Question

How do you write a standard form equation for the hyperbola with `y^2+2y-2x+5=0`?

Answer 1

The given equation is a parabola. It is not a hyperbola.

Explanation:

Here is the graph of the given equation.
graph{y^2+2y-2x+5=0 [-8.375, 11.625, -6.36, 3.64]}
This is a parabola with a horizontal axis of symmetry. As such it's general standard form is
`color(white)("XXX")x=ay^2+by+c` with constants `a, b, c`

Rewriting the given equation:
`color(white)("XXX")y^2+2y-2x+5=0`

`color(white)("XXX")2x=y^2+2y+5`

`color(white)("XXX")x=1/2y^2+1y+5/2`
`color(white)("XXXXXX")`(standard parabolic form)