 1} = y$
$y = - 3 \pm \sqrt{x + 1}$

You will want to notice that the inverse is a relation and not a function because it doesn't pass the vertical line test.