1. Write the standard equation for a hyperbola.
Reminder: Multiplying both sides of each equation by the least common multiple eliminates the fractions. 1. x2 9
– = 1 y2 4
4x2 – 9y2 = 36 2. y2 25
– = 1 x2 16
16y2 – 25x2 = 400
Objectives
Write the standard equation for a hyperbola.
Graph a hyperbola, and identify its center, vertices, co-vertices, foci, and asymptotes.

2. 3. Find the vertices, co-vertices, and asymptotes of , then graph.
Notes A. x2 4
= 1 y2 9
1. Graph the ellipses B.
(x-3)2 16
= 1
(y+1)2 25
2. Graph the hyperbola
(x-3)2 16
= 1
(y+1)2 25
3. Find the vertices, co-vertices, and asymptotes of , then graph.
4. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0). 2

3. What would happen if you pulled the two foci of an ellipse so far apart that they moved outside the ellipse? The result would be a hyperbola, another conic section.
A hyperbola is a set of points P(x, y) in a plane such that the difference of the distances from P to fixed points F1 and F2, the foci, is constant. For a hyperbola, d = |PF1 – PF2 |, where d is the constant difference. You can use the distance formula to find the equation of a hyperbola.

4. The standard form of the equation of a hyperbola depends on whether the hyperbola’s transverse axis is horizontal or vertical.

5. Example 1
Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph x2 49
– = 1 y2 9
Step 1 The vertices are –7, 0) and (7, 0) and the co-vertices are (0, –3) and (0, 3).
Step 2 Draw a box using the vertices and co-vertices. Draw the asymptotes through the corners of the box.
Step 3 Draw the hyperbola by using the vertices and the asymptotes.

6. Example 2: Graphing a Hyperbola
Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.
(x – 3)2 9
– = 1
(y + 5)2 49
Step 1 Center is (3, -5), the vertices are (0, –5) and (6, –5) and the co-vertices are (3, –12) and (3, 2) .
Step 2 The asymptotes cross at (3, -5) and have a
slope = 7 3

7. Example 2B Continued
Step 3 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the box.
Step 4 Draw the hyperbola by using the vertices and the asymptotes.

8. Example 3A: Writing Equations of Hyperbolas
Write an equation in standard form for each hyperbola.
Step 1: The graph opens horizontally, so the equation will be in the form of x2 a2
– = 1 y2 b2 x2 36
– = 1. y2
Step 2: Because a = 6 and b = 6, the equation of the
graph is

9. Example 3B: Writing Equations of Hyperbolas
Write an equation for the hyperbola with center at the origin, vertex (4, 0), and focus (10, 0).
Step 1 Because the vertex and the focus are on the horizontal axis x2 16
– = 1 y2 b2
Step 2 Use focus2 = 16 + other denominator.
Step 3 The equation of the hyperbola is x2 16
– = 1 y2 84

10. 3. Find the vertices, co-vertices, and asymptotes of , then graph.
Notes A. x2 4
= 1 y2 9
1. Graph the ellipses B.
(x-3)2
144
= 1
(y+1)2 25
2. Graph the hyperbola
(x-3)2
144
= 1
(y+1)2 25
3. Find the vertices, co-vertices, and asymptotes of , then graph.
4. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0). 10

11. 3. Find the vertices, co-vertices, and asymptotes of , then graph.
Notes
3. Find the vertices, co-vertices, and asymptotes of , then graph.
asymptotes:
vertices: (–6, ±5); co-vertices (6, 0), (–18, 0); 5 12
y = ± (x + 6)

12. Notes:
4. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0).

13. The values a, b, and c, are related by the equation c2 = a2 + b2
The values a, b, and c, are related by the equation c2 = a2 + b2. Also note that the length of the trans-verse axis is 2a and the length of the conjugate is 2b.

14. As with circles and ellipses, hyperbolas do not have to be centered at the origin.

15. Hyperbolas: Extra Info
The following power-point slides contain extra examples and information.
Reminder: Lesson Objectives
Write the standard equation for a hyperbola.
Graph a hyperbola, and identify its center, vertices, co-vertices, foci, and asymptotes. 15

16. Notice that as the parameters change, the graph of the hyperbola is transformed.

17. Check It Out! Example 3a: Graphing
Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph. x2 16
– = 1 y2 36
Step 1 The equation is in the form so the transverse axis is horizontal with center (0, 0). x2 a2
– = 1 y2 b2

18. Check It Out! Example 3a Continued
Step 2 Because a = 4 and b = 6, the vertices are (4, 0) and (–4, 0) and the co-vertices are (0, 6) and . (0, –6).
Step 3 The equations of the asymptotes are
y = x and y = – x . 3 2

19. Check It Out! Example 3
Step 4 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the box.
Step 5 Draw the hyperbola by using the vertices and the asymptotes.

20. Check It Out! Example 3b: Graphing
Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.
(y + 5)2 1
– = 1
(x – 1)2 9
Step 1 The equation is in the form so the transverse axis is vertical with center (1, –5).
(y – k)2 a2
– = 1
(x – h)2 b2

21. Check It Out! Example 3b Continued
Step 2 Because a = 1 and b =3, the vertices are (1, –4) and (1, –6) and the co-vertices are (4, –5) and (–2, –5).
Step 3 The equations of the asymptotes are
y + 5 = (x – 1) and y + 5 = – (x – 3). 1 3

22. Check It Out! Continued
Step 4 Draw a box by using the vertices and co-vertices. Draw the asymptotes through the corners of the box.
Step 5 Draw the hyperbola by using the vertices and the asymptotes.

23. Check It Out! Example 2a: Writing Equations
Write an equation in standard form for each hyperbola.
Vertex (0, 9), co-vertex (7, 0)
Step 1 Because the vertex is on the vertical axis, the transverse axis is vertical and the equation is in the form y2 a2
– = 1 x2 b2
Step 2 a = 9 and b = 7.
Step 3 Write the equation.
Because a = 9 and b = 7, the equation of the graph is , or y2 92
– = 1 x2 72 81 49

24. Check It Out! Example 2b: Writing Equations
Write an equation in standard form for each hyperbola.
Vertex (8, 0), focus (10, 0) x2 a2
– = 1 y2 b2
Step 1 Because the vertex and the focus are on the horizontal axis, the transverse axis is horizontal and the equation is in the form .

25. Check It Out! Example 2b Continued
Step 2 a = 8 and c = 10; Use c2 = a2 + b2 to solve for b2.
102 = 82 + b2
Substitute 10 for c, and 8 for a.
36 = b2
Step 3 The equation of the hyperbola is x2 64
– = 1 y2 36