2. Hyperbolas
4.4
Chapter 10 – Conics

3. Yesterday’s Exit Slip
Find the coordinates of the center of the graph.
A. (–1, –1) B. (0, 0) C. (1, 0) D. (2, 1)
Find the coordinates of the foci of the graph.
A. (1, 1), (–1, 1) B. (2, 1), (–2, 1) C. (2, 0), (–2, 0) D.
Find the length of the major axis of the graph.
A B C. 12 D. 14
Find the length of the minor axis of the graph.
A B C. 4 D. 2

4. Yesterday’s Exit Slip
Graph the ellipse 5. A. B. C. D.

5. You graphed and analyzed equations of ellipses. (Lesson 10–4)
Write equations of hyperbolas.
Graph hyperbolas.

6. Hyperbolas

7. Graph of a Hyperbola hyperbola transverse axis conjugate axis foci
vertices
co-vertices
constant difference

8. Writing the Equation of a Hyperbola
Write an equation for the hyperbola shown.
The center is the midpoint of the segment connecting the vertices, or (0, 0).
The value of a is the distance from the center to a vertex, or 2 units.
The value of c is the distance from the center to a focus, or 4 units.
Find b: c2 = a2 + b2
The transverse axis
is vertical → the y
comes first when we
write the equation
42 = 22 +b2
16 = 4 + b2
12 = b2

9. Example 1
What is an equation for the hyperbola? A. B. C. D.

10. Graphing Hyperbolas Identify the vertices, foci, and asymptotes.
Step 1 Find the center. The center is at (3, –5).
Step 2 Find a, b, and c. From the equation a2 = 4, so a = 2 and b2 = 9, so b = 3.
c2 = a2 + b
c2 = 4 + 9
c2 = 13
c =
or about 3.61

11. Step 3 Identify the vertices and foci.
The hyperbola is horizontal (x comes first in the equation)
The vertices are 2 units from the center (given by knowing a)
So the vertices are at (1, –5) and (5, –5).
The foci are about 3.61 units from the center.
The foci are at (6.61, –5) and (–0.61, –5).
Step 4 Identify the asymptotes.
Equation for the asymptote of a horizontal hyperbola
a = 2, b = 3, h = 3, and k = –5
The equations for the asymptotes are

12. Step 5 Graph the hyperbola.
Plot the points for the center, vertices, and foci
Use the value for a and b to draw a box
Graph the asymptotes by drawing in the diagonals of the box
Draw symmetrical curves using the asymptotes as your guide

13. Example 2
Identify the asymptotes of the function A. B. C. D.