P.o.D. – Find the standard form equation of the ellipse with the given characteristics.
1.) Vertices (6,0),(-6,0); Foci (2,0),(-2,0).
2.) Foci (2,0),(-2,0); major axis length 8.
3.) Vertices (0,5),(0,-5); passes through (4,2).
4.) major axis vertical; passes through (0,4) and (2,0).
10.4 – Hyperbolas
Learning Target(s): I can write equations of hyperbolas in standard form; find asymptotes; use properties of hyperbolas to solve real-life problems.
Definition:
-The set of all ______in a plane whose ______from two fixed ______is ______.
-Formed by cutting a double-cone ______.
-
Parts of a Hyperbola:
Standard Form of a Horizontal Hyperbola:
Center:
Foci:
Vertices:
Equation of the transverse axis:
Standard Form of a Vertical Hyperbola:
Center:
Foci:
Vertices:
Equation of the transverse axis:
__ is the distance from the center to the ______, or half the length of the ______axis.
__ is half the length of the ______axis.
__ is the distance from the center to a ______, or half the ______length.
EX: Find the equation of the hyperbola with foci at (1,-5) and (1,1) and whose transverse axis is 4 units long.
Begin by finding the center. It will be the ______of the foci.
Next, find __ – the distance from the center to a ______.
Plot the points to determine the direction of the hyperbola. This hyperbola is ______.
Because we know the length of the ______axis, .
We need to find ___.
Substitute into the standard form equation.
EX: Find the equation of the hyperbola with foci at (7,1) and (-3,1) whose transverse axis is 8 units long.
Find the center.
Find a,b, and c.
Write the equation.
Equations of the Asymptotes of a Horizontal Hyperbola:
Equations of the Asymptotes of a Vertical Hyperbola:
EX: Find the coordinates of the center, foci, and vertices and the equations of the asymptotes of the graph of . Then graph the equation.
What do we know from the equation?
Find c.
Find the center.
Find the foci.
Find the vertices.
Find the equations of the asymptotes.
(We could write these equations in slope-intercept form).
Graph on the whiteboard by hand.
Graph using CONICS on the calculator.
EX: Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes for .
What do we know?
Find c.
Find the center.
Find the foci.
Find the vertices.
Find the asymptotes.
Graph on the whiteboard by hand.
Graph on the calculator using CONICS.
EX: Find the coordinates of the center, foci, vertices, and the equations of the asymptotes for .
*We need to ______the ______on both variables.
What do we know?
Find c.
Find the center.
Find the foci.
Find the vertices.
Find the asymptotes.
Graph by hand on the whiteboard.
Graph using CONICS.
EX: Find the center, foci, vertices, and asymptotes for . Then graph.
Complete the square on both variables.
What do we know?
Find c.
Find the center.
Find the foci.
Find the vertices.
Find the asymptotes.
Graph by hand.
Graph using CONICS.
A special type of Hyperbola:
-Whenever we have a ______in the ______.
EX: Graph xy=36.
Show the graph on a calculator.
-This is a hyperbola on a ______(_____).
EX: Find the standard form of the equation of the hyperbola with foci at (0,0) and (8,0) and vertices at (3,0) and (5,0).
Begin by finding the center.
We know that the hyperbola is horizontal, and we also can find a and c.
Now find b.
Substitute.
EX: Put the hyperbola
in standard form.
*We need to complete the square on both variables.
EX: Find the standard form of the equation of the hyperbola having vertices (3,2) and (9,2) and having asymptotes
and .
By plotting the ______, we should recognize that this hyperbola is ______.
We should also realize that the center of this hyperbola is the ______of the ______or where the ______intersect.
We can now find __.
Since this graph is ______, we know that the ______of the ______come from . The given asymptotes have slopes of . Because a=___, we now know that b=___.
We can now substitute into the standard form equation.
General Equation of a Conic:
Classifying a Conic from its General Equation:
Circle:
Parabola:
Ellipse
Hyperbola:
EX: Classify the graph of each equation.
Upon completion of this lesson you should be able to:
- Find the center, vertices, foci, and equations of the asymptotes of a hyperbola.
- Put a hyperbolic equation into standard form.
- Graph hyperbolas.
- Identifying different conic sections.
HWPg.7603-36 3rds, 46-60E, 67-72.
Quiz 10.1-10.4 tomorrow.