2 HyperbolaSet of all points where the difference between the distances to two fixed points (foci) is a positive constant.20 cm12 cm7cm15 cm12 cm4cm10cm2cmFocusFocus
3 Other parts of a hyperbola: Transversal axis can run verticallyAxis (transverse)CenterVerticesFoci
4 Equation of a Hyperbola Similar to ellipse Similarities/Differences:Subtraction between x2 and y2 termsVariable above a determines the direction of axisa is still the distance from center to a vertexc is still the distance from center to a focusc is now larger so…Horizontal axisVertical axis
5 ExampleFind the standard form of the equation with foci (-1, 2) and (5, 2) and vertices (0, 2) and (4, 2).
6 Asymptotes Question: What is b in an hyperbola? It is still a distance from the center in the opposite direction of the axisHow it applies to a parabola has to do with a new part unique to hyperbolas.AsymptotesLines that bound the hyperbolaPass through the diagonals of a rectangle with dimensions 2a and 2b
7 Asymptotes (h, k + b) (h - a, k) (h, k) (h + a, k) (h, k - b) Conjugate axis(h, k + b)(h - a, k)(h, k)(h + a, k)(h, k - b)b is the distance from center to edge of rectangle along conjugate axis
8 Equations for Asymptotes Horizontal hyperbola (transverse axis)Vertical hyperbola (transverse axis)b is up/down – a is left/righta is up/down – b is left/rightslope is rise over run →slope is rise over run →
9 ExampleSketch the graph of 4x2 – y2 = 16, include the asymptotes.
10 ExampleFind the standard form of the hyperbola with vertices (3, -5), (3, 1) and asymptotes
11 Eccentricity of an Hyperbola where e > 1 (since c is larger than a)large e → flatter curvee close to 1 → more curved - pointedProblem Set 9.3