Distinguishing between a parabola and (one branch of) a hyperbola by looking at the graph

Question

If I were to show you this curve that is quite zoomed in is it possible to tell if it is a hyperbola of a parabola

This is a picture my friend sent me and I cannot tell which conic section it is.

Answer 1

This looks like a hyperbola.

For $x > 4$, the gradients of the curve are nearly constant for both positive and negative values of $y$. This suggests it is approaching an asymptote, something which does not happen for a a parabola.

Answer 2

Draw two or more parallel chords (not perpendicular to the axis). If their midpoints are aligned parallel to the axis, then it's a parabola. Otherwise it's a hyperbola and the intersection between the line of the midpoints and the axis is its center.