11.3 Hyperbolas.pptx
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This document discusses hyperbolas and how to sketch them. It defines a hyperbola as the set of points where the difference between the distances to two fixed foci is a constant. It provides the equations of horizontal and vertical hyperbolas with their vertices, asymptotes, and foci. Finally, it outlines the four steps to sketch a hyperbola: 1) sketch the central box, 2) draw the asymptotes, 3) plot the vertices, and 4) sketch each branch approaching the asymptotes.
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- 2. Geometric Definition of a Hyperbola pg. 799 A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points 𝐹1 and 𝐹2 is a constant. These two fixed points are the foci of the hyperbola
- 3. Hyperbola with center at the Origin pg. 800 (horizontal) Equation 𝑥2 𝑎2 − 𝑦2 𝑏2 = 1 𝑎 > 0, 𝑏 > 0 Vertices (±𝑎, 0) Transverse Axis Horizontal, length 2𝑎 Asymptotes 𝑦 = ± 𝑏 𝑎 𝑥 Foci ±𝑐, 0 , 𝑐2 = 𝑎2 + 𝑏2
- 4. Hyperbola with center at the Origin pg. 800 (vertical) Equation 𝑦2 𝑎2 − 𝑥2 𝑏2 = 1 𝑎 > 0, 𝑏 > 0 Vertices (0, ±𝑎) Transverse Axis Vertical, length 2𝑎 Asymptotes 𝑦 = ± 𝑎 𝑏 𝑥 Foci 0, ±𝑐 , 𝑐2 = 𝑎2 + 𝑏2
- 5. How to Sketch a Hyperbola pg. 801 1. Sketch the Central Box This is the rectangle centered at the origin, with sides parallel to the axis, that crosses one axis at ±𝑎 and the other at ±𝑏
- 6. How to Sketch a Hyperbola pg. 801 2. Sketch the Asymptotes These are the lines obtained by extending the diagonals of the central box
- 7. How to Sketch a Hyperbola pg. 801 3. Plot the Vertices These are the two 𝑥- intercepts or the two 𝑦- intercepts 4. Sketch the Hyperbola Start at the vertex, and sketch a branch of the hyperbola, approaching the asymptotes. Sketch the other branch in the same way