1. Problems #1-6 on worksheet
Warm Up
Problems #1-6 on worksheet

2. 6.4 Hyperbolas
Objective: To find equations of hyperbolas and to graph them.

3. The hyperbola is the locus of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points in the plane, the foci, is constant.
P(x, y) F1 F2
| PF1 - PF2| = 2a

4. The Hyperbola F1 and F2 are the foci.
A1 and A2 are called the vertices. c c a a O F1 A1 A2 F2
The distance from the center to either focus is represented by c.

5. These lines are asymptotes. The rectangle has dimensions 2a and 2b.
The Hyperbola Centered at the Origin
The diagram shows a graph of a hyperbola with a rectangle centered at the origin. The points A1, A2 , B1 and B2 are the midpoints of the sides of the rectangles. The hyperbola lies between the lines containing its diagonals. As | x | increases, the hyperbola comes closer to these lines.
These lines are asymptotes.
The rectangle has dimensions 2a and 2b. B1 A1 A2 B2

6. To find the foci use c2 = a2 + b2
The Standard Equation of a Hyperbola
With Center (0, 0) and Foci on the x-axis
Horizontal Hyperbola
The equation of a hyperbola with the center (0, 0) and foci on the x-axis is:
𝐁 𝟏 (0, b)
The length of the rectangle is 2a.
The height of the rectangle is 2b.
The vertices are (a, 0) and (-a, 0).
The foci are (c, 0) and (-c, 0).
The slopes of the asymptotes are
(-c, 0) A1 A2
(c, 0) F1
(-a, 0)
(a, 0) F2
𝐁 𝟐 (0, -b)
To find the foci use c2 = a2 + b2
The equations of the asymptotes are 𝒚= 𝒃 𝒂 𝒙 and 𝒚=− 𝒃 𝒂 𝒙.

7. The equations of the asymptotes are 𝒚= 𝒂 𝒃 𝒙 and 𝒚=− 𝒂 𝒃 𝒙.
The Standard Equation of a Hyperbola with
Center (0, 0) and Foci on the y-axis
Vertical Hyperbola
The equation of a hyperbola with the center (0, 0) and foci on the y-axis is:
F1(0, c)
A1(0, a)
The length of the rectangle is 2b.
The height of the rectangle is 2a.
The vertices are (0, a) and (0, -a).
The foci are (0, c) and (0, -c).
The slopes of the asymptotes are
B1(-b, 0)
B2(b, 0)
A2(0, -a)
F2(0, -c)
The equations of the asymptotes are 𝒚= 𝒂 𝒃 𝒙 and 𝒚=− 𝒂 𝒃 𝒙.

8. Graph the hyperbola. State the coordinates of the vertices, the coordinates of the foci, and the equations of the asymptotes.
The equations of the asymptotes are
For this equation, a = 2 and b = 4.
The vertices are (2, 0) and (-2, 0)
c2 = a2 + b2 =
= 20
The coordinates of the foci are

9. Graph the hyperbola. State the coordinates of the vertices, the coordinates of the foci, and the equations of the asymptotes.
For this equation, a = 5 and b = 3.
The vertices are (0, 5) and (0, -5)
c2 = a2 + b2 =
= 34
The coordinates of the foci are
The equations of the asymptotes are

10. Find the equation of a hyperbola given:
Foci: (-7, 0) and (7, 0)
Vertices: (-5, 0) and (5, 0) 8
Horizontal Hyperbola
a2 = 25 and c2 = 49 F V C V F 10
c2 = a2 + b2
49 = 25 + b2
b2 = 24
𝑥 − 𝑦 =1

11. Classwork
Worksheet #7-12

12. Homework
Page 235 #1,3,5,15,16,19,21,23