Question

How do you graph `x=2y-y^2`?

Answer 1

If you have studied graphing parabolas, then you want to use the standard method for them.

Because the `y` is square, not the `x`, this parabola opens horizontally.
(There's more than one way to proceed. I'll use one of them.)
Get the equation in standard form: `x-h=a(y-k)^2`
`x=-y^2+2y=-(y^2-2y )`
so `x=-(y^2-2y+1-1)=-(y^2-2y+1)+1`. Therefore,
`(x-1)=-(y-1)^2`

The vertex is at `(h, k)=(1, 1)`

The coefficient of the square term is negative, so the parabola opens in the negative direction. For a horizontal parabola, this means it opens to the left.

The axis of symmetry is the line `y=1`.
And if we start at the vertex `(1, 1)` and go `+-1` in the `y` direction (vertically), the we'll go `a=-1` in the x direction (horizontally) This gives us 2 additional points `(0, 0)` and `(0, 2)`.

That's enough to sketch the graph:

graph{x=2y-y^2 [-4.933, 4.932, -2.466, 2.467]}