Hyperbola Graph (x²/a² − y²/b² = 1)
Enter a and b to graph the hyperbola x²/a² − y²/b² = 1. Automatically computes eccentricity, foci, vertices, and asymptotes.
What is a hyperbola graph?
The hyperbola graph is a tool that lets you draw the hyperbola x²/a² − y²/b² = 1 simply by entering the semi-axes a and b, while automatically computing the eccentricity, foci, vertices, and asymptotes. A hyperbola consists of two separate branches. Where an ellipse is defined by the sum of the distances from two foci staying constant, a hyperbola is defined by the opposite property: the difference of those two distances stays constant instead.
The graph also plots the asymptotes y = ±(b/a)x as grey dashed lines, so you can visually confirm which straight lines the hyperbola approaches as it stretches toward infinity. The eccentricity of a hyperbola is always greater than 1, and the larger this value gets, the more sharply the two branches flare open.
Because the tool recalculates every value the moment you change a or b, it works equally well as a quick reference for a single calculation and as a way to build intuition for how the shape of the curve responds to each parameter — something a static formula sheet cannot show on its own.
How to use the hyperbola graph
- Enter semi-axis a The distance from the center to the vertices along the x-axis. Enter a value greater than 0.
- Enter semi-axis b The value along the y-axis that determines the slope of the asymptotes. Also enter a value greater than 0.
- Check the graph The hyperbola, drawn with its two branches, appears together with its asymptotes and foci in a single chart.
- Review the eccentricity, foci, and vertices The automatically calculated eccentricity e, the coordinates of the foci and vertices, and the equations of the asymptotes are all listed for you.
Tips for getting more out of it
- The hyperbola x²/a² − y²/b² = 1 consists of two separate branches. The right branch exists where x ≥ a and the left branch where x ≤ −a.
- The asymptotes y = ±(b/a)x are the lines the hyperbola approaches but never touches. They are shown as grey dashed lines on the graph.
- Eccentricity e = √(1 + b²/a²) is always greater than 1. Larger e values produce more "open" hyperbolas.
- The foci are at (±c, 0) where c = √(a² + b²). They appear as orange diamonds on the graph.
Ways to use the hyperbola graph
Understanding navigation systems (LORAN)
LORAN, a radio navigation system that pinpoints a position from the time difference between signals sent by two stations, relies directly on the hyperbola's defining property that the difference in distances stays constant.
Understanding the secondary mirror of reflecting telescopes
Cassegrain telescopes use a hyperbolic secondary mirror to bring the focal point out behind the instrument. Visualizing the hyperbola's focal property makes this design easier to grasp.
Studying conic sections through contrast with the ellipse
An eccentricity below 1 gives an ellipse, while one above 1 gives a hyperbola. You can compare the two shapes directly by changing the values and watching the graph respond.
Practicing conic sections for school or university
You can build a hands-on understanding of the hyperbola's standard form, its asymptotes, and its relationship to the foci by adjusting the parameters yourself and observing the results.
Glossary
- Eccentricity
- A value always greater than 1 that describes how widely a hyperbola opens. The larger the value, the more sharply the branches flare outward.
- Asymptote
- A straight line that the hyperbola approaches ever more closely as it extends toward infinity (x → ±∞), without ever actually touching it.
- Focus
- One of two special points such that, for every point on the hyperbola, the difference between the distances to these two points stays constant.
- Vertex
- The point where the hyperbola crosses the x-axis (or major axis) — the point on the curve that lies closest to the corresponding focus.
- LORAN
- Short for Long Range Navigation. A 20th-century radio navigation system that used the time difference between radio signals from two stations to place a ship or aircraft on a hyperbola.
FAQ
Side Note — Hyperbolas and navigation
A hyperbola is defined as the set of points where the difference of distances to two fixed foci is constant. This property powered the LORAN (Long Range Navigation) system used during the 20th century: ships located themselves by measuring the time difference between radio signals from two shore stations.
Hyperbolic mirrors appear in Cassegrain telescopes, where a convex hyperbolic secondary mirror reflects light through a hole in the primary mirror, placing the focal point behind the telescope body.