Hyperbolas

A hyperbola's standard equation (x-h)²/a² - (y-k)²/b² = 1 looks almost like an ellipse, but that minus sign changes everything. Solve for y and you get a square root of (x-h)² - a², which only gives real values once x is far enough from h in either direction. That gap is why the two branches never meet: there's a forbidden zone around the center where no point on the curve exists. Each branch opens away from the center, one to the right, one to the left.

The asymptotes fall straight out of the equation because as x grows huge, the -a² inside that square root becomes negligible compared to the rest, and the curve behaves like the simpler line y - k = ±(b/a)(x-h). So without any extra work, a and b hand you the slopes directly: rise b, run a. Sketch a center box of width 2a and height 2b, draw its diagonals, and those diagonals are the asymptotes the branches hug but never touch.

This sits right next to ellipses in the same family of conics built from distances to two foci, but a hyperbola uses a difference of distances instead of a sum. That one sign swap — plus versus minus — is the difference between a closed loop and two endless branches, which is why rewriting a hyperbola as some stretched or flipped ellipse never works. Vertices, center, and axis orientation all transfer over from the ellipse toolkit, but the shape itself is a different species of curve.