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 Circles Ellipses Hyperbolas Recognizing Conic Sections from the General Form
Slide 4 / 160 Midpoint and Distance Formula Return to Table of Contents Midpoint and Distance Formula Slide 5 / 160 The Midpoint Formula Give points A(x1,y1) and B (x2,y2), the point midway between A and B is Examples: Find the midpoint of the segment with the given endpoints. Slide 6 / 160
Midpoint and Distance Formula Slide 7 / 160 1 Find the midpoint of K(1,8) & L(5,2). A (2,3) B (3,5) C (-2,-3) D (-3,-5) Midpoint and Distance Formula Slide 8 / 160 2 Find the midpoint of H(-4, 8) & L(6, 10). A (1,9) B (2,18) C (-2,-18) D (-1,-9) Midpoint and Distance Formula Slide 9 / 160 3 Given the midpoint of a segment is (4, 9) and one endpoint is (-3, 10), find the other midpoint. A (-10, 8) B (11, 8) C (-10, 11) D (.5, 9.5)
Slide 10 / 160 Slide 11 / 160 Midpoint and Distance Formula Slide 12 / 160 4 What is the distance between (2, 4) and (-1, 8)?
Midpoint and Distance Formula Slide 13 / 160 5 What is the distance between (0, 7) and (5, -5)? Midpoint and Distance Formula Slide 14 / 160 Note: The distance between points A and B can be notated as AB Midpoint and Distance Formula Slide 15 / 160 6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x. A -7 B -5 C -3 D -1 E 0 F 1 G 3 H 5 I 7 J 9
Slide 16 / 160 Intro to Conic Sections Return to Table of Contents Intro to Conic Sections Slide 17 / 160 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. Intro to Conic Sections Slide 18 / 160 A Circle comes from cutting parallel to the "base". The term base is mis-leading because cones continue on, like lines.
Intro to Conic Sections Slide 19 / 160 An Ellipse comes from cutting skew (diagonal) to the "base". Intro to Conic Sections Slide 20 / 160 A Parabola comes from cutting the cone an intersecting the "base" and parallel to a side. Intro to Conic Sections Slide 21 / 160 A Hyperbola comes from cutting the cones perpendicular to the "bases". This is the only cross section that intersects both cones.
Slide 22 / 160 Return to Table of Contents Slide 23 / 160 As we've studied earlier, come from a quadratic equation of the form y=ax 2 +bx+c and have a "U" shaped graph. 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 Slide 24 / 160 7 What is the vertex of A (3, 2) B (-3, -2) C (2, 3) D (-2, -3)
Slide 25 / 160 8 What is the vertex of A (3, 2) B (-3, -2) C (2, 3) D (-2, -3) Slide 26 / 160 9 What is the vertex of A (3, 2) B (-3, -2) C (2, -3) D (-2, -3) Slide 27 / 160
Slide 28 / 160 Slide 29 / 160 10 What is the vertex of A (3, 2) B (-3, 2) C (2, 3) D (-2, -3) Slide 30 / 160 11 What is the vertex of A (3, 2) B (-3, -2) C (2, 3) D (-2, -3)
Slide 31 / 160 Slide 32 / 160 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 Example: Find the vertex of the parabola Slide 33 / 160
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Slide 37 / 160 Slide 38 / 160 Slide 39 / 160 17 What is the vertex of y 2-10y - x + 29 = 0? A (4, 5) B (-4, 5) C (-5, 4) D (5, 4)
Slide 40 / 160 18 What is the vertex of A (4, 5) B (-4, 5) C (-5, 4) D (5, 4) Slide 41 / 160 Converting from General Form to Standard Form } +18 } -12 Slide 42 / 160 19 What should be factored out of (4y 2-8y + ) = x - 9 +?
Slide 43 / 160 20 What value completes the square of 4(y 2-2y + ) = x - 9 +? Slide 44 / 160 21 What value should follow "-9" in 4(y 2-2y + ) = x - 9 +? Slide 45 / 160 22 Which is the correct standard form of 4(y 2-2y + ) = x - 9 + A B C D
Slide 46 / 160 23 What should be factored out of (-5x 2-20x + ) = y - 7 +? Slide 47 / 160 24 What value completes the square of -5(x 2 + 4x + ) = y - 7 +? Slide 48 / 160 25 What value should follow "-7" in -5(x 2 + 4x + ) = y - 7 +?
Slide 49 / 160 26 Which is the correct standard form of (-5x 2-20x + ) = y - 7 + A B C D Slide 50 / 160 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. Slide 51 / 160 Eccentricity of a Parabola L 1=L 2 L 1 L 2 Focus Directrix
Slide 52 / 160 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 Directrix Axis of Symmetry Focus Vertex Directrix Axis of Symmetry Slide 53 / 160 Slide 54 / 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?
Slide 55 / 160 Graph the equation from the last example. 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? Slide 57 / 160 Graph
Slide 58 / 160 Slide 59 / 160 Graph Slide 60 / 160 27 Given the following equation, which direction does it open? A B C D UP DOWN LEFT RIGHT
Slide 61 / 160 28 Where is the vertex for the following equation? A (-3, 4) B (3, 4) C (4, 3) D (4, -3) Slide 62 / 160 29 What is the equation of the axis of symmetry for the following equation? A y = 3 B y = -3 C x = 4 D x = -4 Slide 63 / 160 30 What is the focal distance in the following equation?
Slide 64 / 160 31 What is the equation of the directrix for the following equation? A y = 2 B y = -4 C x = 3 D x = -5 Slide 65 / 160 32 Where is the focus for the following equation? A (-3, 5) B (3, 5) C (5, 3) D (5, -3) Slide 66 / 160 33 What is the eccentricity of the following conic section?
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Slide 73 / 160 Slide 74 / 160 Slide 75 / 160 42 Where is the vertex for the following equation? A (0, 4) B (0, -4) C (4, 0) D (-4, 0)
Slide 76 / 160 43 What is the equation of the axis of symmetry for the following equation? A y = 0 B y = -0 C x = 4 D x = -4 Slide 77 / 160 44 What is the focal distance in the following equation? Slide 78 / 160 45 What is the equation of the directrix for the following equation? A y = 0 B y = -4 C x = 8 D x = 0
Slide 79 / 160 46 Where is the focus for the following equation? A (4, 8) B (-4, 4) C (4, 4) D (4, -4) Slide 80 / 160 47 What is the eccentricity of the following conic section? Slide 81 / 160 Circles Return to Table of Contents
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Circles Slide 85 / 160 48 Write the equation of the circle with center (5, 2) and radius 6 A B C D Circles Slide 86 / 160 49 Write the equation of the circle with center (-5, 0) and radius 7 A B C D Circles Slide 87 / 160 50 Write the equation of the circle with center (-2, 1) and radius A B C D
Circles Slide 88 / 160 51 What is the center and radius of the following equation? A B C D Slide 89 / 160 Circles Slide 90 / 160 53 What is the center and radius of the following equation? A B C D
Circles Slide 91 / 160 54 What is eccentricity of a circle? Circles Slide 92 / 160 Ex: Write the equation of the circle that meets the following criteria: Diameter with endpoints (4, 7) and (-2, -1). 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. Circles Slide 93 / 160 Ex: Write the equation of the circle that meets the following criteria: Center (1, -2) and passes through (4, 6) Since we know the center we only need to find the radius. The radius is the distance from the center to the point.
Circles Slide 94 / 160 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. Slide 95 / 160 Circles Slide 96 / 160 Write the equation of the circle in standard form that meets the following criteria: Complete the square for the x's
Circles Slide 97 / 160 55 What is the equation of the circle that has a diameter with endpoints (0, 0) and (16, 12)? A B C D Circles Slide 98 / 160 56 What is the equation of the circle with center (-3, 5) and contains point (1, 3)? A B C D Circles Slide 99 / 160 57 What is the equation of the circle with center (7, -3) and tangent to the x-axis? A B C D
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Ellipses Slide 103 / 160 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. Ellipses Slide 104 / 160 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) Ellipses Slide 105 / 160 60 What letter or letters corresponds with ellipse's center? A B C D B A C E D E
Ellipses Slide 106 / 160 61 What letter or letters corresponds with ellipse's foci? A B C D B A C E D E Ellipses Slide 107 / 160 62 What letter or letters corresponds with ellipse's major axis? A B C D B A C E D E Ellipses Slide 108 / 160 63 Which choice best describes an ellipse's eccentricity? A e = 0 B 0< e < 1 C e = 1 D e > 1
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Ellipses Slide 112 / 160 64 What is the center of A (9, 4) B (5, 6) C (-5, -6) D (3, 2) Ellipses Slide 113 / 160 65 How long is the major axis of A 9 B 4 C 3 D 2 Ellipses Slide 114 / 160 66 How long is the minor axis of A 9 B 4 C 3 D 2
Ellipses Slide 115 / 160 67 Name one foci of A B C D Ellipses Slide 116 / 160 68 Name one foci of A B C D Ellipses Slide 117 / 160 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
Slide 118 / 160 Slide 119 / 160 Ellipses Slide 120 / 160 What is equation of an ellipse with foci (3, -2) and (3, 6) and minor axis of length 8?
Ellipses Slide 121 / 160 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? A (8, 2) B (0, 2) C (0, 1) D (-8, 1) Ellipses Slide 122 / 160 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? A B C D horizontally vertically obliquely it is not elongated Ellipses Slide 123 / 160 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
Ellipses Slide 124 / 160 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? A B C D Ellipses Slide 125 / 160 73 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, find a. A B C D Slide 126 / 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 Slide 127 / 160 Ex: Ex: Slide 128 / 160 Ellipses Slide 129 / 160 75 Convert the following ellipse to standard form. A B C D
Slide 130 / 160 Slide 131 / 160 Slide 132 / 160 Hyperbolas Return to Table of Contents
Hyperbolas Slide 133 / 160 The standard form of a horizontal hyperbola is Hyperbolas Slide 134 / 160 The standard form of a vertical hyperbola is Hyperbolas Slide 135 / 160 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
Hyperbolas Example: Graph Slide 136 / 160 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? Hyperbolas Slide 137 / 160 Example: Graph The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens? Slide 138 / 160
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Slide 145 / 160 Hyperbolas Slide 146 / 160 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 Hyperbolas Slide 147 / 160 85 What is the focal distance for the following equation? A 12 B 13 C 5 D 8
Hyperbolas Slide 148 / 160 86 What is the location of one of the foci for this hyperbola? A (-13,-6) B (10,-6) C (-10,-6) D (13,-6) Hyperbolas Slide 149 / 160 87 What is the location of one of the foci for this hyperbola? A (3,7) B (3,19) C (3,-7) D (3,13) Slide 150 / 160
Slide 151 / 160 Slide 152 / 160 Recognizing Conic Sections from the General Form Return to Table of Contents Slide 153 / 160
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