1. Section 8-3 The Hyperbola

2. Section 8-3 • the geometric definition of a hyperbola • standard form of a hyperbola with a center at (0 , 0) • translating a hyperbola – center at (h , k) • graphing a hyperbola • finding the equations of the asymptotes • finding the equation of a hyperbola • eccentricity and orbits • reflective properties of hyperbola

3. Geometry of a Hyperbola • hyperbola – the set of all points whose distance from two fixed points (the foci) have a constant difference • all the points are coplanar • the line through the foci is called the focal axis • the midway point between the foci is called the center

4. Geometry of a Hyperbola F1 F2 V1 center V2 F1 and F2 are the foci V1 and V2 are the vertices (chord between called the transverse axis)

5. Geometry of a Hyperbola F1 F2 V1 center V2 d2 d1 F1 and F2 are the foci d1 - d2 = constant V1 and V2 are the vertices (chord between called the transverse axis)

6. Standard Form: Center (0 , 0) • 2a = length of the transverse axis (endpoints are the vertices) • 2b = length of the conjugate axis • c = focal radius (distance from the center to each foci) • c2 = a2 + b2 (use to find c)

7. Standard Form: Center (h , k)

8. Graphing a Hyperbola • convert the equation into standard form, if necessary (complete the square) • find and plot the center • use “a” to plot the vertices (same direction as the variable a2 is underneath) • use “b” to plot two other points • draw a rectangle using these four points • draw the diagonals of the rectangle (dashed), these are the asymptotes • draw in the hyperbola (use vertices) • plot the foci using “c” (c is the distance from the center to each focus)

9. Equations of the Asymptotes • the equations of the asymptotes can be found by replacing the 1 on the right-side of the equation with a 0 and then solving for y