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

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Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 2015-03-24 www.njctl.

1 Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 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

2 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

3 Midpoint and Distance Formula Slide 7 / 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 / 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 / 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)

4 Slide 10 / 160 Slide 11 / 160 Midpoint and Distance Formula Slide 12 / What is the distance between (2, 4) and (-1, 8)?

5 Midpoint and Distance Formula Slide 13 / 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 / 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.

6 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

Intro to Conic Sections Slide 20 / 160 A Parabola comes from

Intro to Conic Sections Slide 21 / 160 A Hyperbola comes from

7 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.

8 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 / What is the vertex of A (3, 2) B (-3, -2) C (2, 3) D (-2, -3)

9 Slide 25 / What is the vertex of A (3, 2) B (-3, -2) C (2, 3) D (-2, -3) Slide 26 / What is the vertex of A (3, 2) B (-3, -2) C (2, -3) D (-2, -3) Slide 27 / 160

10 Slide 28 / 160 Slide 29 / What is the vertex of A (3, 2) B (-3, 2) C (2, 3) D (-2, -3) Slide 30 / What is the vertex of A (3, 2) B (-3, -2) C (2, 3) D (-2, -3)

11 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

12 Slide 34 / 160 Slide 35 / 160 Slide 36 / 160

13 Slide 37 / 160 Slide 38 / 160 Slide 39 / 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,

14 Slide 40 / 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 / 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 +?

15 Slide 43 / What value completes the square of 4(y 2-2y + ) = x - 9 +? Slide 44 / What value should follow "-9" in 4(y 2-2y + ) = x - 9 +? Slide 45 / Which is the correct standard form of 4(y 2-2y + ) = x A B C D

16 Slide 46 / What should be factored out of (-5x 2-20x + ) = y - 7 +? Slide 47 / What value completes the square of -5(x 2 + 4x + ) = y - 7 +? Slide 48 / What value should follow "-7" in -5(x 2 + 4x + ) = y - 7 +?

17 Slide 49 / Which is the correct standard form of (-5x 2-20x + ) = y 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

18 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?

19 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

20 Slide 58 / 160 Slide 59 / 160 Graph Slide 60 / Given the following equation, which direction does it open? A B C D UP DOWN LEFT RIGHT

21 Slide 61 / Where is the vertex for the following equation? A (-3, 4) B (3, 4) C (4, 3) D (4, -3) Slide 62 / 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 / What is the focal distance in the following equation?

22 Slide 64 / 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 / Where is the focus for the following equation? A (-3, 5) B (3, 5) C (5, 3) D (5, -3) Slide 66 / What is the eccentricity of the following conic section?

23 Slide 67 / 160 Slide 68 / 160 Slide 69 / 160

24 Slide 70 / 160 Slide 71 / 160 Slide 72 / 160

Slide 73 / 160 Slide 74 / 160 Slide 75 / 160 42 Where is the

25 Slide 73 / 160 Slide 74 / 160 Slide 75 / Where is the vertex for the following equation? A (0, 4) B (0, -4) C (4, 0) D (-4, 0)

26 Slide 76 / 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 / What is the focal distance in the following equation? Slide 78 / What is the equation of the directrix for the following equation? A y = 0 B y = -4 C x = 8 D x = 0

27 Slide 79 / Where is the focus for the following equation? A (4, 8) B (-4, 4) C (4, 4) D (4, -4) Slide 80 / What is the eccentricity of the following conic section? Slide 81 / 160 Circles Return to Table of Contents

28 Slide 82 / 160 Slide 83 / 160 Slide 84 / 160

Circles Slide 85 / 160 48 Write the equation of the circle with center

of the circle with center (-5, 0) and radius 7 A B C D Circles Slide 87 /

29 Circles Slide 85 / Write the equation of the circle with center (5, 2) and radius 6 A B C D Circles Slide 86 / Write the equation of the circle with center (-5, 0) and radius 7 A B C D Circles Slide 87 / Write the equation of the circle with center (-2, 1) and radius A B C D

30 Circles Slide 88 / What is the center and radius of the following equation? A B C D Slide 89 / 160 Circles Slide 90 / What is the center and radius of the following equation? A B C D

31 Circles Slide 91 / 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.

32 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)?

33 Circles Slide 97 / 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 / What is the equation of the circle with center (-3, 5) and contains point (1, 3)? A B C D Circles Slide 99 / What is the equation of the circle with center (7, -3) and tangent to the x-axis? A B C D

34 Slide 100 / 160 Slide 101 / 160 Slide 102 / 160 Ellipses Return to Table of Contents

35 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 / What letter or letters corresponds with ellipse's center? A B C D B A C E D E

36 Ellipses Slide 106 / What letter or letters corresponds with ellipse's foci? A B C D B A C E D E Ellipses Slide 107 / What letter or letters corresponds with ellipse's major axis? A B C D B A C E D E Ellipses Slide 108 / Which choice best describes an ellipse's eccentricity? A e = 0 B 0< e < 1 C e = 1 D e > 1

37 Slide 109 / 160 Slide 110 / 160 Slide 111 / 160

38 Ellipses Slide 112 / What is the center of A (9, 4) B (5, 6) C (-5, -6) D (3, 2) Ellipses Slide 113 / How long is the major axis of A 9 B 4 C 3 D 2 Ellipses Slide 114 / How long is the minor axis of A 9 B 4 C 3 D 2

39 Ellipses Slide 115 / Name one foci of A B C D Ellipses Slide 116 / 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

40 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?

41 Ellipses Slide 121 / 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 / 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 / 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

42 Ellipses Slide 124 / 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 / 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

43 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 / Convert the following ellipse to standard form. A B C D

44 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

45 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?

46 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

47 Slide 139 / 160 Slide 140 / 160 Slide 141 / 160

48 Slide 142 / 160 Slide 143 / 160 Slide 144 / 160

49 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 / What is the focal distance for the following equation? A 12 B 13 C 5 D 8

50 Hyperbolas Slide 148 / 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 / 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

51 Slide 151 / 160 Slide 152 / 160 Recognizing Conic Sections from the General Form Return to Table of Contents Slide 153 / 160

52 Slide 154 / 160 Slide 155 / 160 Slide 156 / 160

53 Slide 157 / 160 Slide 158 / 160 Slide 159 / 160

54 Slide 160 / 160

Pre-Calc Conics

Slide 1 / 160 Slide 2 / 160 Pre-Calc Conics 2015-03-24 www.njctl.org 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