You identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas.
|
|
- Zoe Stokes
- 6 years ago
- Views:
Transcription
1 You identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas.
2 conic section degenerate conic locus parabola focus directrix axis of symmetry vertex latus rectum
3
4 Determine Characteristics and Graph For (y 3) 2 = 8(x + 1), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.
5 For (x + 1) 2 = 4(y 2), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola. A. B. vertex: ( 1, 2); focus: ( 1, 3); directrix: y = 1; axis of symmetry: x = 1 vertex: ( 1, 2); focus: ( 1, 1); directrix: y = 3; axis of symmetry: x = 1 C. D. vertex: ( 1, 2); focus: (0, 2); directrix: x = 2; axis of symmetry: y = 2 vertex: ( 1, 2); focus: ( 2, 2); directrix: x = 0; axis of symmetry: y = 2
6 Characteristics of Parabolas ASTRONOMY The parabolic mirror for the California Institute of Technology s Hale telescope at Mount Palomar has a shape modeled by y 2 = 2668x, where x and y are measured in inches. What is the focal length of the mirror?
7 ASTRONOMY The cross section of the image of a constellation can be modeled by 12(y 6) = x 2, where x and y are measured in centimeters. What is the focal length of the cross section? A. 6 centimeters B. 12 centimeters C. 3 centimeters D.
8 Write in Standard Form Write x 2 8x y = 18 in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.
9 Write in Standard Form
10 Write y x = 55 6y in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola. A. (y + 3) 2 = 16(x 4); vertex: (4, 3); focus: (0, 3); directrix: x = 8; axis of symmetry: y = 3 B. (y + 3) 2 = 16(x 4); vertex: ( 3, 4); focus: ( 3, 0); directrix: y = 8; axis of symmetry: x = 3 C. (y 3) 2 = 16(x 4); vertex: (4, 3); focus: (0, 3); directrix: x = 8; axis of symmetry: y = 3 D. (y 3) 2 = 16(x 4); vertex: (4, 3); focus: (8, 3); directrix: x = 0; axis of symmetry: y = 3
11 Write Equations Given Characteristics A. Write an equation for and graph a parabola with focus (2, 1) and vertex ( 5, 1).
12 Write Equations Given Characteristics B. Write an equation for and graph a parabola with vertex (3, 2) and directrix y = 1.
13 Write Equations Given Characteristics C. Write an equation for and graph a parabola that has focus ( 1, 7), opens up, and contains (3, 7).
14 Write an equation for and graph a parabola with focus ( 2, 5) and directrix x = 4. A. 12(x + 2) = (y 5) 2 B. 12(x 1) = (y 5) 2 C. 12(x 1) = (y 5) 2 D. 2(x + 2) = (y 4.5) 2
15
16 Find a Tangent Line at a Point Write an equation for the line tangent to y = x 2 2 at (2, 2).
17 Write an equation for the line tangent to y 2 = 4x + 4 at (0, 2). A. y = 2x + 2 B. y = x + 2 C. y = x + 2 D. y = 2x + 2
18 You analyzed and graphed parabolas. (Lesson 7 1) Analyze and graph equations of ellipses and circles. Use equations to identify ellipses and circles.
19 ellipse foci major axis center minor axis vertices co-vertices eccentricity
20
21 Graph Ellipses A. Graph the ellipse
22 Graph Ellipses
23 Graph Ellipses B. Graph the ellipse 4x x + y 2 10y 3 = 0.
24 Graph Ellipses
25 Graph the ellipse 144x x + 25y 2 300y 396 = 0. A. C. B. D.
26 Write Equations Given Characteristics A. Write an equation for an ellipse with a major axis from (5, 2) to ( 1, 2) and a minor axis from (2, 0) to (2, 4).
27 Write Equations Given Characteristics B. Write an equation for an ellipse with vertices at (3, 4) and (3, 6) and foci at (3, 4) and (3, 2)
28 Write an equation for an ellipse with co-vertices ( 8, 6) and (4, 6) and major axis of length 18. A. B. C. D.
29
30 Determine the Eccentricity of an Ellipse Determine the eccentricity of the ellipse given by
31 Determine the eccentricity of the ellipse given by 36x x + 49y 2 98y = A B C D. 0.60
32 Use Eccentricity ASTRONOMY The eccentricity of the orbit of Uranus is Its orbit around the Sun has a major axis length of AU (astronomical units). What is the length of the minor axis of the orbit?
33 PARKS A lake in a park is elliptically-shaped. If the length of the lake is 2500 meters and the width is 1500 meters, find the eccentricity of the lake. A. 0.2 B. 0.4 C. 0.6 D. 0.8
34
35 Determine Types of Conics A. Write 9x 2 + 4y 2 + 8y 32 = 0 in standard form. Identify the related conic.
36 Determine Types of Conics B. Write x 2 + 4x 4y + 16 = 0 in standard form. Identify the related conic selection.
37 Determine Types of Conics C. Write x 2 + y 2 + 2x 6y 6 = 0 in standard form. Identify the related conic.
38 Write 16x 2 + y 2 + 4y 60 = 0 in standard form. Identify the related conic. A. B. 16x 2 + (y + 2) 2 = 64; circle C. D. 16x 2 + (y + 2) 2 = 64; ellipse
39 You analyzed and graphed ellipses and circles. (Lesson 7-2) Analyze and graph equations of hyperbolas. Use equations to identify types of conic sections.
40 hyperbola transverse axis conjugate axis
41
42 Graph Hyperbolas in Standard Form A. Graph the hyperbola given by
43 Graph Hyperbolas in Standard Form
44 Graph Hyperbolas in Standard Form B. Graph the hyperbola given by
45 Graph Hyperbolas in Standard Form
46 Graph the hyperbola given by A. B. C. D.
47 Graph a Hyperbola Graph the hyperbola given by 4x2 y2 + 24x + 4y = 28.
48 Graph a Hyperbola
49 Graph the hyperbola given by 3x2 y2 30x 4y = 119. A. B. C. D.
50 Write Equations Given Characteristics A. Write an equation for the hyperbola with foci (1, 5) and (1, 1) and transverse axis length of 4 units.
51 Write Equations Given Characteristics B. Write an equation for the hyperbola with vertices ( 3, 10) and ( 3, 2) and conjugate axis length of 6 units.
52 Write an equation for the hyperbola with foci at (13, 3) and ( 5, 3) and conjugate axis length of 12 units. A. B. C. D.
53 Find the Eccentricity of a Hyperbola
54 A B C D. 1.69
55
56 Identify Conic Sections A. Use the discriminant to identify the conic section in the equation 2x2 + y2 2x + 5xy + 12 = 0.
57 Identify Conic Sections B. Use the discriminant to identify the conic section in the equation 4x2 + 4y2 4x + 8 = 0.
58 Identify Conic Sections C. Use the discriminant to identify the conic section in the equation 2x2 + 2y2 6y + 4xy 10 = 0.
59 Use the discriminant to identify the conic section given by y + y2 = 14x 3x2. A. ellipse B. circle C. hyperbola D. parabola
60 Apply Hyperbolas A. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the equation for the hyperbola on which the ship is located.
61 Apply Hyperbolas
62 Apply Hyperbolas B. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the exact coordinates of the ship if it is 125 miles from the shore.
63 Apply Hyperbolas
64 NAVIGATION Suppose LORAN stations S and T are located 240 miles apart along a straight shore with S due north of T. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 60 miles farther from station T than it is from station S. Find the equation for the hyperbola on which the ship is located. A. B. C. D.
65 You identified and graphed conic sections. (Lessons 7 1 through 7 3) Find rotation of axes to write equations of rotated conic sections. Graph rotated conic sections.
66
67 Write an Equation in the x y -Plane Use θ = 90 to write x 2 + 3xy y 2 = 3 in the x y -plane. Then identify the conic.
68 Use θ = 60 to write 4x 2 + 6xy + 9y 2 = 12 in the x y -plane. Then identify the conic. A. B. C. D.
69
70 Write an Equation in Standard Form Using a suitable angle of rotation for the conic with equation x 2 4xy 2y 2 6 = 0, write the equation in standard form.
71 Write an Equation in Standard Form
72 A. B. C. D.
73
74 Write an Equation in the xy-plane
75 Write an Equation in the xy-plane
76 ASTRONOMY A sensor on a satellite is modeled by after a 60 rotation. Find the equation for the sensor in the xy-plane. A. B. C. D.
77 Graph a Conic Using Rotations
78 Graph a Conic Using Rotations
79 A. B. C. D.
80 Graph a Conic in Standard Form Use a graphing calculator to graph the conic section given by 8x2 + 5xy 4y2 = 2.
81 Use a graphing calculator to graph the conic section given by 3x2 6xy + 8y2 + 4x 2y = 0. A. B. C. D.
82 You modeled motion using quadratic functions. (Lesson 1 5) Graph parametric equations. Solve problems related to the motion of projectiles.
83 parametric equation parameter orientation parametric curve
84
85 Sketch Curves with Parametric Equations
86 Sketch Curves with Parametric Equations
87 Sketch the curve given by x = 2t 6 and y = t2 3 over 3 t 3. A. B. C. D.
88 Write in Rectangular Form Write y = 2t and x = t2 + 2 in rectangular form.
89 Write y = 4t 2 and x = 2t 4 in rectangular form. A. y = (x + 4)2 B. C. y = 2x + 8 D. y = x2
90 Rectangular Form with Domain Restrictions
91 A. y = (x 5)2, x < 5 B. y = (x + 5)2, x > 5 C. y = (x + 5)2, x < 0 D. y = (x 5)2, x > 0
92 Rectangular Form with θ as Parameter Write y = 5 sin θ and x = 3 cos θ in rectangular form. Then graph the equation.
93 Write y = 9 sin θ and x = 5 cos θ in rectangular form. Then graph the equation. A. B. C. D.
94 Write Parametric Equations from Graphs A. Use the parameter t = x 1 to write the parametric equations that can represent y = x Then graph the equation, indicating the speed and orientation.
95 Write Parametric Equations from Graphs B. Use the parameter t = 2x to write the parametric equations that can represent y = x Then graph the equation, indicating the speed and orientation.
96 Write Parametric Equations from Graphs
97 A. B. C. D.
98
99 Projectile Motion FOOTBALL Shane Lechler of the Oakland Raiders has the record career punting average with an average of yards. Suppose that he kicked the ball with an initial velocity of 26 yards per second at an angle of 72. How far will the ball travel horizontally if he punts it with an initial height of 1 yard?
100 Projectile Motion
101 SOFTBALL Kensey hits a softball with an initial velocity of 83 feet per second at an angle of 34. How far will the ball travel horizontally if Kensey s bat was 3.5 feet from the ground at the time of impact? A. 103 ft B. 205 ft C. 255 ft D. 657 ft
Algebra II B Review 3
Algebra II B Review 3 Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the equation. Describe the graph and its lines of symmetry. 1. a. c. b. graph
More information2.3: The Human Cannonball
2.3: The Human Cannonball Parabola Equations and Graphs As a human cannonball Rosa is shot from a special cannon. She is launched into the air by a spring. Rosa lands in a horizontal net 150 ft. from the
More information(3,4) focus. y=1 directrix
Math 153 10.5: Conic Sections Parabolas, Ellipses, Hyperbolas Parabolas: Definition: A parabola is the set of all points in a plane such that its distance from a fixed point F (called the focus) is equal
More informationRECTANGULAR EQUATIONS OF CONICS. A quick overview of the 4 conic sections in rectangular coordinates is presented below.
RECTANGULAR EQUATIONS OF CONICS A quick overview of the 4 conic sections in rectangular coordinates is presented below. 1. Circles Skipped covered in MAT 124 (Precalculus I). 2. s Definition A parabola
More informationHyperbolas Graphs, Equations, and Key Characteristics of Hyperbolas Forms of Hyperbolas p. 583
C H A P T ER Hyperbolas Flashlights concentrate beams of light by bouncing the rays from a light source off a reflector. The cross-section of a reflector can be described as hyperbola with the light source
More informationThis early Greek study was largely concerned with the geometric properties of conics.
4.3. Conics Objectives Recognize the four basic conics: circle, ellipse, parabola, and hyperbola. Recognize, graph, and write equations of parabolas (vertex at origin). Recognize, graph, and write equations
More informationThe Ellipse. PF 1 + PF 2 = constant. Minor Axis. Major Axis. Focus 1 Focus 2. Point 3.4.2
Minor Axis The Ellipse An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, the foci, is constant. Focus 1 Focus 2 Major Axis Point PF
More informationC.2 Equations and Graphs of Conic Sections
0 section C C. Equations and Graphs of Conic Sections In this section, we give an overview of the main properties of the curves called conic sections. Geometrically, these curves can be defined as intersections
More informationPre-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
More informationChapter 4: The Ellipse
Chapter 4: The Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza Chapter 4: The Ellipse Lecture 1: Introduction to Ellipse Lecture 13: Converting
More informationPre Calc. Conics.
1 Pre Calc Conics 2015 03 24 www.njctl.org 2 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
More informationPolar Conics TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System
Math Objectives Students will understand that the equations for conics can be expressed in polar form. Students will be able to describe the relationship between eccentricity and the type of conic section.
More information7.1 Solving Quadratic Equations by Graphing
Math 2201 Date: 7.1 Solving Quadratic Equations by Graphing In Mathematics 1201, students factored difference of squares, perfect square trinomials and polynomials of the form x 2 + bx + c and ax 2 + bx
More informationCONIC SECTIONS. Teacher's Guide
CONIC SECTIONS Teacher's Guide This guide is designed for use with Conic Sections, a series of three programs produced by TVOntario, the television service of the Ontario Educational Communications Authority.
More informationPre-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
More informationThe Geometric Definitions for Circles and Ellipses
18 Conic Sections Concepts: The Origin of Conic Sections Equations and Graphs of Circles and Ellipses The Geometric Definitions for Circles and Ellipses (Sections 10.1-10.3) A conic section or conic is
More informationRAKESH JALLA B.Tech. (ME), M.Tech. (CAD/CAM) Assistant Professor, Department Of Mechanical Engineering, CMR Institute of Technology. CONICS Curves Definition: It is defined as the locus of point P moving
More informationOn the. Geometry. of Orbits
On the Geometry of Orbits The Possible Orbits The Possible Orbits circle The Possible Orbits ellipse The Possible Orbits parabola The Possible Orbits hyperbola Speed and Distance 4000 mi 17,600 mph 1.4
More informationCONIC SECTIONS 1. Inscribe a parabola in the given rectangle, with its axis parallel to the side AB
Inscribe a parabola in the given rectangle, with its parallel to the side AB A D 1 1 2 2 3 3 B 3 2 1 1 2 3 C Inscribe a parabola in the rectangle below, with its vertex located midway along the side PQ.
More informationPre-Calc. Midpoint and Distance Formula. Slide 1 / 160 Slide 2 / 160. Slide 4 / 160. Slide 3 / 160. Slide 5 / 160. Slide 6 / 160.
Slide 1 / 160 Slide 2 / 160 Pre-alc onics 2015-03-24 www.njctl.org Slide 3 / 160 Slide 4 / 160 Table of ontents click on the topic to go to that section Review of Midpoint and istance Formulas Intro to
More informationUNIT I PLANE CURVES AND FREE HAND SKETCHING CONIC SECTIONS
UNIT I PLANE CURVES AND FREE HAND SKETCHING CONIC SECTIONS Definition: The sections obtained by the intersection of a right circular cone by a cutting plane in different positions are called conic sections
More informationEngineering Graphics, Class 5 Geometric Construction. Mohammad I. Kilani. Mechanical Engineering Department University of Jordan
Engineering Graphics, Class 5 Geometric Construction Mohammad I. Kilani Mechanical Engineering Department University of Jordan Conic Sections A cone is generated by a straight line moving in contact with
More informationConceptual Explanations: Analytic Geometry or Conic Sections
Conceptual Explanations: Analytic Geometry or Conic Sections So far, we have talked about how to graph two shapes: lines, and parabolas. This unit will discuss parabolas in more depth. It will also discuss
More informationYou may recall from previous work with solving quadratic functions, the discriminant is the value
8.0 Introduction to Conic Sections PreCalculus INTRODUCTION TO CONIC SECTIONS Lesson Targets for Intro: 1. Know and be able to eplain the definition of a conic section.. Identif the general form of a quadratic
More informationChapter 9. Conic Sections and Analytic Geometry. 9.1 The Ellipse. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 9 Conic Sections and Analytic Geometry 9.1 The Ellipse Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Graph ellipses centered at the origin. Write equations of ellipses in standard
More information10.1 Curves defined by parametric equations
Outline Section 1: Parametric Equations and Polar Coordinates 1.1 Curves defined by parametric equations 1.2 Calculus with Parametric Curves 1.3 Polar Coordinates 1.4 Areas and Lengths in Polar Coordinates
More informationFor Questions 1-15, NO CALCULATOR!
For Questions 1-15, NO CALCULATOR! 1. Identify the y-intercept: Identify the vertex: 2. The revenue, R(x), generated by an increase in price of x dollars for an item is represented by the equation Identify
More informationDESIGN & COMMUNICATION GRAPHICS Conic Sections 1
The projections of a right cone are shown below. The traces of a simply inclined plane VTH are also given. The plane is parallel to an element of the cone. The intersection of a plane and a right cone
More information2. Polar coordinates:
Section 9. Polar Coordinates Section 9. Polar Coordinates In polar coordinates ou do not have unique representation of points. The point r, can be represented b r, ± n or b r, ± n where n is an integer.
More informationFOUR CONIC SECTIONS. Sections of a Cone
Conic Sections FOUR CONIC SECTIONS 1 Sections of a Cone The circle, ellipse, parabola and hyperbola are known as conic sections Circle Ellipse Parabola Hyperbola All four curves are obtained by slicing
More informationM.V.S.R. ENGINEERING COLLEGE, NADERGUL HYDERABAD B.E. I/IV I - Internal Examinations (November 2014)
Sub: Engineering Graphics Branches: Civil (1&2), IT-2 Time: 1 Hr 15 Mins Max. Marks: 40 Note: Answer All questions from Part-A and any Two from Part B. Assume any missing data suitably. 1. Mention any
More informationUnit 6 Task 2: The Focus is the Foci: ELLIPSES
Unit 6 Task 2: The Focus is the Foci: ELLIPSES Name: Date: Period: Ellipses and their Foci The first type of quadratic relation we want to discuss is an ellipse. In terms of its conic definition, you can
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK Subject Code : Engineering Graphics& Design Course & Branch : B.Tech ALL Year & Sem : I B.Tech & I Sem
More informationGroup assignments affect the grade of all members in the group Individual assignments only affect the grade of the individual
CONIC PROJECT Algebra H DUE DATE: Friday March 15, 013. This project is in place of a test. Projects are to be turned in during your period, handed to the teacher. Projects may be turned in early (They
More informationUp and Down or Down and Up
Lesson.1 Assignment Name Date Up and Down or Down and Up Exploring Quadratic Functions 1. The citizens of Herrington County are wild about their dogs. They have an existing dog park for dogs to play, but
More informationENGINEERING CURVES (Week -2)
UNIT 1(a) CONIC SECTIONS ENGINEERING CURVES (Week -2) These are non-circular curves drawn by free hand. Sufficient number of points are first located and then a smooth curve passing through them are drawn
More informationJUNIOR CERTIFICATE 2009 MARKING SCHEME TECHNICAL GRAPHICS HIGHER LEVEL
. JUNIOR CERTIFICATE 2009 MARKING SCHEME TECHNICAL GRAPHICS HIGHER LEVEL Sections A and B Section A any ten questions from this section Q1 12 Four diagrams, 3 marks for each correct label. Q2 12 2 marks
More informationLearn new definitions of familiar shapes such as parabolas, hyperbolas, and circles.
CHAPTER 11 To begin this chapter, you will revisit the parabola by investigating the principle that makes a satellite dish work. You will discover a new way to define a parabola and will use that new definition
More informationUnit 8 Trigonometry. Math III Mrs. Valentine
Unit 8 Trigonometry Math III Mrs. Valentine 8A.1 Angles and Periodic Data * Identifying Cycles and Periods * A periodic function is a function that repeats a pattern of y- values (outputs) at regular intervals.
More informationCLEMSON ALGEBRA PROJECT UNIT 14: CONIC SECTIONS
CLEMSON ALGEBRA PROJECT UNIT 14: CONIC SECTIONS PROBLEM 1: LORAN - LONG-DISTANCE RADIO NAVIGATION LORAN, long-distance radio navigation for aircraft and ships, uses synchronized pulses transmitted by widely
More information11.5 Conic Sections. Objective A. To graph a parabola
Section 11.5 / Conic Sections 11.5/1 11.5 Conic Sections Objective A To graph a parabola VIDEO & DVD CD TUTOR WEB SSM Point of Interest Hpatia (c. 3 15) is considered the first prominent woman mathematician.
More informationLecture 3: Geometrical Optics 1. Spherical Waves. From Waves to Rays. Lenses. Chromatic Aberrations. Mirrors. Outline
Lecture 3: Geometrical Optics 1 Outline 1 Spherical Waves 2 From Waves to Rays 3 Lenses 4 Chromatic Aberrations 5 Mirrors Christoph U. Keller, Leiden Observatory, keller@strw.leidenuniv.nl Lecture 3: Geometrical
More informationMath 1330 Section 8.2 Ellipses
Math 1330 Section 8.2 Ellipses To form a conic section, we ll take this double cone and slice it with a plane. When we do this, we ll get one of several different results. 1 Part 1 - The Circle Definition:
More informationFolding Activity 1. Colored paper Tape or glue stick
Folding Activity 1 We ll do this first activity as a class, and I will model the steps with the document camera. Part 1 You ll need: Patty paper Ruler Sharpie Colored paper Tape or glue stick As you do
More informationUNIT 2: FACTOR QUADRATIC EXPRESSIONS. By the end of this unit, I will be able to:
UNIT 2: FACTOR QUADRATIC EXPRESSIONS UNIT 2 By the end of this unit, I will be able to: o Represent situations using quadratic expressions in one variable o Expand and simplify quadratic expressions in
More informationCHAPTER 10 Conics, Parametric Equations, and Polar Coordinates
CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... Section. Parametric Equations and Calculus............
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission. Leaving Certificate Marking Scheme. Design and Communication Graphics.
Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate 2016 Marking Scheme Design and Communication Graphics Higher Level Note to teachers and students on the use of published
More informationVOCABULARY WORDS. quadratic equation root(s) of an equation zero(s) of a function extraneous root quadratic formula discriminant
VOCABULARY WORDS quadratic equation root(s) of an equation zero(s) of a function extraneous root quadratic formula discriminant 1. Each water fountain jet creates a parabolic stream of water. You can represent
More informationMathematics Algebra II Unit 11: Conic Sections
Mathematics Algebra II Unit 11: Conic Sections 2013 201 1 What conic section is formed when a plane is passed through a cone parallel to its base? 5 raph the following: (x 3) 2 (y + 2) 2 = 36 2 Complete
More informationAlgebra 2 Conic Sections Study Guide
ALGEBRA 2 CONIC SECTIONS STUDY GUIDE PDF - Are you looking for algebra 2 conic sections study guide Books? Now, you will be happy that at this time algebra 2 conic sections study guide PDF is available
More informationAlgebra 2 Conic Sections Packet Answers
ALGEBRA 2 CONIC SECTIONS PACKET ANSWERS PDF - Are you looking for algebra 2 conic sections packet answers Books? Now, you will be happy that at this time algebra 2 conic sections packet answers PDF is
More information6.1 - Introduction to Periodic Functions
6.1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that
More informationLength of a Side (m)
Quadratics Day 1 The graph shows length and area data for rectangles with a fixed perimeter. Area (m ) 450 400 350 300 50 00 150 100 50 5 10 15 0 5 30 35 40 Length of a Side (m) 1. Describe the shape of
More informationOn Surfaces of Revolution whose Mean Curvature is Constant
On Surfaces of Revolution whose Mean Curvature is Constant Ch. Delaunay May 4, 2002 When one seeks a surface of given area enclosing a maximal volume, one finds that the equation this surface must satisfy
More informationDiscussion 8 Solution Thursday, February 10th. Consider the function f(x, y) := y 2 x 2.
Discussion 8 Solution Thursday, February 10th. 1. Consider the function f(x, y) := y 2 x 2. (a) This function is a mapping from R n to R m. Determine the values of n and m. The value of n is 2 corresponding
More informationWaves & Oscillations
Physics 42200 Waves & Oscillations Lecture 33 Geometric Optics Spring 2013 Semester Matthew Jones Aberrations We have continued to make approximations: Paraxial rays Spherical lenses Index of refraction
More informationPrecalculus Second Semester Final Review
Precalculus Second Semester Final Review This packet will prepare you for your second semester final exam. You will find a formula sheet on the back page; these are the same formulas you will receive for
More informationINSTITUTE OF AERONAUTICAL ENGINEERING
Course Name Course Code Class Branch INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad - 500 043 MECHANICAL ENGINEERING TUTORIAL QUESTION BANK : ENGINEERING DRAWING : A10301 : I - B. Tech : Common
More informationActivity 1 A D V A N C E D H O M E W O R K 1
Activity 1 A D V A N C E D H O M E W O R K 1 A D V A N C E D H O M E W O R K 2 Activity 2 Research Required: Recursive Functions Activity 3 A D V A N C E D H O M E W O R K 3 A D V A N C E D H O M E W O
More informationPre-Calculus Notes: Chapter 6 Graphs of Trigonometric Functions
Name: Pre-Calculus Notes: Chapter Graphs of Trigonometric Functions Section 1 Angles and Radian Measure Angles can be measured in both degrees and radians. Radian measure is based on the circumference
More informationCONIC SECTIONS. Our starting point is the following definition sketch-
CONIC SECTIONS One of the most important areas of analtic geometr involves the concept of conic sections. These represent d curves formed b looking at the intersection of a transparent cone b a plane tilted
More informationTechnical Graphics Higher Level
Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2005 Technical Graphics Higher Level Marking Scheme Sections A and B Section A Q1. 12 Four diagrams, 3 marks for
More informationConic and Quadric Surface Lab page 4. NORTHEASTERN UNIVERSITY Department of Mathematics Fall 03 Conic Sections and Quadratic Surface Lab
Conic and Quadric Surface Lab page 4 NORTHEASTERN UNIVERSITY Department of Mathematics Fall 03 Conic Sections and Quadratic Surface Lab Goals By the end of this lab you should: 1.) Be familar with the
More informationDetermine if the function is even, odd, or neither. 1) f(x) = 8x4 + 7x + 5 A) Even B) Odd C) Neither
Assignment 6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the function is even, odd, or neither. 1) f(x) = 8x4 + 7x + 5 1) A)
More informationTechnical Drawing Paper 1 - Higher Level (Plane and Solid Geometry)
Coimisiún na Scrúduithe Stáit State Examinations Commission 2008. M81 Leaving Certificate Examination 2008 Technical Drawing Paper 1 - Higher Level (Plane and Solid Geometry) (200 Marks) Friday 13 June
More informationB.E. 1 st year ENGINEERING GRAPHICS
B.E. 1 st year ENGINEERING GRAPHICS Introduction 1. What is an Engineering Graphics and its requirements? A standardized graphic representation of physical objects and their relationship is called Engineering
More informationTest Yourself. 11. The angle in degrees between u and w. 12. A vector parallel to v, but of length 2.
Test Yourself These are problems you might see in a vector calculus course. They are general questions and are meant for practice. The key follows, but only with the answers. an you fill in the blanks
More informationPractice problems from old exams for math 233
Practice problems from old exams for math 233 William H. Meeks III January 14, 2010 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These
More information1.6. QUADRIC SURFACES 53. Figure 1.18: Parabola y = 2x 2. Figure 1.19: Parabola x = 2y 2
1.6. QUADRIC SURFACES 53 Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces Figure 1.19: Parabola x = 2y 2 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more
More information3D VISUALIZATION OF CONIC SECTIONS IN XNA GAME PROGRAMMING FRAMEWORK. A Thesis. Presented to the. Faculty of. San Diego State University
3D VISUALIZATION OF CONIC SECTIONS IN XNA GAME PROGRAMMING FRAMEWORK A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of
More informationPrecalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor
Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor As we studied last section points may be described in polar form or rectangular form. Likewise an equation may be written using either
More informationLecture 2: Geometrical Optics. Geometrical Approximation. Lenses. Mirrors. Optical Systems. Images and Pupils. Aberrations.
Lecture 2: Geometrical Optics Outline 1 Geometrical Approximation 2 Lenses 3 Mirrors 4 Optical Systems 5 Images and Pupils 6 Aberrations Christoph U. Keller, Leiden Observatory, keller@strw.leidenuniv.nl
More informationEngineering Graphics. Practical Book. Government Engineering College Bhuj (Kutch - Gujarat) Department of Mechanical Engineering
Engineering Graphics Practical Book ASHISH J. MODI Department of Mechanical Engineering Government Engineering College Bhuj 370 001 (Kutch - Gujarat) SYLLABUS (AS PER GUJARAT TECHNOLOGICAL UNIVERSITY,
More information12-6 Circular and Periodic Functions
26. CCSS SENSE-MAKING In the engine at the right, the distance d from the piston to the center of the circle, called the crankshaft, is a function of the speed of the piston rod. Point R on the piston
More informationJUNIOR CERTIFICATE 2008 MARKING SCHEME TECHNICAL GRAPHICS HIGHER LEVEL
JUNIOR CERTIFICATE 2008 MARKING SCHEME TECHNICAL GRAPHICS HIGHER LEVEL Sections A and B Section A - any ten questions from this Section Q1 12 Four diagrams, 3 marks for each correct label. Q2 12 3 height
More informationNow we are going to introduce a new horizontal axis that we will call y, so that we have a 3-dimensional coordinate system (x, y, z).
Example 1. A circular cone At the right is the graph of the function z = g(x) = 16 x (0 x ) Put a scale on the axes. Calculate g(2) and illustrate this on the diagram: g(2) = 8 Now we are going to introduce
More informationLecture 2: Geometrical Optics. Geometrical Approximation. Lenses. Mirrors. Optical Systems. Images and Pupils. Aberrations.
Lecture 2: Geometrical Optics Outline 1 Geometrical Approximation 2 Lenses 3 Mirrors 4 Optical Systems 5 Images and Pupils 6 Aberrations Christoph U. Keller, Leiden Observatory, keller@strw.leidenuniv.nl
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle. We noticed how the x and y
More informationSM3 Lesson 2-3 (Intercept Form Quadratic Equation)
SM3 Lesson 2-3 (Intercept Form Quadratic Equation) Factor the following quadratic expressions: x 2 + 11x + 30 x 2 10x 24 x 2 8x + 15 Standard Form Quadratic Equation (x + 5)(x + 6) (x 12)(x + 2) (x 5)(x
More informationtechnical drawing
technical drawing school of art, design and architecture nust spring 2011 http://www.youtube.com/watch?v=q6mk9hpxwvo http://www.youtube.com/watch?v=bnu2gb7w4qs Objective abstraction - axonometric view
More informationCHAPTER 10 Conics, Parametric Equations, and Polar Coordinates
CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... 8 Section. Parametric Equations and
More information2. (8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given
Trigonometry Joysheet 1 MAT 145, Spring 2017 D. Ivanšić Name: Covers: 6.1, 6.2 Show all your work! 1. 8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given that sin
More informationPART I: Emmett s teacher asked him to analyze the table of values of a quadratic function to find key features. The table of values is shown below:
Math (L-3a) Learning Targets: I can find the vertex from intercept solutions calculated by quadratic formula. PART I: Emmett s teacher asked him to analyze the table of values of a quadratic function to
More information5-5 Multiple-Angle and Product-to-Sum Identities
Find the values of sin 2, cos 2, and tan 2 for the given value and interval. 1. cos =, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 and a distance
More informationStereometry Day #1. Stereometry Day #2
8 th Grade Stereometry and Loci Lesson Plans February 2008 Comments: Stereometry is the study of 3-D solids, which includes the Platonic and Archimedean solids. Loci is the study of 2-D curves, which includes
More informationthe input values of a function. These are the angle values for trig functions
SESSION 8: TRIGONOMETRIC FUNCTIONS KEY CONCEPTS: Graphs of Trigonometric Functions y = sin θ y = cos θ y = tan θ Properties of Graphs Shape Intercepts Domain and Range Minimum and maximum values Period
More informationExam 2 Review Sheet. r(t) = x(t), y(t), z(t)
Exam 2 Review Sheet Joseph Breen Particle Motion Recall that a parametric curve given by: r(t) = x(t), y(t), z(t) can be interpreted as the position of a particle. Then the derivative represents the particle
More information5.3-The Graphs of the Sine and Cosine Functions
5.3-The Graphs of the Sine and Cosine Functions Objectives: 1. Graph the sine and cosine functions. 2. Determine the amplitude, period and phase shift of the sine and cosine functions. 3. Find equations
More informationUNIT I PLANE CURVES AND FREE HAND SKETCHING 15
Importance of graphics in engineering applications Use of drafting instruments BIS conventions and specifications Size, layout and folding of drawing sheets Lettering and dimensioning. UNIT I PLANE CURVES
More informationMath 165 Section 3.1 Linear Functions
Math 165 Section 3.1 Linear Functions - complete this page Read the book or the power point presentations for this section. Complete all questions on this page Also complete all questions on page 6 1)
More informationAppendix: Sketching Planes and Conics in the XYZ Coordinate System
Appendi: D Sketches Contemporar Calculus Appendi: Sketching Planes and Conics in the XYZ Coordinate Sstem Some mathematicians draw horrible sketches of dimensional objects and the still lead productive,
More informationMath 5BI: Problem Set 1 Linearizing functions of several variables
Math 5BI: Problem Set Linearizing functions of several variables March 9, A. Dot and cross products There are two special operations for vectors in R that are extremely useful, the dot and cross products.
More informationCross Sections of Three-Dimensional Figures
Domain 4 Lesson 22 Cross Sections of Three-Dimensional Figures Common Core Standard: 7.G.3 Getting the Idea A three-dimensional figure (also called a solid figure) has length, width, and height. It is
More informationMath 148 Exam III Practice Problems
Math 48 Exam III Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationENGINEERING GRAPHICS (Engineering Drawing is the language of Engineers)
ENGINEERING GRAPHICS (Engineering Drawing is the language of Engineers) UNIT 1 Conic Section (Ellipse, Parabola & Hyperbola) - Cycloids, epicycloids, hypocycloids & Involutes around circle and square scales
More informationa) 2, 4, 8, 14, 22, b) 1, 5, 6, 10, 11, c) 3, 9, 21, 39, 63, d) 3, 0, 6, 15, 27, e) 3, 8, 13, 18, 23,
Pre-alculus Midterm Exam Review Name:. Which of the following is an arithmetic sequence?,, 8,,, b),, 6, 0,, c), 9,, 9, 6, d), 0, 6,, 7, e), 8,, 8,,. What is a rule for the nth term of the arithmetic sequence
More information5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs
Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2
More informationChapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Section 8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry
More informationMATH Exam 2 Solutions November 16, 2015
MATH 1.54 Exam Solutions November 16, 15 1. Suppose f(x, y) is a differentiable function such that it and its derivatives take on the following values: (x, y) f(x, y) f x (x, y) f y (x, y) f xx (x, y)
More informationc. Using the conditions described in Part b, how far does Mario travel each minute?
Trig. Modeling Short Answer 1. Mario's bicycle has 42 teeth in the crankset attached to the pedals. It has three sprockets of differing sizes connected to the rear wheel. The three sprockets at the rear
More information