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

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

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Answer 1 · 2014-08-22T05:39:09

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

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

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

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

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

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