1. LIAL HORNSBY SCHNEIDER
COLLEGE ALGEBRA
LIAL
HORNSBY
SCHNEIDER

2. Hyperbolas 6.3 Equations and Graphs of Hyperbolas
Translated Hyperbolas
Eccentricity

3. Equations and Graphs of Hyperbolas
An ellipse was defined as the set of all points in a plane the sum of whose distances from two fixed points is a constant. A hyperbola is defined similarly.

4. Hyperbola
A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points is constant. The two
fixed points are called the foci of the hyperbola.

5. Equations and Graphs of Hyperbolas
Suppose a hyperbola has center at the origin and foci at F′(– c, 0) and F(c, 0). The midpoint of the segment F′F is the center of the hyperbola and the points V′(– a, 0) and V(a, 0) are the vertices of the hyperbola.
The line segment V′V is the transverse axis of the hyperbola.

6. Equations and Graphs of Hyperbolas
As with the ellipse,
so the constant in the definition is 2a, and
for any point P(x, y) on the hyperbola.

7. Equations and Graphs of Hyperbolas
The distance formula and algebraic manipulation
similar to that used for finding an equation for an ellipse produce the result

8. Equations and Graphs of Hyperbolas
Letting b2 = c2 – a2 gives
as an equation of the hyperbola. Letting y = 0 shows that the x-intercepts are a.

9. Equations and Graphs of Hyperbolas
If x = 0, the equation becomes
Let x = 0.
Multiply by – b2.
which has no real number solutions, showing that this hyperbola has no
y-intercepts.

10. Equations and Graphs of Hyperbolas
Starting with the equation for a hyperbola and solving for y, we get
Subtract 1;
add

11. Equations and Graphs of Hyperbolas
Write the left side as a single fraction.
Remember both the positive and negative square roots.
Take square roots on both sides; multiply by b.

12. Equations and Graphs of Hyperbolas
If x2 is very large in comparison to a2, the difference x2 – a2 would be very close to x2. If this happens, then the points satisfying the final equation above
would be very close to one of the lines

13. Equations and Graphs of Hyperbolas
Thus, as x gets larger and larger, the points of the hyperbola
get closer and closer to the lines

14. Equations and Graphs of Hyperbolas
These lines, called asymptotes of the
hyperbola, are useful when sketching the graph.

15. USING ASYMPTOTES TO GRAPH A HYPERBOLA
Example 1
Graph
Sketch the asymptotes, and find the coordinates of the vertices and foci. Give the domain and range.
Algebraic Solution
For this hyperbola, a = 5 and b = 7.

16. y = 7. Choosing x = – 5 also gives y = 7.
USING ASYMPTOTES TO GRAPH A HYPERBOLA
Example 1
With these values
Asymptotes
If we choose x = 5, then
y = 7. Choosing x = – 5 also gives y = 7.
These four points — (5, 7), (5,– 7), (– 5, 7) and
(– 5, – 7) — are the corners of the rectangle shown here.

17. USING ASYMPTOTES TO GRAPH A HYPERBOLA
Example 1
The extended diagonals of this rectangle, called the fundamental rectangle, are the asymptotes of the hyperbola. Since a = 5, the vertices of the hyperbola are (5, 0) and (– 5, 0).

18. We find the foci by letting
USING ASYMPTOTES TO GRAPH A HYPERBOLA
Example 1
We find the foci by letting so
Therefore, the foci are
and
The domain is
(– , – 5]  [5, ), and the range is (– , ).

19. Note
When graphing hyperbolas, remember that the fundamental rectangle and the asymptotes are not actually parts of the graph. They are simply aids in sketching the graph.

20. USING ASYMPTOTES TO GRAPH A HYPERBOLA
While a > b for an ellipse, examples would show that for hyperbolas, it is
possible that a > b, a < b, or a = b. If the foci of a hyperbola are on the y-axis,
the equation of the hyperbola has the form
with asymptotes

21. Caution
If the foci of a hyperbola are on the x-axis, the asymptotes have equations while foci on the y-axis lead to asymptotes To
avoid errors, write the equation of the hyperbola in either the form
and replace 1 with 0. Solving the resulting equation for y produces the proper equations for the asymptotes. or

22. The hyperbola with center at the origin and equation
STANDARD FORMS OF EQUATIONS FOR HYPERBOLAS
The hyperbola with center at the origin and equation
has vertices (a, 0), asymptotes
and foci (c, 0), where
c2 = a2 +b 2.

23. The hyperbola with center at the origin and equation
STANDARD FORMS OF EQUATIONS FOR HYPERBOLAS
The hyperbola with center at the origin and equation
has vertices (0, a), asymptotes
and foci (0, c), where
c2 = a2 +b 2.

24. Give the domain and range.
GRAPHING A HYPERBOLA
Example 2
Graph
Give the domain and range.
Solution
Divide by 100; standard form.
This hyperbola is centered at the origin, has foci on the y-axis, and has vertices
(0, 2) and (0, – 2). To find the equations of the asymptotes, replace 1 with 0.

25. Remember both the positive and negative square roots..
GRAPHING A HYPERBOLA
Example 2
Add
Remember both the positive and negative square roots..
Multiply by 4.
Square root property.

26. GRAPHING A HYPERBOLA
Example 2
To graph the asymptotes,
use the points (5, 2), (5,– 2),
(– 5, 2) and (– 5, – 2) to determine the fundamental rectangle. The diagonals of
this rectangle are the asymptotes for the graph, as
shown here. The domain of the relation is (– , ), and
the range is (–  ,– 2]  [2,  ).

27. Translated Hyperbolas
In the graph of each hyperbola considered so far, the center is the origin and the asymptotes pass through the origin. This feature holds in general; the asymptotes of any hyperbola pass through the center of the hyperbola. Like an ellipse, a hyperbola can have its center translated
away from the origin.

28. Solution Example 3 Graph Give the domain and range.
GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN
Example 3
Graph
Give the domain and range.
Solution
This equation represents a hyperbola centered at
(– 3, – 2). For this vertical hyperbola, a = 3 and
b = 2. The x-values of the vertices are – 3. Locate the y-values of the vertices by taking the y-value of the center, – 2 , and adding and subtracting 3. Thus, the vertices are (– 3, 1) and (– 3, – 5). The asymptotes have slopes and pass through the center (– 3,– 2).

29. Solution Example 3 Graph Give the domain and range.
GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN
Example 3
Graph
Give the domain and range.
Solution
The equations of the asymptotes can be found either by using the point-slope form of the equation of a line or by replacing 1 with 0 in the equation of the hyperbola as was done in Example 2.

30. Solution Example 3 Graph Give the domain and range.
GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN
Example 3
Graph
Give the domain and range.
Solution
Using the point-slope form, we get
Point-slope form.
Solve for y.

31. Solution Example 3 Graph Give the domain and range.
GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN
Example 3
Graph
Give the domain and range.
Solution
The graph is shown here. The domain of the relation is (– , ), and the range is
(–  ,– 5]  [1,  ).

32. Eccentricity
If we apply the definition of eccentricity from the previous section to the hyperbola, we get
Since c > a, we have e > 1. Narrow hyperbolas have e near 1, and wide hyperbolas have large e. See the figures on the next slide.

33. Eccentricity

34. Solution Example 4 Find the eccentricity of the hyperbola
FINDING ECCENTRICITY FROM THE EQUATION OF A HYPERBOLA
Example 4
Find the eccentricity of the hyperbola
Solution
Here a2 = 9; thus, a = 3,
and

35. Solution FINDING THE EQUATION OF A HYPERBOLA Example 5
Find the equation of the hyperbola with eccentricity 2 and foci at (– 9, 5) and (– 3, 5).
Solution
Since the foci have the same y-coordinate, the line through them, and therefore the hyperbola, is horizontal. The center of the hyperbola is halfway between the two foci at (– 6, 5). The distance from each focus to the center is c = 3.

36. Solution FINDING THE EQUATION OF A HYPERBOLA Example 5
Find the equation of the hyperbola with eccentricity 2 and foci at (– 9, 5) and (– 3, 5).
Solution
Since
Thus,

37. Solution FINDING THE EQUATION OF A HYPERBOLA Example 5
Find the equation of the hyperbola with eccentricity 2 and foci at (– 9, 5) and (– 3, 5).
Solution
The equation of the hyperbola is or
Simplify complex fractions.

38. Summary Conic Eccentricity Parabola e = 1 Circle e = 0 Ellipse
Hyperbola
and
and