Parts of a Hyperbola – Vertices, Foci and Asymptotes

All 7 parts on the diagram

Show the list

Points 3

Center The midpoint between the two vertices, where the asymptotes cross.
Vertices The point on each branch nearest the center. One is a vertex.
Foci Two fixed points, one inside each branch. One is a focus.

Lines and curves 4

Transverse axis The segment joining the two vertices.
Conjugate axis The segment through the center at right angles to the transverse axis.
Asymptotes Two lines the curves get closer and closer to. That happens far from the center.
Branches The two separate curves of the hyperbola, mirror images of each other.

How this list was compiled: Every name checked on 2026-09-30 in two independent places. (1) Wikipedia, Hyperbola, https://en.wikipedia.org/wiki/Hyperbola (two branches, each the mirror image of the other; the asymptotes meet at the center of symmetry; the vertices lie at distance a from the center on the line through the foci; the transverse axis contains the segment of length 2a that ends at the vertices and the conjugate axis the segment of length 2b perpendicular to it with its midpoint at the center; the difference of the distances from a point of the curve to the two foci is constant). (2) Percey F. Smith and Arthur S. Gale, The Elements of Analytic Geometry (1904), public domain, https://archive.org/details/elementsofanalyt00smitrich (center, foci, the transverse axis 2a and the conjugate axis 2b, the asymptotes, a curve having two branches, c squared = a squared + b squared). (3) Paul Dawkins, Paul’s Online Notes, Algebra: Hyperbolas, Lamar University, https://tutorial.math.lamar.edu/Classes/Alg/Hyperbolas.aspx (each piece has a vertex; the asymptotes are lines the graph gets closer and closer to and are not part of the graph; where the two asymptotes cross is the center). Licence: own work

The drawing: Drawn for this site on 2026-09-30 from plain geometry: the curve x squared over a squared minus y squared over b squared equals 1 with a = 2.2 and b = 2.4. Own original drawing; nothing is traced or copied. Licence: own work

How it works

The parts of a hyperbola are its two branches, the center between them, a vertex on each branch, two foci, the transverse and conjugate axes, and the two asymptotes.

Read the full guide

Start from the center

The center is the point midway between the branches. The transverse axis runs through it from one vertex to the other. The conjugate axis crosses it at right angles. Each focus lies on the line of the transverse axis, farther out than the vertex.

What the asymptotes do

The asymptotes are two straight lines through the center. Far from the center each branch gets closer and closer to them. They guide the sketch but are not part of the curve itself.

Every point on the curve is nearer one focus than the other, and the difference between the two distances is always the same.

Questions and answers

What are the parts of a hyperbola?

Two branches, a center, two vertices, two foci, a transverse axis, a conjugate axis and two asymptotes.

What is the difference between the transverse and conjugate axis?

The transverse axis joins the two vertices, so it runs the way the hyperbola opens. The conjugate axis passes through the center at right angles to it and does not touch the curve.

Where are the foci of a hyperbola?

On the line through the vertices, one inside each branch. If a is the distance from the center to a vertex and b is half the conjugate axis, each focus is c from the center, where c² = a² + b².

Does a hyperbola ever touch its asymptotes?

No. Each branch gets closer and closer to the asymptotes as it runs outward, but the lines are not part of the graph.