Download 9.3 Ellipses 1. Describe the equation for an ellipse. How does it

9.3 Ellipses
1. Describe the equation for an ellipse. How does it
differ from the equation for a circle?
2. In the general equation for an ellipse (p.470),
what does rx stand for? What does ry stand for? What
do h and k stand for?
Example 1. Following the example in the text,
sketch the graph of the following ellipse
16x 2 + 96x + 4 y 2 −16y + 96 = 0
3. Write the geometric definition of an ellipse.
4. What are the foci?
5. Define focal radius. How are rx and ry related to
the focal radius?
Example 2. For the ellipse you sketched in example
1 find the foci and the focal radius.
6. Recall the general equation for a quadratic
relation from 9.1. What would the coefficients look
like if the relation was an ellipse?
9.4 Hyperbolas
1. How does the equation for a hyperbola differ
from the equation for an ellipse?
2. Explain why the graph of a hyperbola has two
branches.
3. How can you tell from the equation in which
direction the hyperbola opens?
4. How do you determine the diagonal asymptotes of
a hyperbola?
5. What is the x radius (rx) of a hyperbola? What is
the y radius (ry)?
6. What is the transverse axis? What is the
conjugate axis?
7. What are the equations of the asymptotes?
8. What are conjugate hyperbolas?
9. Write the geometric definition of a hyperbola.
Example 1. Sketch the graph of
−16x 2 + 64 x + 9y 2 − 54 y −127 = 0
Example 2: Find and plot the foci of the hyperbola
in Example 1.
9.5 Parabolas
1. Review the general forms of equations for
parabolas. What does standard form look like?
What does factored form look like? What does
vertex form look like?
2. What is the formula for the x coordinate of the
vertex? How do you find the y coordinate of the
vertex?
Example (review) For the following parabola find
the vertex, y intercept, symmetric point, x intercepts,
and draw the graph. y = 3x 2 + 4 x − 4
4. Consider the general equation for a quadratic
relation. How does the equation for a parabola differ
from the equation for a hyperbola and the equation
for an ellipse?
5. How does a parabola with a y2 term differ from a
parabola with an x2 term?
Example. Graph the following parabola. x=2y26y+3
9.6, 9.7 Recognizing Conic Sections
Example: Follow the example on p. 493 Find the
equation of the parabola whose focus is (-3,1) and
whose directrix is y=-4.
Dealing with general form quadratic equations with
xy terms is a bit more complex. We will not plot
equations with xy terms, but when you encounter
one you need to recognize what type of conic section
it is.
Example: Which conic section will the following
equation be? x 2 + 4 xy + y 2 −10x − 8y + 16 = 0
9.8 Systems of Quadratics
Review the three methods for solving systems of
linear equations. (graphs, elimination, and linear
combination)
1. What methods could you use to solve a system of
3 equations and 2 unknowns in which one or both of
the equations is a quadratic polynomial?
4 x 2 + y 2 = 104
Example 1. Solve the system
2x 2 − 3y 2 = 38
3x 2 − 4 y 2 + 7y = 33
Example 2 Solve the system
12x 2 + 4 y 2 = 228
Example 3: Check the answer to example 2 by
graphing
x 2 + y 2 − 2x + 3y = 8
Example 4. Solve the system
2x + 4 y = −4