Question

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

Answer 1

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+6x+8`. We start by swapping `x` with `y`:

`x=y^2+6y+8`
`x=(y^2+6y)+8` start completing the square for `y`
`x=(y^2+6y+9)+8-9`
`x=(y+3)^2-1`
`x+1=(y+3)^2`
`+-sqrt(x+1)=y+3`
`-3+-sqrt(x+1)=y`
`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.