Conic Sections & Introduction to Limits

Hyperbolas

Graphing hyperbolas from their standard-form equations.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Parabolas & Ellipses

One Sign Change, a Completely Different Shape

An ellipse's equation is x²/a² + y²/b² = 1. Before reading on, predict what changes visually if that plus sign becomes a minus sign: x²/a² − y²/b² = 1. Does the shape still look like a stretched circle, or something else entirely?

Definition — Hyperbola

A hyperbola is the set of all points where the difference of the distances to two fixed foci stays constant. Standard form: x²/a² − y²/b² = 1, with two separate branches opening left and right (or, for y²/a² − x²/b² = 1, opening up and down), asymptotes y = ±(b/a)x, and c² = a² + b² giving the distance from center to each focus.

The single sign change produces two completely disconnected branches instead of one closed loop, and it also introduces something an ellipse never has: asymptotes. Reasoning about the equation for extremely large x: the constant '1' on the right side becomes negligible compared to the two growing squared terms, so the equation behaves increasingly like x²/a² − y²/b² ≈ 0, which rearranges to y ≈ ±(b/a)x — the graph settles closer and closer to those two straight lines the farther out it goes, without ever quite reaching them.

Worked Example — Identifying a Hyperbola's Features

Identify the vertices, foci, and asymptotes of x²/4 − y²/9 = 1. Here a² = 4 (a = 2) and b² = 9 (b = 3). Vertices at (±2, 0), since the x² term is positive (branches open left/right). Asymptotes: y = ±(3/2)x. Foci: c² = a² + b² = 4 + 9 = 13, so c = √13, giving foci at (±√13, 0).

Worked Example — Distinguishing an Ellipse from a Hyperbola by Sign

Without graphing, classify x²/16 + y²/25 = 1 and x²/16 − y²/25 = 1. The first has a plus sign between the two squared terms — an ellipse. The second has a minus sign — a hyperbola. The sign alone, not the numbers, decides which family the equation belongs to.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

Sketch a hyperbola's asymptotes first as two straight guide lines, then draw each branch approaching but never touching them — the asymptotes make the overall shape far easier to sketch accurately than plotting points alone.

Common Mistakes

Using c² = a² − b² for a hyperbola's foci, the ellipse relationship, instead of c² = a² + b².
A hyperbola's foci always use c² = a² + b² — unlike an ellipse, a and b can be any positive values here without one needing to be larger than the other.
Assuming a hyperbola's graph touches or crosses its asymptotes at some point.
A hyperbola's branches get arbitrarily close to their asymptotes but never actually touch them, no matter how far the branch is followed — that's exactly what makes them asymptotes.

Key Takeaways

A hyperbola is the set of points where the difference of distances to two foci stays constant, producing two separate branches.
A hyperbola's asymptotes arise because the constant term becomes negligible for very large x, leaving the equation behaving like its two squared terms alone.
A hyperbola's foci use c² = a² + b², distinguishing it from an ellipse's c² = a² − b².

Summary

A hyperbola's asymptote reasoning — what an equation approaches as x grows without bound — is a genuine preview of the idea the final lesson makes precise: the limit.

Track your progress

← Parabolas & Ellipses Back to Conic Sections & Introduction to Limits Estimating Limits Graphically→