Vertices of an hyperbola

High School

Definition

The vertices on a hyperbola refers to the point of maximum curvature. On a hyperbola, each curve will have one vertex, and is always aligned with the line of symmetry as well as the X axis

Worked examples

\(\frac{x^2}{9} - \frac{y^2}{16} = 1\)

This hyperbola opens left-right; vertices are at \((3, 0)\) and \((-3, 0)\), where \(a = 3\).

\(\frac{y^2}{25} - \frac{x^2}{4} = 1\)

This hyperbola opens up-down; vertices are at \((0, 5)\) and \((0, -5)\), where \(a = 5\).

Common mistakes

Vertices are at \((\pm a, \pm b)\) → Vertices are at \((\pm a, 0)\) or \((0, \pm a)\) depending on orientation Only the 'a' term (under the positive squared term) determines vertex distance; b relates to the conjugate axis.
The foci and vertices are the same points → Vertices are closer to center than foci; \(c > a\) always Foci lie farther out on the transverse axis since \(c^2 = a^2 + b^2\).
A hyperbola has four vertices → A hyperbola has exactly two vertices, one on each branch The endpoints of the conjugate axis are co-vertices, not vertices.

Where you'll use it next

Vertices anchor the graph of a hyperbola and appear in every problem involving hyperbola equations, eccentricity, and asymptotes in precalculus and analytic geometry.

Found in 1 StudyPug lesson

Conics - Hyperbola: Mastering the Art of Curved Geometry

Grade 12 Math

Dive into the world of hyperbolas! Discover key components, graphing techniques, and real-world applications. Perfect for students seeking to excel in conic sections and analytical geometry.

12th Grade12th