Rotation of Axes 1. Rotation of Axes. At the beginning of Chapter 5 we stated that all equations of the form


 Erin Kennedy
 2 years ago
 Views:
Transcription
1 Rotation of Axes 1 Rotation of Axes At the beginning of Chapter we stated that all equations of the form Ax + Bx + C + Dx + E + F =0 represented a conic section, which might possibl be degenerate. We saw in Section. that the graph of the quadratic equation Ax + C + Dx + E + F =0 is a parabola when A =0orC = 0, that is, when AC = 0. In Section.3 we found that the graph is an ellipse if AC > 0, and in Section.4 we saw that the graph is a hperbola when AC < 0. So we have classified the situation when B = 0. Degenerate situations can occur; for example, the quadratic equation x = 0 has no solutions, and the graph of x = 0 is not a hperbola, but the pair of lines with equations = ±x. But excepting this tpe of situation, we have categorized the graphs of all the quadratic equations with B =0. When B 0, a rotation can be performed to bring the conic into a form that will allow us to compare it with one that is in standard position. To set up the discussion, letusfirstsupposethatasetofxcoordinate axes has been rotated about the origin b an angle θ, where 0 <θ<π/, to form a new set of ˆxaxes, as shown in Figure 1(a). We would like to determine the coordinates for a point P in the plane relative to the two coordinate sstems. P(x, ) P(x, ˆ ) ˆ P(x, ) P(x, ˆ ) ˆ r B r f B u A x u A x (a) (b) Figure 1
2 CHAPTER Conic Sections, Polar Coordinates, and Parametric Equations First introduce a new pair of variables r and φ to represent, respectivel, the distance from P to the origin and the angle formed b the ˆxaxis and the line connecting the origin to P, as shown in Figure 1(b). From the right triangle OBP shown in Figure (a) we see that ˆx = r cos φ and = r sin φ, and from the right triangle OAP shown in Figure (b) we see that x = r cos(φ + θ) and = r sin(φ + θ). P(x, ˆ ) ˆ P(x, ) f r B r O O f 1 u x A (a) (b) Figure we have Using the fundamental trigonometric identities for the sum of the sine and the cosine x = r cos(φ + θ) =r cos φ cos θ r sin φ sin θ and = r sin(φ + θ) =r cos φ sin θ + r sin φ cos θ. Since ˆx = r cos φ and = r sin φ, we have the following result. The second pair of equations is derived b solving for ˆx and in the first pair.
3 Rotation of Axes 3 Coordinate Rotation Formulas If a rectangular xcoordinate sstem is rotated through an angle θ to form an ˆxcoordinate sstem, then a point P (x, ) will have coordinates P (ˆx, ) in the new sstem, where (x, ) and(ˆx, ) are related b and x =ˆx cos θ sin θ and =ˆx sin θ + cos θ. ˆx = x cos θ + sin θ and = x sin θ + cos θ. EXAMPLE 1 Show that the graph of the equation x = 1 is a hperbola b rotating the xaxes through an angle of π/4. Solution: Denoting a point in the rotated sstem b (ˆx, ), we have and x =ˆx cos π 4 sin π 4 = (ˆx ) =ˆx sin π 4 + cos π 4 = (ˆx +). Substituting these expressions into the original equation x = 1 produces the equation (ˆx ) (ˆx +) =1 or ˆx =. In the ˆxcoordinate sstem, then, we have a standard position hperbola whose asmptotes are = ±ˆx. These are the same lines as the x andaxes, as seen in Figure 3.
4 4 CHAPTER Conic Sections, Polar Coordinates, and Parametric Equations x 1 ˆ d x Figure 3 In Example 1 the appropriate angle of rotation was provided to eliminate the ˆxterm from the equation. Such an angle θ can alwas be found so that when the coordinate axes are rotated through this angle, the equation in the new coordinate sstem will not involve the product ˆx. To determine the angle, suppose we have the general seconddegree equation Ax + Bx + C + Dx + E + F =0, where B 0. Introduce the variables x =ˆx cos θ sin θ and =ˆx sin θ + cos θ, and substitute for x and in the original equation. This gives us the new equation in ˆx and : A(ˆx cos θ sin θ) + B(ˆx cos θ sin θ)(ˆx sin θ + cos θ)+c(ˆx sin θ + cos θ) + D(ˆx cos θ sin θ)+e(ˆx sin θ + cos θ)+f =0. Performing the multiplication and collecting the similar terms gives ˆx ( A(cos θ) + B(cos θ sin θ)+c(sin θ) ) +ˆx [ A cos θ sin θ + B ( (cos θ) (sin θ) ) +C sin θ cos θ ] + ( A(sin θ) B(sin θ cos θ)+c(cos θ) ) +ˆx(D cos θ + E sin θ)+( D sin θ + E cos θ)+f =0.
5 Rotation of Axes To eliminate the ˆxterm from this equation, choose θ so that the coefficient of this term is zero, that is, so that A cos θ sin θ + B ( (cos θ) (sin θ) ) +C sin θ cos θ =0. Simplifing this equation, we have and B ( (cos θ) (sin θ) ) =(A C)cosθ sin θ (cos θ) (sin θ) cosθ sin θ = A C B. Using the double angle formulas for sine and cosine gives cos θ sin θ = A C B, or cot θ = A C B. Rotating Quadratic Equations To eliminate the xterm in the general quadratic equation Ax + Bx + C + Dx + E + F =0, where B 0, rotate the coordinate axes through an angle θ that satisfies cot θ = A C B. EXAMPLE Sketch the graph of 4x 4x +7 4 = 0. Solution: To eliminate the xterm we first rotate the coordinate axes through the angle θ where cot θ = A C B = = 3 4 From the triangle in Figure 4, we see that cos θ = 3.
6 6 CHAPTER Conic Sections, Polar Coordinates, and Parametric Equations Using the half angle formulas we have 1+ 3 cos θ = =, and sin θ = 1 3 =. cot u! cos u E 4 u 3 Figure 4 The rotation of axes formulas then give x = ˆx and = ˆx +. Substituting these into the original equation gives ( ) ( 4 ˆx )( 4 ˆx ˆx + ) ( +7 ˆx + ) 4 = 0. After simplifing we have ˆx +8 =4 or ˆx =1. This is the graph of an ellipse in standard position with respect to the ˆx coordinate sstem. The graph of the equation 4x 4x +7 =4 is consequentl the graph of an ellipse that has been rotated through an angle θ = arcsin 6, asshowninfigure.
7 Rotation of Axes 7 4 u arcsin(œ/) 4 4 x 4x 4x Figure There is an easil applied formula that can be used to determine which conic will be produced once the rotation has been performed. Suppose that a rotation θ of the coordinate axes changes the equation Ax + Bx + C + Dx + E + F =0 into the equation Âˆx + ˆBˆx + Ĉ + ˆDˆx + Ê + ˆF =0. It is not hard to show, although it is algebraicall tedious, that the coefficients of these two equations satisf B 4AC = ˆB 4ÂĈ. When the particular angle chosen is such that ˆB =0,wehave B 4AC = 4ÂĈ. However, except for the degenerate cases we know that the new equation Âˆx + Ĉ + ˆDˆx + Ê + ˆF =0 will be: i) An ellipse if ÂĈ >0; ii) A hperbola if ÂĈ <0; iii) A parabola if ÂĈ =0. Appling this result to the original equation gives the following.
8 8 CHAPTER Conic Sections, Polar Coordinates, and Parametric Equations Classification of Conic Curves Except for degenerate cases, the equation Ax + Bx + C + Dx + E + F =0 will be: i) An ellipse if B 4AC = 4ÂĈ <0; ii) A hperbola if B 4AC = 4ÂĈ >0; iii) A parabola if B 4AC = ÂĈ =0. The beaut of this result is that we do not need to know the angle that produces the elimination of the ˆxterms. We onl need to know that such an angle exists, and we have alread established this fact. If we appl the result to the examples in this section we see that: Example 1. The equation x =1has B 4AC =1 4(0)(0) = 1 > 0 and is therefore a hperbola. Example. The equation 4x 4x +7 4 = 0 has B 4AC =( 4) 4(4)(7) = 6 < 0 and is therefore an ellipse. We strongl recommend that ou appl this simple test when rotation is required to graph a general quadratic equation. It provides a final check on a result produced from a considerable amount of algebraic and trigonometric computation. While it will not ensure that ou are correct, it can tell ou when ou are certainl wrong, and this is often more important.
9 Rotation of Axes 9 Rotation of Axes Exercises In Exercises 1 4, determine the ˆxcoordinates of the given point if the coordinate axes are rotated through the given angle θ. 1. (0, 1), θ=60. (1, 1), θ=4 3. ( 3, 1), θ=30 4. (, ), θ=90 In Exercises 1, (a) determine whether the conic section is an ellipse, hperbola, or parabola, and (b) perform a rotation, and if necessar a translation, and sketch the graph.. x x + = 6. x +4x + = x 6x +9 =0 8. x 3x =1 9. 4x 4x +7 =4 10. x 3x + = 11. 6x +4 3x + 9x = 0 1. x + 3x +3 +(4+ 3)x +(4 3 ) +1=0 13. Find an equation of the parabola with axis = x passing through the points (1, 0), (0, 1), and (1, 1) (a) in ˆxcoordinates, and (b) in xcoordinates.
Ax 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0 P(X, Y) X
Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general seconddegree
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More information17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style
Conic Sections 17.1 Introduction The conic sections (or conics)  the ellipse, the parabola and the hperbola  pla an important role both in mathematics and in the application of mathematics to engineering.
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationSection 10.5 Rotation of Axes; General Form of a Conic
Section 10.5 Rotation of Axes; General Form of a Conic 8 Objective 1: Identifying a Nonrotated Conic. The graph of the equation Ax + Bxy + Cy + Dx + Ey + F = 0 where A, B, and C cannot all be zero is
More information6.3 Polar Coordinates
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More information7.4 Applications of Eigenvalues and Eigenvectors
37 Chapter 7 Eigenvalues and Eigenvectors 7 Applications of Eigenvalues and Eigenvectors Model population growth using an age transition matri and an age distribution vector, and find a stable age distribution
More informationChange of Coordinates in Two Dimensions
CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More informationAlgebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2
1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationUnit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks
Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationPrime Coe cients Formula for Rotating Conics
Prime Coe cients Formula for Rotating Conics David Rose Polk State College David Rose (Institute) Prime Coe cients Formula for Rotating Conics 1 / 18 It is known that the graph of a conic section in the
More informationParametric Curves. EXAMPLE: Sketch and identify the curve defined by the parametric equations
Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationParametric Surfaces. Solution. There are several ways to parameterize this. Here are a few.
Parametric Surfaces 1. (a) Parameterie the elliptic paraboloid = 2 + 2 + 1. Sketch the grid curves defined b our parameteriation. Solution. There are several was to parameterie this. Here are a few. i.
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry
1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years
More informationUse Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures.
Learning Objectives Loci and Conics Lesson 2: The Circle Level: Precalculus Time required: 90 minutes Students are now acquainted with the idea of locus, and how Geometry Expressions can be used to explore
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationComplex Numbers. w = f(z) z. Examples
omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If
More informationChapter 10: Topics in Analytic Geometry
Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationREVIEW OF CONIC SECTIONS
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
More informationJune 2011 PURDUE UNIVERSITY Study Guide for the Credit Exams in Single Variable Calculus (MA 165, 166)
June PURDUE UNIVERSITY Stud Guide for the Credit Eams in Single Variable Calculus (MA 65, 66) Eam and Eam cover respectivel the material in Purdue s courses MA 65 (MA 6) and MA 66 (MA 6). These are two
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _i FIGURE Complex numbers as points in the Arg plane i _i +i i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationUnit 10: Quadratic Relations
Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationLecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...
CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper 1 Practice Paper A Time allowed 1 hour 0 minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions 1 0 (40 marks) Instructions
More informationSection 105 Parametric Equations
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More information29 Wyner PreCalculus Fall 2016
9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationObjective 1: Identify the characteristics of a quadratic function from its graph
Section 8.2 Quadratic Functions and Their Graphs Definition Quadratic Function A quadratic function is a seconddegree polynomial function of the form, where a, b, and c are real numbers and. Every quadratic
More informationApr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa
Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,
More informationCHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS
CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRECALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems
More information105 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
105 Parabolas Warm Up Lesson Presentation Lesson Quiz 2 Warm Up 1. Given, solve for p when c = Find each distance. 2. from (0, 2) to (12, 7) 13 3. from the line y = 6 to (12, 7) 13 Objectives Write the
More informationSECTION 91 Conic Sections; Parabola
66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships
More informationThe Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More information95 Complex Numbers and De Moivre's Theorem
Graph each number in the complex plane and find its absolute value 1 z = 4 + 4i 3 z = 4 6i For z = 4 6i, (a, b) = ( 4, 6) Graph the point ( 4, 6) in the complex plane For z = 4 + 4i, (a, b) = (4, 4) Graph
More informationAdvanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More information3.1 Quadratic Functions
Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More informationSolving Systems of Equations
Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that
More informationTrigonometry Hard Problems
Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.
More informationYears t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304
Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More informationPARABOLAS AND THEIR FEATURES
STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationMoore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS
Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS The following is a list of terms and properties which are necessary for success in Math Concepts and College Prep math. You will
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More information