On an (x, y) graph a hyperbola can have the shape shown to the left.
Here's a link to EZ Graph - Hyperbola. This is a graphic calculator presenting functions that draw hyperbolas.
If a plane intersects the cone when it is parallel to the axis, the conic section is a hyperbola. This is shown to the left.
Click this link for an interactive 3D animation showing how a cone and plane intersect to form a hyperbola.
Here is a video that shows several possibilities for forming a hyperbola by intersecting a cone and a plane.