1. Chapter 8: Conic Sections
Algebra II
Chapter 8: Conic Sections

2. 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”

3. 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

4. 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:

5. 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)

6. “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?

7. “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.

8. 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:

9. 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)

10. 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)

11. Summary: Learn the midpoint and distance formulas
Be able to answer any question that may involve them.
Questions?

12. 8.2: Parabolas
Write equations of parabola in standard form and vertex form
Graph parabolas

13. Equations of Parabolas
Standard Form
y = ax2 + bx + c
Vertex Form
y = a(x – h)2 + k

14. Example 1:
Write y = 3x2 + 24x + 50 in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

15. Example 1:
Write y = -x2 – 2x + 3 in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

16. 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

17. Concept Summary (pg 422) Form of Equation y = a(x – h)2 + k
x = a(y – k)2 + h
Vertex
(h, k)
Axis of Symmetry
x = h
y = k
Focus
Directrix
Direction of Opening
upward if a > 0
downward if a < 0
right if a > 0
left if a < 0
Example

18. 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

19. 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

20. Example 3:
Graph:
y = ½(x – 1)2 + 2
Graph:
x = -2(y + 1)2 - 3

21. Classwork/Homework Workbook Section 8.1 Section 8.2
1, 3, 5, 11, 17, 19, 21, 31, 32
Section 8.2
1 – 6 (all)

22. 8.3: Circles
Write equations of circles
Graph circles

23. 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

24. Example One:
Write an equation for the circle that satisfies each set of conditions:
Center (8, -3), Radius 6
Center (5, -6), Radius 4

25. 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)

26. 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)

27. 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

28. Example Two:
Find the center and radius of the circle given the equation. Then graph the circle
(x – 3)2 + y2 = 9

29. Example Two:
Find the center and radius of the circle given the equation. Then graph the circle
(x – 1)2 + (y + 3)2 = 25

30. Example Two:
Find the center and radius of the circle given the equation. Then graph the circle
x2 + y2 – 10x + 8y + 16 = 0

31. Example Two:
Find the center and radius of the circle given the equation. Then graph the circle
x2 + y2 – 4x + 6y = 12

32. Classwork/Homework
Workbook
Lesson 8.3
1 – 13 (all)

33. Homework Answers: Workbook 8.3
(x + 4)2 + (y – 2)2 = 64
x2 + y2 = 16
(x + ¼)2 + (y )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

34. Homework Review

35. 8.4: Ellipses
Write equations of ellipses
Graph ellipses

36. 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

37. Ellipses Chart (pg 434) Standard Form of Equation Center
(h,k)
Direction of Major Axis
Horizontal
Vertical
Foci
(h + c, k), (h – c, k)
(h, k + c), (h, k – c)
Length of Major Axis
2a units
Length of Minor Axis
2b units

38. Example One:
Graph the ellipse

39. Your Turn:
Graph the ellipse

40. Example Two:
Graph the ellipse

41. Your Turn:
Graph the ellipse

42. Example Three:
Write the equation of the ellipse in the graph:

43. Your Turn:
Write the equation of the ellipse in the graph:

44. Example Four:
Write the equation of the ellipse in the graph:

45. Your Turn:
Write the equation of
the ellipse in the graph:

46. 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

47. 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

48. Classwork

52. Hyperbolas Chart Standard Form of Equation
Direction of Transverse Axis
Horizontal
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
Length of Conjugate Axis
2b units
Equations of Asymptotes

53. Example One:
Graph the
hyperbola

54. Your Turn:
Graph the
hyperbola

55. Example Two:
Graph the
hyperbola

56. Your Turn:
Graph the
hyperbola

57. Example Three:
Write the equation of the hyperbola in the graph:

58. Your Turn:
Write the equation of the hyperbola in the graph:

59. 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

60. 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

62. 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.

63. Standard Forms (you already know )
Conic Section
Standard Form of Equation
Parabola
y = a(x – h)2 + k x = a(y – k)2 + h
Circle
(x – h)2 + (y – k)2 = r2
Ellipse
Hyperbola

64. Identifying Conic Sections
Relationship of A and C
Type of Conic Section
Only x2 or y2
Parabola
Same number in front of x2 and y2
Circle
Different number in front of x2 and y2 with plus sign
Ellipse
Different number in front of x2 and y2 with plus sign or minus sign
Hyperbola

65. 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

66. 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

67. 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 = 0
x2 – 4y2 – 16x + 24y – 36 = 0
x2 + 2y2 + 8x + 4y + 2 = 0

68. Classwork/Homework
Workbook
Page 56
1 – 3, 8 – 10
Page 57
1 – 12