Transcript Ellipses and Hyperbolas

Algebra II
Chapter 8: Conic Sections
Cheat Sheet
• In chapter 8 you are allowed a “cheat sheet”
• You are to bring in a tissue box, that has not been
opened, and cover it with paper.
• You are allowed to write anything on this box that you so
choose.
• You may use it on your quiz and chapter test.
• When we are done with Chapter 8, you must give the
tissue box to me.
• It is your decision to do this, you may not have any
other form a “cheat sheet”
8.1: Midpoint and Distance Formulas
• Find the midpoint of a segment on the
coordinate plane
• Find the distance between two points on
the coordinate plane
The Midpoint Formula
• The midpoint is the point in the middle
of a segment
• Definition: M is the midpoint of PQ if M is
between P and Q and PM = MQ.
• Formula:
•
Example 1:
• Find the midpoint of each line segment
with endpoints at the given coordinates:
• (12, 7) and (-2, 11)
• (-8, -3) and (10, 9)
• (4, 15) and (10, 1)
• (-3, -3) and (3, 3)
“Curveball Problem” Example
2:
• Segment MN has a midpoint P. If M has
coordinates (14, -3) and P has coordinates (8, 6), what are the coordinates of N?
• Circle R has a diameter ST. If R has
coordinates (-4, -8) and S has coordinates
(1, 4), what are the coordinates of T?
“Curveball Problem” Example
2:
• Circle Q has a diameter AB. If A is at
(-3,-5) and the center is at (2, 3), find
the coordinates of the B.
The Distance Formula
• Distance is always a positive number
• You can find distance using the
Pythagorean Theorem or using a formula
derived from it
• Formula:
•
Example 3:
• Find the distance between each pair of
points with the given coordinates
• (3, 7) and (-1, 4)
• (-2, -10) and (10, -5)
• (6, -6) and (-2, 0)
Example 4:
• Rectangle ABCD has vertices A(1, 4), B(3,
1), C(-3, -2), and D(-5, 1). Find the
perimeter and area of ABCD
• Circle R has diameter ST with endpoints
S(4, 5) and T(-2, -3). What are the
circumference and are of the circle? (Round
to two decimal places)
Summary:
• Learn the midpoint and distance
formulas
• Be able to answer any question that may
involve them.
• Questions?
8.2: Parabolas
• Write equations of parabola in standard
form and vertex form
• Graph parabolas
Equations of Parabolas
• Standard Form
• y = ax2 + bx + c
• Vertex Form
• y = a(x – h)2 + k
Example 1:
• Write y = 3x2 + 24x + 50 in vertex form.
Identify the vertex, axis of symmetry, and
direction of opening of the parabola.
Example 1:
• Write y = -x2 – 2x + 3 in vertex form.
Identify the vertex, axis of symmetry,
and direction of opening of the
parabola.
Graph Parabola
• You must always graph:
•
•
•
•
Vertex
Axis of Symmetry
Five points on the graph (this is to get the shape)
Focus: point in which all points in a parabola are
equidistant
• Directrix: line that the parabola will never cross
Concept Summary (pg 422)
y = a(x – h)2 + k
x = a(y – k)2 + h
Vertex
(h, k)
(h, k)
Axis of Symmetry
x=h
y=k
upward if a > 0
downward if a < 0
right if a > 0
left if a < 0
Form of Equation
Focus
Directrix
Direction of
Opening
Example
Example 2:
• Identify the coordinates of the vertex and focus, the
equations of the axis of symmetry and directrix, and the
direction of opening of the parabola
• y = x2 + 6x – 4
• x = y2 – 8y + 6
Example 2:
• Identify the coordinates of the vertex and focus, the
equations of the axis of symmetry and directrix, and the
direction of opening of the parabola
• y = 8x – 2x2 + 10
• x = -y2 – 4y – 1
Example 3:
• Graph:
y = ½(x – 1)2 + 2
• Graph:
x = -2(y + 1)2 - 3
Classwork/Homework
• Workbook
– Section 8.1
• 1, 3, 5, 11, 17, 19, 21, 31, 32
– Section 8.2
• 1 – 6 (all)
8.3: Circles
• Write equations of circles
• Graph circles
Circle
• A circle is the set of all point in a plane
that are equidistant from a given point
in the plane, called the center.
• Equation of a circle:
• (x – h)2 + (y – k)2 = r2
r = radius
(h, k)
center
Example One:
• Write an equation for the circle that
satisfies each set of conditions:
• Center (8, -3), Radius 6
• Center (5, -6), Radius 4
Example One:
• Write an equation for the circle that
satisfies each set of conditions:
• Center (-5, 2) passes through (-9, 6)
• Center (7, 7) passes through (12, 9)
Example One:
• Write an equation for the circle that
satisfies each set of conditions:
• Endpoints of a diameter are (-4, -2) and (8, 4)
• Endpoints of a diameter are (-4, 3) and (6, -8)
Graph circles
• Make sure the equation is in standard form
• Graph the center
• Use the length of the radius to graph four
points on the circle (up, down, left, right)
• Connect the dots to create the circle
Example Two:
• Find the center and radius of the circle
given the equation. Then graph the
circle
• (x – 3)2 + y2 = 9
Example Two:
• Find the center and radius of the circle
given the equation. Then graph the
circle
• (x – 1)2 + (y + 3)2 = 25
Example Two:
• Find the center and radius of the circle
given the equation. Then graph the circle
• x2 + y2 – 10x + 8y + 16 = 0
Example Two:
• Find the center and radius of the circle
given the equation. Then graph the circle
• x2 + y2 – 4x + 6y = 12
Classwork/Homework
• Workbook
• Lesson 8.3
• 1 – 13 (all)
Homework Answers: Workbook 8.3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
(x + 4)2 + (y – 2)2 = 64
x2 + y2 = 16
(x + ¼)2 + (y + 3 )2 = 50
(x – 2.5)2 + (y – 4.2)2 = 0.81
(x + 1)2 + (y + 7)2 = 5
(x + 9)2 + (y + 12)2 = 74
(x + 6)2 + (y – 5)2 = 25
(-3, 0); r = 4
(0, 0); r = 2
(-1, -3); r = 6
(1, -2); r = 4
(3, 0); r = 3
(-1, -3); r = 3
Homework Review
8.4: Ellipses
• Write equations of ellipses
• Graph ellipses
Ellipse
• An ellipse is like an oval.
• Every ellipse has two axes of symmetry
• Called the major axis and the minor axis
• The axes intersect at the center of the ellipse
• The major axis is bigger than the minor axis
• We use c2 = a2 – b2 to find c
• a is always greater b
• The equation is always equal to 1
Ellipses Chart (pg 434)
2
2
Standard Form  x - h 2  y - k 2
y - k   x - h

+
=1
+
=1
2
2
2
2
a
b
a
b
of Equation
(h,k)
(h,k)
Center
Horizontal
Vertical
Direction of
Major Axis
(h + c, k), (h – c, k) (h, k + c), (h, k – c)
Foci
2a units
2a units
Length of Major
Axis
2b units
2b units
Length of Minor
Axis
Example One:
Graph the ellipse
x
2
- 2
4
+
y
2
+ 5
1
=1
Your Turn:
Graph the ellipse
x
2
+ 2
81
+
y
2
- 5
16
=1
Example Two:
Graph the ellipse
y
2
- 4
64
+
x
2
- 2
4
=1
Your Turn:
Graph the ellipse
y
2
- 2
36
+
x
2
- 4
9
=1
Example Three:
Write the equation of the ellipse in the
graph:
Your Turn:
Write the equation of the ellipse in the
graph:
Example Four:
Write the equation of the ellipse in the
graph:
Your Turn:
Write the equation of
the ellipse in the graph:
Standard Form
Find the coordinates of the center and
foci and the lengths of the major and
minor axes of the ellipse with equation:
x2 + 4y2 + 24y = -32
Standard Form
Find the coordinates of the center and
foci and the lengths of the major and
minor axes of the ellipse with equation:
9x2 + 6y2 – 36x + 12y = 12
Classwork
Hyperbolas Chart
Standard Form
of Equation
Direction of
Transverse Axis
x
2
- h
2
a
-
y
2
- k
2
b
Horizontal
=1
y
2
- k
2
a
-
x
2
- h
2
b
=1
Vertical
Foci
(h + c, 0), (h - c, 0) (0, h + c), (0, h - c)
Vertices
(h + a, 0), (h - a, 0) (0, h + a), (0, h - a)
Length of
Transverse Axis
2a units
2a units
Length of
Conjugate Axis
2b units
2b units
Equations of
Asymptotes
y - k = ±
b
a
(x - h) y - k = ± (x - h)
a
b
Example One:
Graph the
hyperbola
x2
4
y2
= 1
9
Your Turn:
Graph the
hyperbola
x2
1
y2
= 1
4
Example Two:
Graph the
hyperbola
x
2
- 4
9
-
y
2
+ 2
16
Your Turn:
Graph the
hyperbola
x
2
- 3
9
-
y
2
+ 5
25
Example Three:
Write the equation of the hyperbola in
the graph:
Your Turn:
Write the equation of the hyperbola in
the graph:
Standard Form:
Find the coordinates of the vertices and
foci and the equations of the
asymptotes for the hyperbola with
equation
4x2 – 9y2 – 32x – 18y + 19 = 0
Standard Form:
Find the coordinates of the vertices and
foci and the equations of the
asymptotes for the hyperbola with
equation
x2 – y2 + 6x + 10y – 17 = 0
8.6 Conic Sections
The equation of any conic section can be
written in the general quadratic
equation:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
where A, B, and C ≠ 0
If you are given an equation in this
general form, you can complete the
square to write the equation in one of
the standard forms you have already
learned.
Standard Forms (you already know )
Conic
Section
Parabola
Circle
Ellipse
Standard Form of Equation
y = a(x – h)2 + k
x = a(y – k)2 + h
(x – h)2 + (y – k)2 = r2
x
2
- h
+
2
a
Hyperbola
x
2
- h
2
a
y
2
- k
2
b
=1
2
-
y - k
2
b
=1
y
2
- k
+
2
a
y
2
- k
2
a
-
x
2
- h
2
b
x
=1
2
- h
2
b
=1
Identifying Conic Sections
Relationship of A and C
Type of Conic Section
Only x2 or y2
Parabola
Same number in front of x2 and Circle
y2
Different number in front of x2 Ellipse
and y2 with plus sign
Different number in front of x2 Hyperbola
and y2 with plus sign or minus
sign
Example One:
Write each equation in standard form. Then
state whether the graph of the equation is a
parabola, circle, ellipse, and hyperbola.
y = x2 + 4x + 1
x2 + y2 = 4x + 2
y2 – 2x2 – 16 = 0
x2 + 4y2 + 2x – 24y + 33 = 0
Example Two:
Without writing the equation in standard form, state
whether the graph of each equation is a parabola,
circle, ellipse, and hyperbola.
x2 + 2y2 + 6x – 20y + 53 = 0
x2 + y2 – 4x – 14y + 29 = 0
3y2 + x – 24y + 46 = 0
6x2 – 5y2 + 24x + 20y – 56 = 0
Your Turn:
Without writing the equation in standard form, state
whether the graph of each equation is a parabola,
circle, ellipse, and hyperbola.
x2 + y2 – 6x + 4y + 3 = 0
6x2 – 60x – y + 161 = 0
x2 – 4y2 – 16x + 24y – 36 = 0
x2 + 2y2 + 8x + 4y + 2 = 0
Classwork/Homework
Workbook
Page 56
1 – 3, 8 – 10
Page 57
1 – 12