Question

What is the graph of the conic section defined by `x^2 + 4x - y + 9 = 0`?

Answer 1

Express `y` as a quadratic in `x` - `y` is a parabola

Explanation:

`x^2+4x-y+9=0`

Express `y` as a quadratic in `x`:
`y=x^2+4x+9`

Since the coefficient of `x^2` is `>0`, `y` has a minimum value

`y'=2x+4 -> y` has a minimum value at `x=-2` where `y=5`

The parabolic graph of `y` is shown below.

graph{x^2+4x+9 [-12.74, 12.59, -0.77, 11.9]}