1. Hyperbolas

2. Hyperbolas
Like an ellipse but instead of the sum of distances it is the difference
A hyperbola is the set of all points P such that the differences from P to two fixed points, called foci, is constant
The line thru the foci intersects the two points (the vertices)
The line segment joining the vertices is the transverse axis, and it’s midpoint is the center of the hyperbola.
Has 2 branches(curves) and 2 asymptotes (guide lines)
The asymptotes contain the diagonals of a rectangle centered at the hyperbolas center

3. To Graph an Hyperbola Determine the center
Determine the horizontal and vertical distances. Mark those distances and make a box.
Sketch Asymptotes through the corners of the box.
Mark the vertices.
Determine which way it opens.
Sketch the branches using the vertices and asymptotes.

4. (0,b) (0,-b) Asymptotes Vertex (a,0) Vertex (-a,0) Focus Focus
This is an example of a horizontal transverse axis (a, the biggest number, is under the x2 term with the minus before the y)

5. Vertical transverse axis

6. Example 1: 1. Center? 2. Horizontal - Plot 3. Vertical - Plot
4. Draw Box
5. Draw Asymptotes
6. Which are Vertices?
7. Sketch Branches
8. Find and plot Foci

7. Standard Form of Hyperbola w/ center @ origin
Equation
Transverse Axis
Asymptotes
Vertices
Horizontal
(a, 0)
Vertical
(0, a)
Foci lie on transverse axis, c units from the center c2 = a2+b2

8. Write the equation in standard form

9. Write the equation in standard form

10. Identify the vertices and foci of the hyperbola.

11. Identify the vertices and foci of the hyperbola.

12. Graph the equation. Identify the foci and asymptotes.

13. Graph the equation. Identify the foci and asymptotes.