Algebra II. Slide 1 / 181. Slide 2 / 181. Slide 3 / 181. Conic Sections Table of Contents

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1 Slide 1 / 181 Algebra II Slide 2 / 181 Conic Sections Table of Contents click on the topic to go to that section Slide 3 / 181 Review of Midpoint and Distance Formulas Introduction to Conic Sections Parabolas Circles Ellipses Hyperbolas Recognizing Conic Sections from General Form

2 Slide 4 / 181 Midpoint and Distance Formula Return to Table of Contents Slide 5 / 181 What is the midpoint of segment AB? A (-3, 6) (x,y) B (5, 2) Look at this segment - the midpoint is halfway. To find the coordinates of the midpoint, find the average of the x-values and the average of the y-values. The Midpoint Formula Given points A(x1,y1) and B (x2,y2), the midpoint of AB is Slide 6 / 181 Examples: Find the midpoint of the segment with the given endpoints.

3 Slide 7 / Find the midpoint of K(1,8) & L(5,2). Slide 8 / 181 A (2,3) B (3,5) C (-2,-3) D (-3,-5) 2 Find the midpoint of H(-4, 8) & L(6, 10). Slide 9 / 181 A (5,9) B (-1,9) C (1,9) D (5,1)

4 3 If the midpoint of a segment is (4,9) and one endpoint is (-3,10), find the other endpoint. Slide 10 / 181 A (-10,8) B (11,8) C (-10,11) D (.5,9.5) Slide 11 / 181 A (1, 4) C (1,-2) B (7, -2) How far apart are points A and B? The Distance Formula is derived from the Pythagorean Theorem, a 2 + b 2 = c 2. In this example, AC 2 + CB 2 =AB = AB 2 72 = AB 2 AB = = Slide 12 / 181

5 Slide 13 / What is the distance between (2, 4) and (-1, 8)? Slide 14 / What is the distance between (0, 7) and (5, -5)? Slide 15 / 181

6 6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x. Slide 16 / 181 A -7 B -5 C -3 D -1 E 0 F 1 G 3 H 5 I 7 J 9 7 If the distance between (4,5) and (x,-2) is 10, what are the possible values of x? Slide 17 / 181 A B C D Slide 18 / 181 Introduction to Conic Sections Return to Table of Contents

7 Conic Sections are created by intersecting a set of double cones with a plane. Slide 19 / 181 Discussion Question: Which conic sections are functions? More Info About Conics Slide 20 / 181 Click the link below for a YouTube video that demonstrates the cutting of the cones. SalMathGuy Conics Video Click on the title below to take you to a webpage for more background information about conic sections: "The Occurrence of the Conics", by Dr. Jill Britton The Circle Slide 21 / 181 A Circle comes from cutting parallel to the "base". The term base is misleading because like lines and planes, conic sections continue on forever.

8 Slide 22 / 181 The Ellipse An Ellipse comes from cutting skew to the "base". The Parabola Slide 23 / 181 A Parabola comes from intersecting the cone with a plane that is parallel to a side of the cone. The Hyperbola Slide 24 / 181 A Hyperbola comes from cutting the cones perpendicular to the "bases". This is the only cross section that intersects both cones.

9 Slide 25 / 181 Parabolas Return to Table of Contents This is the graph of y = x 2. Complete the table below: Slide 26 / 181 x y Discuss the patterns that you observe. A graph that has this shape is called a parabola. y = x 2 is the "parent function". The equation of a parabola can be written in two forms: y = ax 2 + bx + c (the General Form) Slide 27 / 181 y = a(x - h) 2 + k (Standard Form) where (h,k) is the vertex. This is also called Vertex Form. Example: Name the vertex of each equation: A) y= -3(x - 4) B) y= 2(x + 7) C) y= (x -3) 2

10 Match each equation to its parabola - drag the number of the graph to its equation. Slide 28 / f(x) = (x - 3)² - 2 g(x) = -2(x + 1)² h(x) = 2 / 3 (x + 5)² Slide 29 / What is the vertex of? Slide 30 / 181 A (3, 2) B (-3, -2) C (2, 3) D (-2, -3)

11 10 What is the vertex of? Slide 31 / 181 A (2, -3) B (-3, -2) C (2, 3) D (-2, -3) Slide 32 / 181 Slide 33 / 181

12 11 What is the vertex of? Slide 34 / 181 A (-3, 2) B (-3, -2) C (2, 3) D (-2, -3) 12 What is the vertex of? Slide 35 / 181 A (3, 2) B (-3, -2) C (2, 3) D (-2, -3) Slide 36 / 181

13 Converting from General Form to Standard Form Slide 37 / 181 Fill in the blank to complete the square: Half of 6 is 3, 3 2 = 9 Converting from General Form to Standard Form Slide 38 / 181 y = x 2-8x + 5 y = (x 2-8x + ) What number completes the square in the parenthesis above? Slide 39 / 181

14 Slide 40 / 181 Slide 41 / 181 Slide 42 / 181

15 Slide 43 / What is the vertex of x = y 2-10y + 29? A (4, 5) B (-4, 5) C (-5, 4) D (5, 4) Slide 44 / What is the vertex of y= x 2-8x +21? Slide 45 / 181 A (4, 5) B (-4, 5) C (-5, 4) D (5, 4)

16 Slide 46 / 181 Converting from General Form to Standard Form Slide 47 / What should be factored out of Slide 48 / 181 x = (4y 2-8y + )+ 9 -?

17 21 What value completes the square of Slide 49 / 181 x = 4(y 2-2y + )+ 9 -? 22 What value should follow "+ 9" in Slide 50 / 181 x = 4(y 2-2y + ) + 9? 23 Which is the correct standard form of Slide 51 / 181 x = (4y 2-8y + )+ 9 -? A x = 4(y - 1) B x = 4(y + 1) C x = 4(y - 1) D x = 4(y + 1) 2 + 5

18 24 What should be factored out of Slide 52 / 181 y = (-5x 2-20x + )+ 7 -? 25 What value completes the square of Slide 53 / 181 click to reveal y = -5(x 2 + 4x + )+ 7 -? 26 What value should follow "+7" in Slide 54 / 181 click y to = reveal -5(x 2 + 4x + )+ 7?

19 27 Which is the correct standard form of Slide 55 / 181 y = (-5x 2-20x + )+ 7 -? A y = -5(x - 2) B y = -5(x + 2) C y = -5(x - 2) 2-13 D y = -5(x - 2) Geometric Definition A parabola is a locus* of points equidistant from a fixed point, the focus, and a fixed line, the directrix. Slide 56 / 181 *locus is just a fancy word for set. Every parabola is symmetric with respect to a line through the focus and perpendicular to the directrix. The vertex of the parabola is the "turning point" and is on the axis of symmetry. Focus and Directrix of a Parabola Slide 57 / 181 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

20 Eccentricity of a Parabola Slide 58 / 181 L 1=L 2 L 1 L 2 Focus Directrix All parabolas have an eccentricity of 1. Parts of a Parabola Parts are the same for all parabolas, regardless of the direction in which they open. Slide 59 / 181 y=ax 2 +bx+c x=ay 2 +by+c Focus Vertex Directrix Axis of Symmetry Focus Vertex Directrix Axis of Symmetry Compare the graphs below: What makes the graph more "narrow" or "wide"? Slide 60 / 181 y = 2x 2 y = x 2 y =.5x 2

21 28 Which of the parabolas below are narrower than their parent functions? Slide 61 / 181 A B C D Focal Distance Slide 62 / 181 To calculate: focal distance = The distance from the vertex to the focus is 1. The distance from the vertex to the directrix is 1. Parabola Summary Slide 63 / 181 General Form y= ax 2 + bx + c x= ay 2 +by + c Standard Form y= a(x - h) 2 +k x= a(y - k) 2 + h Opens a>0 opens up a<0 opens down a>0 opens to the right a<0 opens to the left Axis of Symmetry x = h y = k Vertex (h, k) (h, k) Focal Distance Directrix Focus Eccentricity 1 1

22 Slide 64 / 181 Graph the equation from the last example. Slide 65 / 181 Directrix Focus Axis of Symmetry 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 66 / 181

23 Graph Slide 67 / 181 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 68 / 181 Step 1: Convert the equation from general to standard form. Step 2: 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 69 / 181

24 Graph Slide 70 / Given the following equation, which direction does it open? Slide 71 / 181 A B C D Up Down Left Right 30 How does the following equation compare to the parent function Slide 72 / 181 A Is narrower B Is wider C Is the same width

25 31 Where is the vertex for the following equation? Slide 73 / 181 A (-3, 4) B (3, 4) C (4, 3) D (4, -3) 32 What is the equation of the axis of symmetry for the following equation? Slide 74 / 181 A y = 3 B y = -3 C x = 4 D x = What is the focal distance in the following equation? Slide 75 / 181

26 34 What is the equation of the directrix for the following equation? Slide 76 / 181 A y = 2 B y = -4 C x = 3 D x = Where is the focus for the following equation? Slide 77 / 181 A (-3, 5) B (3, 5) C (5, 3) D (5, -3) 36 What is the eccentricity of the following conic section? Slide 78 / 181

27 Slide 79 / 181 Slide 80 / 181 Slide 81 / 181

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29 Slide 85 / 181 Slide 86 / 181 Slide 87 / 181

30 Slide 88 / 181 Slide 89 / 181 Slide 90 / 181

31 Slide 91 / 181 Slide 92 / What is the equation of the parabola with vertex (2,3) and directrix y = 4? Slide 93 / 181 A y = 4(x - 2) B C D y = -1/4(x - 2) x = 4(y - 2) x = 1/4(y - 2) 2 + 3

32 Challenge Problem The St. Louis Arch is 630 feet tall and 630 feet wide at the base. Write an equation to represent the shape of the arch. Slide 94 / 181 Answer on next page... Slide 95 / 181 Slide 96 / 181 Circles Return to Table of Contents

33 A circle is a locus of points in a plane that are equidistant from a given point. Slide 97 / 181 Radius (x,y) Center (h,k) The distance from the center to a point on the circle is Slide 98 / 181 Slide 99 / 181

34 Slide 100 / Write the equation of the circle with center (5, 2) and radius 6 Slide 101 / 181 A B C D 53 Write the equation of the circle with center (-5,0) and radius 7 Slide 102 / 181 A B C D

35 54 Write the equation of the circle with center (-2,1) and radius Slide 103 / 181 A B C D 55 What is the center and radius of the following equation? Slide 104 / 181 A B C D Slide 105 / 181

36 57 What is the center and radius of the following equation? Slide 106 / 181 A B C D 58 What is eccentricity of a circle? Slide 107 / 181 Write the equations for each part of this unfortunate snowman. Slide 108 / 181

37 Write the equation of the circle that meets the following criteria: Slide 109 / 181 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. The equation of the circle is: Write the equation of the circle that meets the following criteria: Diameter with endpoints (4, 7) and (-2, -1). Slide 110 / 181 Write the equation of the circle that meets the following criteria: Slide 111 / 181 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.

38 Slide 112 / 181 Write the equation of the circle in standard form that meets the following criteria: Slide 113 / 181 Complete the square for the x's. (Remember, the y-term is 0y.) 59 What is the equation of the circle that has a diameter with endpoints (0,0) and (16,12)? Slide 114 / 181 A B C D

39 60 What is the equation of the circle with center (-3,5) that contains the point (1,3)? Slide 115 / 181 A B C D 61 What is the equation of the circle with center (7,-3) and tangent to the x-axis? Slide 116 / 181 A B C D Slide 117 / 181

40 Slide 118 / 181 Challenge Question: What is the equation of a circle that passes through the three points (2,3), (2,-2), and (5,-3)? Slide 119 / 181 Remember that the distance from the radius to the circle is the same for every radius. Let (x,y) be the center and use the distance formula twice. Slide 120 / 181 Ellipses Return to Table of Contents

41 An ellipse is a focus of points in a plane that are each the same total distance from 2 fixed points, called the foci (plural of focus). P 1 P 2 Slide 121 / 181 F 1 F 2 For example, P 1F 1 + P 1F 2 = P 2F 1 + P 2F 2 Slide 122 / 181 Eccentricity of an Ellipse 0 < e < 1 The eccentricity of an ellipse is a number between 0 and 1. 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. The major axis, AB, is the segment through both foci whose endpoints are on the ellipse. The minor axis, CD, is perpendicular to the major axis through the center, O. The vertices of an ellipse are the endpoints of the major axis, points A and B. The co-vertices are the endpoints of the minor axis, points C and D. Slide 123 / 181 C A O B D

42 Parts of an Ellipse Slide 124 / 181 Horizontal ellipse Vertical ellipse Minor axis Major axis Vertex Co-vertex Vertex Co-vertex Major axis Minor axis Co-vertex Vertex Focus The length of the major axis is 2a. The length of the minor axis is 2b. 64 What letter or letters corresponds with ellipse's center? Slide 125 / 181 A B C C D E A E F G B F G D 65 What letter or letters corresponds with ellipse's foci? Slide 126 / 181 A C B C D A E F G B E F G D

43 66 What letter or letters corresponds with ellipse's major axis? Slide 127 / 181 A E B F I C D G H E D B A C G F I H 67 Which choice best describes an ellipse's eccentricity? Slide 128 / 181 A e = 0 B 0< e < 1 C e = 1 D e > 1 68 Which of the ellipses has the greater eccentricity? Slide 129 / 181 A B A B

44 Slide 130 / 181 Finding the foci: (Note that in this case, a represents the hypotenuse of the triangle.) Slide 131 / 181 a b a a c In this ellipse, a = 5 and b = 4, so c = 3. The coordinates of the foci are (3-3,2) and (3+3,2) or (0,2) and (6,2) 69 What is the center of A (9, 4) B (5, 6) C (-5, -6) D (3, 2)? Slide 132 / 181

45 70 How long is the major axis of A 9 B 6 C 3 D 2? Slide 133 / How long is the minor axis of A 9 B 4 C 3 D 2 Slide 134 / Name one foci of A B C D Slide 135 / 181

46 73 Name one foci of Slide 136 / 181 A B C D Graphing an Ellipse Slide 137 / 181 Find and graph the center Find the length and direction of the major and minor axes Draw the major and minor axes Draw the ellipse The center is (4, -2) The major axis is 6 units and horizontal The minor axis is 4 units and vertical Slide 138 / 181

47 Slide 139 / 181 Slide 140 / Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, what is the center of the ellipse? Slide 141 / 181 A (8, 2) B (0, 2) C (0, 1) D (-8, 1)

48 75 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, in which direction is the ellipse elongated? Slide 142 / 181 A B C D horizontally vertically obliquely it is not elongated 76 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? Slide 143 / 181 A 10 B 100 C 5 D 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? Slide 144 / 181 A B C D

49 78 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, find b. Slide 145 / 181 A B C D Slide 146 / Given that an ellipse has foci (4,-4) and (4,2) and minor axis of length 8, which is the equation of the Slide 147 / 181 ellipse?

50 Steps for Converting the ellipse from General Form to Standard Form Slide 148 / 181 factor the x's and y's divide by the constant complete the square for x and/or y Slide 149 / 181 Slide 150 / 181

51 80 Convert the following ellipses to standard form. Slide 151 / 181 A B C D 81 Convert the following ellipses to standard form. Slide 152 / 181 A B C D Slide 153 / 181

52 Slide 154 / 181 Hyperbolas Return to Table of Contents Like the ellipse, the hyperbola is a set of points at a given distance from two foci. In the case of the hyperbola, the absolute value of the difference of the distances from a point to the foci is constant. Slide 155 / 181 a F1 b d c F2 a - b = c - d (Don't worry so much about this definition - it is just to put things in perspective.) Horizontal Hyperbola Asymptote Asymptote Slide 156 / 181 b Focus a Vertex Center b a Focus Vertex Equation: Vertices: move a units to the left and right of the center Foci: move c units to the left and right of the center, where Asymptotes: slope = ± b/a (The asymptotes are lines that pass through the vertices of the rectangle between the vertices with length 2a and width 2b. An asymptote is a line that the graph approaches but never touches.)

53 Vertical Hyperbola Asymptote Asymptote Slide 157 / 181 Equation: Vertices: move a units up and down from the center Foci: move c units up and down from the center, where Focus Vertex a b b Center a Vertex Focus Asymptotes: slope = ± a/b To graph a hyperbola in standard form: Find and graph the center Plot points a right and left of the center, and b up and down for horizontal, or b right and left, and a up and down for vertical Make a rectangle through the four points from previous step Draw asymptotes that contain the diagonals of the rectangle Sketch the graph of the hyperbola Slide 158 / 181 Center: (-1,2) Example: Graph Slide 159 / 181 click The center of the rectangle is ( -5, 4 ) From the center move left/right 2 From the center move up/down 3 The hyperbola opens What are the slopes of the asymptotes? up and down Answer How does this relate to a and b? Why?

54 Example: Graph Slide 160 / 181 The center of the rectangle is ( 6, 0 ) From the center move left/right 4 From the center move up/down 5 The hyperbola opens click left and right Slide 161 / 181 Slide 162 / 181

55 Slide 163 / 181 Slide 164 / 181 Slide 165 / 181

56 Slide 166 / 181 Slide 167 / What is the equation of a hyperbola that has vertices (±6,0) and foci (±10,0)? Slide 168 / 181 A B C D

57 Slide 169 / 181 Convert to standard form: Slide 170 / 181 Slide 171 / 181

58 Slide 172 / 181 Recognizing Conic Sections from General Form Return to Table of Contents General Form: ax 2 + bx + cy 2 + dy + e = 0 Slide 173 / 181 This form could represent any conic under the following conditions: In a parabola, either a=0 or b=0. In a circle, a=c and both a and c are positive. ax 2 + bx + dy +e =0 cy 2 + dy + bx + e=0 ax 2 + bx + cy 2 + dy + e = 0 In an ellipse, a and c are both positive, and a c. * ax 2 + bx + cy 2 + dy + e = 0 In a hyperbola, either a<0 ax 2 + bx - cy 2 + dy + e = 0 and c>0 or a>0 and c<0. cy 2 + dy - ax 2 + bx + e = 0 * A circle is a special type of ellipse in which a = c. Slide 174 / 181

59 Slide 175 / 181 Slide 176 / 181 Slide 177 / 181

60 Slide 178 / 181 Slide 179 / 181 Slide 180 / 181

61 Eccentricity of Conic Sections Slide 181 / 181 Ellipse 0<e<1 Parabola e = 1 Circle e=0 Hyperbola e > 1 This picture depicts the comparative eccentricity of conic sections. Eccentricity (e) is a measure of "unroundness". A circle is round, so has e=0. For an ellipse, as the ellipse becomes more elongated, e increases from 0 to 1, not-including 1. A parabola has e=1, and for a hyperbola e>1.