9/22/2015Math KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

CH 9KM & PP AIM22 Sections of a Cone

CH 9KM & PP AIM23 Sections of a Cone... continued

CH 9KM & PP AIM24 Degenerate Conic Sections

9/22/2015Math KM5 9.1

9/22/2015Math KM6 The Parabola 9.1

9/22/2015Math KM7 A Parabolic Reflector For a Microphone Can You Hear a Pin Drop? 9.1

9/22/2015Math KM8 A Parabolic Archway Architectural Parabola 9.1

9/22/2015Math KM9 A Parabolic Headlight Shine Your Light Forward 9.1

9/22/2015Math KM10 Parabolic Shadows 9.1

9/22/2015Math KM11 y = ax 2 + bx + c a > 0 a < 0 x = ay 2 + by + c a > 0 a < The Basic Ideas 9.1

9/22/2015Math KM12 {-4,2} {5/3} { } {3} Vertex: (-2, -3) Opens upwards (narrow) Axis of symmetry: x = -2 y -intercept: (0,5) 9.1 Ex 1: y = 2x 2 + 8x

9/22/2015Math KM13 {-4,2} {5/3} { } {3} Vertex: (-2, -3) Opens upwards (narrow) Axis of symmetry: x = -2 y -intercept: (0,5) 9.1 Ex 1: y = 2x 2 + 8x + 5 alternate method 9.1

9/22/2015Math KM14 {-4,2} {5/3} { } {3} Vertex: (1, 4) Opens downward (narrow) Axis of symmetry: x = 1 y -intercept: (0,-2) 9.1 Ex 2: y = -6x x

9/22/2015Math KM15 Vertex: (1, 4) Opens downward (narrow) Axis of symmetry: x = 1 y -intercept: (0-2) 9.1 Ex 2: y = -6x x – 2 alternate method 9.1

9/22/2015Math KM16 Vertex: (-3, 2) Opens to the right (narrow) Axis of symmetry: y = 2 x – intercept: (5, 0) 9.1 Ex 3: x = 2y 2 – 8y

9/22/2015Math KM17 Vertex: (-3, 2) Opens to the right (narrow) Axis of symmetry: y = 2 x – intercept: (5, 0) 9.1 Ex 3: x = 2y 2 – 8y + 5 alternate method 9.1

9/22/2015Math KM18 Vertex: (-1, -1) Opens to the left (narrow) Axis of symmetry: y = -1 x – intercept: (-3, 0) 9.1 Ex 4: x = -2y 2 – 4y

9/22/2015Math KM19 Vertex: (-1, -1) Opens to the left (narrow) Axis of symmetry: y = -1 x – intercept: (-3, 0) 9.1 Ex 4: x = -2y 2 – 4y – 3 alternate method 9.1

9/22/2015Math KM20 c a b The Distance Formula 9.1

9/22/2015Math KM21 Determine the distance from P 1 to P 2. P 1 (-2, 3) P 2 (2, 0) P 1 (5, -2) P 2 (-3, -1) 9.1 Distance Formula Examples 9.1

9/22/2015Math KM MIDPOINT 9.1

9/22/2015Math KM23 AVERAGE ! 9.1 Average the Coordinates! 9.1

9/22/2015Math KM24 Determine the midpoint of P 1 P 2. P 1 (-2, 3) P 2 (2, 0) P 1 (5, -2) P 2 (-3, -1) 9.1 Midpoint Examples 9.1

9/22/2015 9:03 PM krm With a COMPASS How do I make a circle ? 9.1 Circles 9.1

9/22/2015 9:03 PM krm The set of all points in a plane that are at a fixed distance, r, called the radius from a fixed point, (h, k), called the center. 9.1 Circle: Center (h,k) Radius r 9.1

9/22/2015Math KM x 2 + y 2 = 1 9.1

9/22/2015Math KM (x + 2) 2 + (y – 4) 2 =

9/22/2015Math KM x 2 + (y + 4) 2 =

9/22/2015 9:03 PM krm Write the equation of the circle with radius 7 and center (-5, 8). 9.1

9/22/2015 9:03 PM krm Look for ax 2 + ay 2 How do I know it’s a circle ? The Equation of a Circle 9.1

9/22/2015 9:03 PM krm Write the equation of the circle in standard form and sketch the graph: x 2 + y 2 - 6x + 10y + 25 = 0 Circle: Standard Form 9.1

9/22/2015Math KM The Ellipse 9.2

9/22/2015Math KM34 x-intercepts (+ a, 0) y-intercepts (0, + b) 9.2 Ellipse (it fits in a box!) 9.2

9/22/2015Math KM Example: Horizontal Major Axis 9.2

9/22/2015Math KM Example: Vertical Major Axis 9.2

9/22/2015Math KM Example: center not at the origin 9.2

9/22/2015Math KM Example: Put in Standard Form First 9.2

9/22/2015Math KM Example continued: Put in Standard Form First 9.2

9/22/2015Math KM The Hyperbola it fits outside the box 9.3

9/22/2015Math KM The Hyperbola STANDARD FORM 9.3

9/22/2015Math KM42 1.Fundamental Rectangle 2.Asymptotes 3.Vertices (if x 2 – y 2 …) 4.Sketch 9.3 Hyperbola: x 2 is first 9.3

9/22/2015Math KM Example x 2 is first 9.3

9/22/2015Math KM44 1.Fundamental Rectangle 2.Asymptotes 3.Vertices (if y 2 – x 2 …) 4.Sketch 9.3 Hyperbola: y 2 is first 9.3

9/22/2015Math KM Example y 2 is first 9.3

9/22/2015Math KM The Hyperbola NONSTANDARD FORM 9.3

9/22/2015Math KM The Hyperbola NONSTANDARD FORM Example 1 xy N

9/22/2015Math KM The Hyperbola NONSTANDARD FORM Example 2 xy N

9/22/2015Math KM49 “Conic sections are among the oldest curves, and is an oldest math subject studied systematically and thoroughly. The conics seems to have been discovered by Menaechmus (a Greek, c BC), tutor to Alexander the Great. They were conceived in an attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to an element of the cone. (An element of a cone is any line that makes up the cone) Depending the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively. Appollonius (estimated c BC) (known as The Great Geometer) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Quote from Morris Kline: "As an achievement it [Appollonius' Conic Sections] is so monumental that it practically closed the subject to later thinkers, at least from the purely geometrical standpoint." Book VIII of Conic Sections is lost to us. Appollonius' Conic Sections and Euclid's Elements may represent the quintessence of Greek mathematics. Appolloniuswas the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola. A brief explanation of the naming can be found in Howard Eves, An Introduction to the History of Math. 6th ed. page 172. In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, espcially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.” From the website: Conics years old?

9/22/2015Math KM50 9.4

9/22/2015Math KM51  Think of the Possibilities! 9.4

9/22/2015Math KM52 Where will they meet? 9.4

9/22/2015Math KM53 Where will they meet - exactly? 9.4

9/22/2015Math KM54 Where will they meet - exactly? 9.4

9/22/2015Math KM55 Where will they meet - exactly? 9.4

9/22/2015Math KM56 How about a really tough one? 9.4

9/22/2015Math KM57 How about a really tough one? Continued

9/22/2015Math KM58 How about a really tough one? Continued

9/22/2015Math KM59 That’s All for Now!