Pre-Calc Conics

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2 Slide 2 / 160 Pre-Calc Conics

3 Slide 3 / 160 Table of Contents click on the topic to go to that section Review of Midpoint and Distance Formulas Intro to Conic Sections Parabolas Circles Ellipses Hyperbolas Recognizing Conic Sections from the General Form

4 Slide 4 / 160 Midpoint and Distance Formula Return to Table of Contents

5 Slide 5 / 160 Midpoint and Distance Formula The Midpoint Formula Give points A(x 1,y 1 ) and B (x 2,y 2 ), the point midway between A and B is Teacher Examples: Find the midpoint of the segment with the given endpoints.

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7 Slide 7 / 160 Midpoint and Distance Formula 1 Find the midpoint of K(1,8) & L(5,2). Teacher A (2,3) B (3,5) C (-2,-3) D (-3,-5)

8 Slide 8 / 160 Midpoint and Distance Formula 2 Find the midpoint of H(-4, 8) & L(6, 10). A (1,9) Teacher B (2,18) C (-2,-18) D (-1,-9)

9 Slide 9 / 160 Midpoint and Distance Formula 3 Given the midpoint of a segment is (4, 9) and one endpoint is (-3, 10), find the other midpoint. A (-10, 8) Teacher B (11, 8) C (-10, 11) D (.5, 9.5)

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12 Slide 12 / 160 Midpoint and Distance Formula 4 What is the distance between (2, 4) and (-1, 8)? Teacher

13 Slide 13 / 160 Midpoint and Distance Formula 5 What is the distance between (0, 7) and (5, -5)? Teacher

14 Slide 14 / 160 Midpoint and Distance Formula Note: The distance between points A and B can be notated as AB

15 Slide 15 / 160 Midpoint and Distance Formula 6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x. Teacher A -7 B -5 C -3 D -1 E 0 F 1 G 3 H 5 I 7 J 9

16 Slide 16 / 160 Intro to Conic Sections Return to Table of Contents

17 Slide 17 / 160 Intro to Conic Sections Conic Sections come from cutting through 2 cones, which is called taking cross sections. Conic Sections are often times not functions because they do not pass the Vertical Line Test.

18 Slide 18 / 160 Intro to Conic Sections A Circle comes from cutting parallel to the "base". The term base is mis-leading because cones continue on, like lines.

19 Slide 19 / 160 Intro to Conic Sections An Ellipse comes from cutting skew (diagonal) to the "base".

20 Slide 20 / 160 Intro to Conic Sections A Parabola comes from cutting the cone an intersecting the "base" and parallel to a side.

21 Slide 21 / 160 Intro to Conic Sections A Hyperbola comes from cutting the cones perpendicular to the "bases". This is the only cross section that intersects both cones.

22 Slide 22 / 160 Parabolas Return to Table of Contents

23 Slide 23 / 160 Parabolas As we've studied earlier, Parabolas come from a quadratic equation of the form y=ax 2 +bx+c and have a "U" shaped graph. Teacher Another helpful form of the equation is called Standard Form. Standard Form is (x - h) 2 = 4p(y - k), where (h,k) is the vertex. This is also called Vertex Form. Example: What is the vertex of: (x - 4) 2 = -3(y - 5) (x + 7) 2 = 2(y - 2) (x -3) 2 = y

24 Slide 24 / 160 Parabolas 7 What is the vertex of Teacher A (3, 2) B (-3, -2) C (2, 3) D (-2, -3)

25 Slide 25 / 160 Parabolas 8 What is the vertex of A (3, 2) Teacher B (-3, -2) C (2, 3) D (-2, -3)

26 Slide 26 / 160 Parabolas 9 What is the vertex of A (3, 2) Teacher B (-3, -2) C (2, -3) D (-2, -3)

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29 Slide 29 / 160 Parabolas 10 What is the vertex of A (3, 2) Teacher B (-3, 2) C (2, 3) D (-2, -3)

30 Slide 30 / 160 Parabolas 11 What is the vertex of A (3, 2) Teacher B (-3, -2) C (2, 3) D (-2, -3)

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32 Slide 32 / 160 Parabolas Converting from General Form to Standard Form Note: To convert into Standard Form, we use a process called Completing the Square. Steps: 1) Group the quadratic and its linear term on one side, and move the other linear and constant terms to the other side. 2) If there is a number in front of the quadratic, factor it out of the group. 3) Take the number in front of the linear term, divide it in half and square it. 4) Add this number inside the parenthesis; multiply it by the number you factored out in step two, and add it to the other side of the equation as well. 5) Factor the quadratic function inside the parenthesis

33 Slide 33 / 160 Parabolas Example: Find the vertex of the parabola Teacher

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39 Slide 39 / 160 Parabolas 17 What is the vertex of y 2-10y - x + 29 = 0? A (4, 5) Teacher B (-4, 5) C (-5, 4) D (5, 4)

40 Slide 40 / 160 Parabolas 18 What is the vertex of A (4, 5) Teacher B (-4, 5) C (-5, 4) D (5, 4)

41 Slide 41 / 160 Parabolas Converting from General Form to Standard Form } +18 } -12

42 Slide 42 / 160 Parabolas 19 What should be factored out of (4y 2-8y + ) = x - 9 +? Teacher

43 Slide 43 / 160 Parabolas 20 What value completes the square of 4(y 2-2y + ) = x - 9 +? Teacher

44 Slide 44 / 160 Parabolas 21 What value should follow "-9" in 4(y 2-2y + ) = x - 9 +? Teacher

45 Slide 45 / 160 Parabolas 22 Which is the correct standard form of 4(y 2-2y + ) = x Teacher A B C D

46 Slide 46 / 160 Parabolas 23 What should be factored out of (-5x 2-20x + ) = y - 7 +? Teacher

47 Slide 47 / 160 Parabolas 24 What value completes the square of -5(x 2 + 4x + ) = y - 7 +? Teacher

48 Slide 48 / 160 Parabolas 25 What value should follow "-7" in -5(x 2 + 4x + ) = y - 7 +? Teacher

49 Slide 49 / 160 Parabolas 26 Which is the correct standard form of (-5x 2-20x + ) = y Teacher A B C D

50 Slide 50 / 160 Parabolas Focus and Directrix of a Parabola Every point on the parabola is the same distance from the directrix and the focus. L 1 =L 2 L 1 L 2 Focus Axis of Symmetry Directrix The focal distance is the distance from the vertex to the focus, which is the same as the distance from the vertex to the directrix.

51 Slide 51 / 160 Parabolas Eccentricity of a Parabola L 1 =L 2 L 1 L 2 Focus Directrix

52 Slide 52 / 160 Parabolas Parts of a Parabola Whether a quadratic has the x 2 or y 2, they have the same parts. ax 2 +bx+dy+e=0 cy 2 +dy+bx+e=0 Focus Vertex Axis of Symmetry Vertex Focus Directrix Directrix Axis of Symmetry

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54 Slide 54 / 160 Parabolas Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? Teacher

55 Slide 55 / 160 Parabolas Graph the equation from the last example.

56 Parabolas Slide 56 / 160 Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? Teacher

57 Slide 57 / 160 Parabolas Graph Teacher

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59 Slide 59 / 160 Parabolas Graph Teacher

60 Slide 60 / 160 Parabolas 27 Given the following equation, which direction does it open? A B C D UP DOWN LEFT RIGHT Teacher

61 Slide 61 / 160 Parabolas 28 Where is the vertex for the following equation? A (-3, 4) Teacher B (3, 4) C (4, 3) D (4, -3)

62 Slide 62 / 160 Parabolas 29 What is the equation of the axis of symmetry for the following equation? Teacher A y = 3 B y = -3 C x = 4 D x = -4

63 Slide 63 / 160 Parabolas 30 What is the focal distance in the following equation? Teacher

64 Slide 64 / 160 Parabolas 31 What is the equation of the directrix for the following equation? Teacher A y = 2 B y = -4 C x = 3 D x = -5

65 Slide 65 / 160 Parabolas 32 Where is the focus for the following equation? Teacher A (-3, 5) B (3, 5) C (5, 3) D (5, -3)

66 Slide 66 / 160 Parabolas 33 What is the eccentricity of the following conic section? Teacher

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75 Slide 75 / 160 Parabolas 42 Where is the vertex for the following equation? A (0, 4) Teacher B (0, -4) C (4, 0) D (-4, 0)

76 Slide 76 / 160 Parabolas 43 What is the equation of the axis of symmetry for the following equation? Teacher A y = 0 B y = -0 C x = 4 D x = -4

77 Slide 77 / 160 Parabolas 44 What is the focal distance in the following equation? Teacher

78 Slide 78 / 160 Parabolas 45 What is the equation of the directrix for the following equation? Teacher A y = 0 B y = -4 C x = 8 D x = 0

79 Slide 79 / 160 Parabolas 46 Where is the focus for the following equation? A (4, 8) Teacher B (-4, 4) C (4, 4) D (4, -4)

80 Slide 80 / 160 Parabolas 47 What is the eccentricity of the following conic section? Teacher

81 Slide 81 / 160 Circles Return to Table of Contents

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85 Slide 85 / 160 Circles 48 Write the equation of the circle with center (5, 2) and radius 6 Teacher A B C D

86 Slide 86 / 160 Circles 49 Write the equation of the circle with center (-5, 0) and radius 7 Teacher A B C D

87 Slide 87 / 160 Circles 50 Write the equation of the circle with center (-2, 1) and radius Teacher A B C D

88 Slide 88 / 160 Circles 51 What is the center and radius of the following equation? Teacher A B C D

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90 Slide 90 / 160 Circles 53 What is the center and radius of the following equation? Teacher A B C D

91 Slide 91 / 160 Circles 54 What is eccentricity of a circle? Teacher

92 Slide 92 / 160 Circles Ex: Write the equation of the circle that meets the following criteria: Diameter with endpoints (4, 7) and (-2, -1). Teacher Since the midpoint of the diameter is the center use the midpoint formula. The radius is distance from the center to either of the given points.

93 Slide 93 / 160 Circles Ex: Write the equation of the circle that meets the following criteria: Center (1, -2) and passes through (4, 6) Teacher Since we know the center we only need to find the radius. The radius is the distance from the center to the point.

94 Slide 94 / 160 Circles Ex: Write the equation of the circle that meets the following criteria: Center at (-5, 6) and tangent to the y-axis. "Tangent to the y-axis" means the circle only touches the y-axis at one point. Look at the graph. Teacher

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96 Slide 96 / 160 Circles Write the equation of the circle in standard form that meets the following criteria: Teacher Complete the square for the x's

97 Slide 97 / 160 Circles 55 What is the equation of the circle that has a diameter with endpoints (0, 0) and (16, 12)? Teacher A B C D

98 Slide 98 / 160 Circles 56 What is the equation of the circle with center (-3, 5) and contains point (1, 3)? Teacher A B C D

99 Slide 99 / 160 Circles 57 What is the equation of the circle with center (7, -3) and tangent to the x-axis? A B C D Teacher

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102 Slide 102 / 160 Ellipses Return to Table of Contents

103 Slide 103 / 160 Ellipses An ellipse is the set of points the same total distance from 2 points. In this example, P As point moves along the ellipse, L 1 and L 2 will change but their sum will stay ten.

104 Slide 104 / 160 Ellipses In this graph F 1 and F 2 are foci. (Plural of focus) They lie on the major axis. (The longest distance) The shortest distance is the minor axis. Where the axes intersect is the ellipse's center. P The more elongated the ellipse the closer the eccentricity is to 1. The closer an ellipse is to being a circle, the closer the eccentricity is to 0. (0 < e < 1)

105 Slide 105 / 160 Ellipses 60 What letter or letters corresponds with ellipse's center? A B C D B A C E Teacher D E

106 Slide 106 / 160 Ellipses 61 What letter or letters corresponds with ellipse's foci? Teacher A B C D B A C E D E

107 Slide 107 / 160 Ellipses 62 What letter or letters corresponds with ellipse's major axis? A Teacher B C D B A C E D E

108 Slide 108 / 160 Ellipses 63 Which choice best describes an ellipse's eccentricity? A e = 0 Teacher B 0< e < 1 C e = 1 D e > 1

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112 Slide 112 / 160 Ellipses 64 What is the center of A (9, 4) Teacher B (5, 6) C (-5, -6) D (3, 2)

113 Slide 113 / 160 Ellipses 65 How long is the major axis of A 9 Teacher B 4 C 3 D 2

114 Slide 114 / 160 Ellipses 66 How long is the minor axis of A 9 Teacher B 4 C 3 D 2

115 Slide 115 / 160 Ellipses 67 Name one foci of A Teacher B C D

116 Slide 116 / 160 Ellipses 68 Name one foci of Teacher A B C D

117 Slide 117 / 160 Ellipses Graphing an Ellipse Find and graph the center Find the length and direction of the major and minor axes From the center go half the length the axis from the center for each Graph the ellipse The center is (4, -2) The major axis is 6 units and horizontal The minor axis is 4 units and vertical

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120 Slide 120 / 160 Ellipses What is equation of an ellipse with foci (3, -2) and (3, 6) and minor axis of length 8? Teacher

121 Slide 121 / 160 Ellipses 69 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, what is the center of the ellipse? Teacher A (8, 2) B (0, 2) C (0, 1) D (-8, 1)

122 Slide 122 / 160 Ellipses 70 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, in which direction is the ellipse elongated? Teacher A B C D horizontally vertically obliquely it is not elongated

123 Slide 123 / 160 Ellipses 71 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, how far is it from the center to an endpoint of the major axis? A 10 B 100 C 5 D 25 Teacher

124 Slide 124 / 160 Ellipses 72 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis? Teacher A B C D

125 Slide 125 / 160 Ellipses 73 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, find a. Teacher A B C D

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127 Slide 127 / 160 Ellipses Converting to Standard Form complete the square for x and/or y factor the x's and y's divide by the constant Teacher Ex: Ex:

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129 Slide 129 / 160 Ellipses 75 Convert the following ellipse to standard form. Teacher A B C D

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132 Slide 132 / 160 Hyperbolas Return to Table of Contents

133 Slide 133 / 160 Hyperbolas The standard form of a horizontal hyperbola is

134 Slide 134 / 160 Hyperbolas The standard form of a vertical hyperbola is

135 Slide 135 / 160 Hyperbolas To graph a hyperbola in standard form: graph (h,k) as center of graph go a right and left of the center, and b up and down make a rectangle through the four points from previous step draw asymptotes that contain the diagonals of the rectangle decide if hyperbola goes left & right or up & down left & right: the "x term" is first up & down: the "y term" is first graph hyperbola

136 Slide 136 / 160 Hyperbolas Example: Graph Teacher The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens? What are the slopes of the asymptotes? How does this relate to a and b? Why?

137 Slide 137 / 160 Hyperbolas Example: Graph Teacher The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens?

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Slide 146 / 160 Hyperbolas Standard Form of an Hyperbola The Foci are equidistant from the center in the horizontal direction if the x-term comes first, or in

146 Slide 146 / 160 Hyperbolas Standard Form of an Hyperbola The Foci are equidistant from the center in the horizontal direction if the x-term comes first, or in the vertical direction if the y-term comes first. The distance from the center to the foci is In this example, the focal distance is And their location is at and

147 Slide 147 / 160 Hyperbolas 85 What is the focal distance for the following equation? Teacher A 12 B 13 C 5 D 8

148 Slide 148 / 160 Hyperbolas 86 What is the location of one of the foci for this hyperbola? A (-13,-6) B (10,-6) C (-10,-6) Teacher D (13,-6)

149 Slide 149 / 160 Hyperbolas 87 What is the location of one of the foci for this hyperbola? A (3,7) Teacher B (3,19) C (3,-7) D (3,13)

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152 Slide 152 / 160 Recognizing Conic Sections from the General Form Return to Table of Contents

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Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas

Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 2015-03-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 160 Review of Midpoint and Distance Formulas Intro to Conic Sections