Hyperbola Overview and Components

Aug 25, 2025

Overview

This lecture introduces the hyperbola, covering its standard equations, key features such as axes, vertices, foci, and asymptotes, and explains how to identify and graph its components.

Hyperbola: Basic Concepts

  • A hyperbola is the set of all points where the absolute difference of distances to two fixed points (foci) is constant.
  • A hyperbola consists of two separate branches which can open left/right or up/down.

Standard Forms of Hyperbola

  • The standard equation for a hyperbola with a horizontal transverse axis:
    ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 )
  • For a vertical transverse axis:
    ( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 )
  • For hyperbolas centered at (h, k):
    ( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 ) (horizontal)
    ( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 ) (vertical)

Key Components of Hyperbola

  • The transverse axis is the axis along which the branches open (horizontal or vertical).
  • The center of the hyperbola is the midpoint between the vertices.
  • Each branch has a vertex; vertices are points where the hyperbola intersects the transverse axis.
  • The foci are located at a distance ( c = \sqrt{a^2 + b^2} ) from the center along the transverse axis.
  • The conjugate axis is perpendicular to the transverse axis.
  • The fundamental rectangle helps in sketching the hyperbola and its asymptotes.

Asymptotes and Graphing

  • Asymptotes are lines the hyperbola approaches but never touches, passing through the center and forming diagonals of the fundamental rectangle.
  • For ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ), asymptote equations are ( y = \frac{b}{a}x ) and ( y = -\frac{b}{a}x ).

Key Terms & Definitions

  • Hyperbola — the set of points where the difference of distances to two foci is constant.
  • Transverse Axis — the axis along which the branches of the hyperbola open.
  • Conjugate Axis — the axis perpendicular to the transverse axis.
  • Vertex — endpoint of the transverse axis where the hyperbola curves most.
  • Focus (plural: foci) — fixed points used to define the hyperbola.
  • Center — midpoint between the two vertices.
  • Asymptote — a line the hyperbola approaches but never reaches.

Action Items / Next Steps

  • Practice identifying components of hyperbolas in equations.
  • Sketch the graph of a hyperbola using its standard form, axes, vertices, foci, and asymptotes.