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.