The Geometry. Mathematics 15: Lecture 20. Dan Sloughter. Furman University. November 6, 2006
|
|
- Eileen Howard
- 6 years ago
- Views:
Transcription
1 The Geometry Mathematics 15: Lecture 20 Dan Sloughter Furman University November 6, 2006 Dan Sloughter (Furman University) The Geometry November 6, / 18
2 René Descartes Dan Sloughter (Furman University) The Geometry November 6, / 18
3 René Descartes Central figure in the move from medieval to modern mathematics, as well as from medieval to modern philosophy Dan Sloughter (Furman University) The Geometry November 6, / 18
4 Geometry and algebra Al-Khowarizmi (780? - 850?) Dan Sloughter (Furman University) The Geometry November 6, / 18
5 Geometry and algebra Al-Khowarizmi (780? - 850?) Wrote Al-jabr wa l muqabalah Dan Sloughter (Furman University) The Geometry November 6, / 18
6 Geometry and algebra Al-Khowarizmi (780? - 850?) Wrote Al-jabr wa l muqabalah Our word algorithm comes from his name Dan Sloughter (Furman University) The Geometry November 6, / 18
7 Geometry and algebra Al-Khowarizmi (780? - 850?) Wrote Al-jabr wa l muqabalah Our word algorithm comes from his name As an example of techniques found in Al-Khowarizmi s work, consider the problem of solving the quadratic equation x x = 39. Dan Sloughter (Furman University) The Geometry November 6, / 18
8 Geometry and algebra Al-Khowarizmi (780? - 850?) Wrote Al-jabr wa l muqabalah Our word algorithm comes from his name As an example of techniques found in Al-Khowarizmi s work, consider the problem of solving the quadratic equation x x = 39. Al-Khowarizmi constructed the following diagram: 5 5x 25 x x 2 x 5x 5 Dan Sloughter (Furman University) The Geometry November 6, / 18
9 Geometry and algebra Al-Khowarizmi (780? - 850?) Wrote Al-jabr wa l muqabalah Our word algorithm comes from his name As an example of techniques found in Al-Khowarizmi s work, consider the problem of solving the quadratic equation x x = 39. Al-Khowarizmi constructed the following diagram: 5 5x 25 x x 2 x 5x 5 Note: he has completed the square by adding the square of area 25 to the figure which has area x x, which, in the given equation, is supposed to have area 39. Dan Sloughter (Furman University) The Geometry November 6, / 18
10 Geometry and algebra (cont d) That is, we have (x + 5) 2 = x 2 + 5x + 5x + 25 = x x + 25 = = 64. Dan Sloughter (Furman University) The Geometry November 6, / 18
11 Geometry and algebra (cont d) That is, we have (x + 5) 2 = x 2 + 5x + 5x + 25 = x x + 25 = = 64. Hence x + 5 = 8, and so x = 3. Dan Sloughter (Furman University) The Geometry November 6, / 18
12 Geometry and algebra (cont d) That is, we have (x + 5) 2 = x 2 + 5x + 5x + 25 = x x + 25 = = 64. Hence x + 5 = 8, and so x = 3. Note: Al-Khowarizmi did not consider negative solutions. Dan Sloughter (Furman University) The Geometry November 6, / 18
13 Geometric algebra of Descartes Multiplying numbers as lines (see page 240): Dan Sloughter (Furman University) The Geometry November 6, / 18
14 Geometric algebra of Descartes Multiplying numbers as lines (see page 240): BE BD = BC BA Dan Sloughter (Furman University) The Geometry November 6, / 18
15 Geometric algebra of Descartes Multiplying numbers as lines (see page 240): BE BD = BC BA Since BA = 1, we have either BE = BD BC or, equivalently, BE BD = BC. Dan Sloughter (Furman University) The Geometry November 6, / 18
16 Geometric algebra of Descartes (cont d) Finding a square root (see page 240): Dan Sloughter (Furman University) The Geometry November 6, / 18
17 Geometric algebra of Descartes (cont d) Finding a square root (see page 240): (GI ) 2 = (KI ) 2 (GK) 2 = (KI GK)(KI + GK) = KH + GK = GH Dan Sloughter (Furman University) The Geometry November 6, / 18
18 Geometric algebra of Descartes (cont d) Finding a square root (see page 240): (GI ) 2 = (KI ) 2 (GK) 2 = (KI GK)(KI + GK) = KH + GK = GH So GI is the square root of GH. Dan Sloughter (Furman University) The Geometry November 6, / 18
19 Geometric algebra of Descartes(cont d) Solving the quadratic equation z 2 = az + b 2 (see page 248): Dan Sloughter (Furman University) The Geometry November 6, / 18
20 Geometric algebra of Descartes(cont d) Solving the quadratic equation z 2 = az + b 2 (see page 248): OM LM = LM PM, so OM PM = (LM)2. Dan Sloughter (Furman University) The Geometry November 6, / 18
21 Geometric algebra of Descartes(cont d) Solving the quadratic equation z 2 = az + b 2 (see page 248): OM LM = LM PM, so OM PM = (LM)2. That is, z(z a) = b 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
22 Geometric algebra of Descartes(cont d) Solving the quadratic equation z 2 = az + b 2 (see page 248): OM LM = LM PM, so OM PM = (LM)2. That is, z(z a) = b 2. Equivalently, z 2 = az + b 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
23 Geometric algebra of Descartes(cont d) Solving the quadratic equation z 2 = az + b 2 (see page 248): OM LM = LM PM, so OM PM = (LM)2. That is, z(z a) = b 2. Equivalently, z 2 = az + b 2. Then z = ON + NM = 1 2 a a2 + b 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
24 The Cartesian plane Cartesian plane: identify every point in the plane, uniquely, with an ordered pair of numbers which give the horizontal and vertical distances of the point from two perpendicular axes y (x, y) x Dan Sloughter (Furman University) The Geometry November 6, / 18
25 The Cartesian plane Cartesian plane: identify every point in the plane, uniquely, with an ordered pair of numbers which give the horizontal and vertical distances of the point from two perpendicular axes y (x, y) x Note: the first coordinate of a point to the left of the vertical axis is negative, as is the second coordinate of a point which lies below the horizontal axis. Dan Sloughter (Furman University) The Geometry November 6, / 18
26 The Cartesian plane (cont d) Important idea: certain geometric relationships may be expressed in the language of algebra. Dan Sloughter (Furman University) The Geometry November 6, / 18
27 The Cartesian plane (cont d) Important idea: certain geometric relationships may be expressed in the language of algebra. Example: If P = (x 1, y 1 ) and Q = (x 2, y 2 ) are two points in the plane, then Dan Sloughter (Furman University) The Geometry November 6, / 18
28 The Cartesian plane (cont d) Important idea: certain geometric relationships may be expressed in the language of algebra. Example: If P = (x 1, y 1 ) and Q = (x 2, y 2 ) are two points in the plane, then x 2 x 1 represents the horizontal displacement of Q from P, Dan Sloughter (Furman University) The Geometry November 6, / 18
29 The Cartesian plane (cont d) Important idea: certain geometric relationships may be expressed in the language of algebra. Example: If P = (x 1, y 1 ) and Q = (x 2, y 2 ) are two points in the plane, then x 2 x 1 represents the horizontal displacement of Q from P, y 2 y 1 represents the vertical displacement of Q from P, and, Dan Sloughter (Furman University) The Geometry November 6, / 18
30 The Cartesian plane (cont d) Important idea: certain geometric relationships may be expressed in the language of algebra. Example: If P = (x 1, y 1 ) and Q = (x 2, y 2 ) are two points in the plane, then x 2 x 1 represents the horizontal displacement of Q from P, y 2 y 1 represents the vertical displacement of Q from P, and, by the Pythagorean theorem, PQ = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
31 The Cartesian plane (cont d) Important idea: certain geometric relationships may be expressed in the language of algebra. Example: If P = (x 1, y 1 ) and Q = (x 2, y 2 ) are two points in the plane, then x 2 x 1 represents the horizontal displacement of Q from P, y 2 y 1 represents the vertical displacement of Q from P, and, by the Pythagorean theorem, PQ = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. Example: The distance between the points ( 1, 2) and (3, 4)is (3 ( 1)) 2 + (4 ( 2)) 2 = = 52 = 4 13 = Dan Sloughter (Furman University) The Geometry November 6, / 18
32 Circles Note: curves may now be expressed by compact algebraic expressions. Dan Sloughter (Furman University) The Geometry November 6, / 18
33 Circles Note: curves may now be expressed by compact algebraic expressions. Example: A circle of radius r centered at the point (a, b) consists of the set of all points (x, y) which satisfy (x a) 2 + (y b) 2 = r 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
34 Example (x 2) 2 + (y 3) 2 = 4 is the equation of a circle of radius 2 centered at (2, 3) Dan Sloughter (Furman University) The Geometry November 6, / 18
35 Lines Example Dan Sloughter (Furman University) The Geometry November 6, / 18
36 Lines Example Consider a line through the origin with slope m: 1 m x y Dan Sloughter (Furman University) The Geometry November 6, / 18
37 Lines Example Consider a line through the origin with slope m: 1 m x y That is, for every change of one unit in the x direction, the line changes by m units in the y direction. Dan Sloughter (Furman University) The Geometry November 6, / 18
38 Lines Example Consider a line through the origin with slope m: 1 m x y That is, for every change of one unit in the x direction, the line changes by m units in the y direction. Then if (x, y) is a point on the line, we must have, by similar triangles, m 1 = y x. Dan Sloughter (Furman University) The Geometry November 6, / 18
39 Lines Example Consider a line through the origin with slope m: 1 m x y That is, for every change of one unit in the x direction, the line changes by m units in the y direction. Then if (x, y) is a point on the line, we must have, by similar triangles, Hence y = mx. m 1 = y x. Dan Sloughter (Furman University) The Geometry November 6, / 18
40 Lines (cont d) Note: in general, y = mx + b is the equation of a line with slope m which passes through (0, b). Dan Sloughter (Furman University) The Geometry November 6, / 18
41 Lines (cont d) Note: in general, y = mx + b is the equation of a line with slope m which passes through (0, b). Example: y = 2x + 3 is the equation of a line which passes through the point (0, 3) with slope Dan Sloughter (Furman University) The Geometry November 6, / 18
42 Parabolas Geometric definition: given a line l, called the directrix, and a point F, called the focus, a parabola is the set of all points P such that the distance from P to F is the same as the distance from P to l. Dan Sloughter (Furman University) The Geometry November 6, / 18
43 Parabolas Geometric definition: given a line l, called the directrix, and a point F, called the focus, a parabola is the set of all points P such that the distance from P to F is the same as the distance from P to l. Let C be the parabola with focus at (0, d) and directrix the line y = d. Dan Sloughter (Furman University) The Geometry November 6, / 18
44 Parabolas Geometric definition: given a line l, called the directrix, and a point F, called the focus, a parabola is the set of all points P such that the distance from P to F is the same as the distance from P to l. Let C be the parabola with focus at (0, d) and directrix the line y = d. Then a point P = (x, y) is on C if and only if the distance from (x, y) to (0, d) is the same as the distance from (x, y) to (x, d). Dan Sloughter (Furman University) The Geometry November 6, / 18
45 Parabolas Geometric definition: given a line l, called the directrix, and a point F, called the focus, a parabola is the set of all points P such that the distance from P to F is the same as the distance from P to l. Let C be the parabola with focus at (0, d) and directrix the line y = d. Then a point P = (x, y) is on C if and only if the distance from (x, y) to (0, d) is the same as the distance from (x, y) to (x, d). That is, (x, y) is on C if and only if x 2 + (y d) 2 = y + d. Dan Sloughter (Furman University) The Geometry November 6, / 18
46 Parabolas Geometric definition: given a line l, called the directrix, and a point F, called the focus, a parabola is the set of all points P such that the distance from P to F is the same as the distance from P to l. Let C be the parabola with focus at (0, d) and directrix the line y = d. Then a point P = (x, y) is on C if and only if the distance from (x, y) to (0, d) is the same as the distance from (x, y) to (x, d). That is, (x, y) is on C if and only if x 2 + (y d) 2 = y + d. Hence x 2 + (y d) 2 = (y + d) 2, and so x 2 + y 2 2dy + d 2 = y 2 + 2dy + d 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
47 Parabolas Geometric definition: given a line l, called the directrix, and a point F, called the focus, a parabola is the set of all points P such that the distance from P to F is the same as the distance from P to l. Let C be the parabola with focus at (0, d) and directrix the line y = d. Then a point P = (x, y) is on C if and only if the distance from (x, y) to (0, d) is the same as the distance from (x, y) to (x, d). That is, (x, y) is on C if and only if x 2 + (y d) 2 = y + d. Hence x 2 + (y d) 2 = (y + d) 2, and so x 2 + y 2 2dy + d 2 = y 2 + 2dy + d 2. It follows that (x, y) is on C if and only if y = 1 4d x 2. Dan Sloughter (Furman University) The Geometry November 6, / 18
48 Example The curve with equation y = x 2 is a parabola with focus at ( 0, 1 ) 4 and directrix y = Dan Sloughter (Furman University) The Geometry November 6, / 18
49 Geometry to algebra With the introduction of rectangular coordinate systems by Descartes and Fermat ( ), it is possible to describe curves algebraically. Dan Sloughter (Furman University) The Geometry November 6, / 18
50 Geometry to algebra With the introduction of rectangular coordinate systems by Descartes and Fermat ( ), it is possible to describe curves algebraically. At first, only curves described geometrically are accepted as legitimate, but, with time, it is accepted that a curve may be defined first by an algebraic equation. Dan Sloughter (Furman University) The Geometry November 6, / 18
51 Problems 1. Use the geometric method of Al-Khowarizmi to find solutions for the following equations. a. x x = 108 b. x 2 + 8x = Suppose p > 0 and q > 0. a. Using the geometric method of Al-Khowarizmi, show that one solution of x 2 + 2px = q is given by x = p 2 + q p. b. Use the quadratic formula to verify this result. 3. Plot the points (2, 1), ( 2, 3), ( 4, 1), and (5, 3) in the Cartesian plane. Dan Sloughter (Furman University) The Geometry November 6, / 18
52 Problems (cont d) 4. Draw the circles in the Cartesian plane with the following equations: a. x 2 + y 2 = 25 b. (x 1) 2 + (y 1) 2 = 1 c. (x + 2) 2 + (y 3) 2 = 4 5. Draw the lines in the Cartesian plane with the following equations: a. y = 3x b. y = x + 4 c. y = 5 x d. y = 2x 4 e. y = 4 f. x = 2 6. Identify the directrix and focus of each of the following parabolas. a. y = 4x 2 b. y = 1 4 x 2 c. y = 1 16 x 2 d. y = 12x 2 Dan Sloughter (Furman University) The Geometry November 6, / 18
Calculus: Area. Mathematics 15: Lecture 22. Dan Sloughter. Furman University. November 12, 2006
Calculus: Area Mathematics 15: Lecture 22 Dan Sloughter Furman University November 12, 2006 Dan Sloughter (Furman University) Calculus: Area November 12, 2006 1 / 7 Area Note: formulas for the areas of
More informationCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole
More informationMathematics 13: Lecture 4
Mathematics 13: Lecture Planes Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 1 / 10 Planes in R n Suppose v and w are nonzero
More informationMATH 680 : History of Mathematics
1 MATH 680 : History of Mathematics Term: Spring 013 Instructor: S. Furino Assignment 5: Analytic Geometry Weight: 8% Due: 3:55 (11:55PM) Waterloo time on 18 June 013 When typesetting your solutions use
More informationA plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.
Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their
More informationMathematics 13: Lecture 10
Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, 2008 1 / 19 Matrices Recall: A matrix is a
More informationy d y b x a x b Fundamentals of Engineering Review Fundamentals of Engineering Review 1 d x y Introduction - Algebra Cartesian Coordinates
Fundamentals of Engineering Review RICHARD L. JONES FE MATH REVIEW ALGEBRA AND TRIG 8//00 Introduction - Algebra Cartesian Coordinates Lines and Linear Equations Quadratics Logs and exponents Inequalities
More information9.1 Circles and Parabolas. Copyright Cengage Learning. All rights reserved.
9.1 Circles and Parabolas Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in
More informationSampling Distributions
Sampling Distributions Mathematics 47: Lecture 9 Dan Sloughter Furman University March 16, 2006 Dan Sloughter (Furman University) Sampling Distributions March 16, 2006 1 / 10 Definition We call the probability
More informationChange of Variables: Indefinite Integrals
Change of Variables: Indefinite Integrals Mathematics 11: Lecture 39 Dan Sloughter Furman University November 29, 2007 Dan Sloughter (Furman University) Change of Variables: Indefinite Integrals November
More informationMathematics 22: Lecture 7
Mathematics 22: Lecture 7 Separation of Variables Dan Sloughter Furman University January 15, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 7 January 15, 2008 1 / 8 Separable equations
More informationPractice Assessment Task SET 3
PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,
More informationCHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg:
CHAPTER ONE FUNCTIONS AND GRAPHS 1.0 Introduction to Functions In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed
More informationMATHS (O) NOTES. SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly. The Institute of Education Topics Covered: Complex Numbers
MATHS (O) NOTES The Institute of Education 07 SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly Topics Covered: COMPLEX NUMBERS Strand 3(Unit ) Syllabus - Understanding the origin and need for complex
More informationChapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
More informationCIRCLES: #1. What is an equation of the circle at the origin and radius 12?
1 Pre-AP Algebra II Chapter 10 Test Review Standards/Goals: E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form. E.3.b.: I can graph circles
More informationy 1 x 1 ) 2 + (y 2 ) 2 A circle is a set of points P in a plane that are equidistant from a fixed point, called the center.
Ch 12. Conic Sections Circles, Parabolas, Ellipses & Hyperbolas The formulas for the conic sections are derived by using the distance formula, which was derived from the Pythagorean Theorem. If you know
More informationMATH 241 FALL 2009 HOMEWORK 3 SOLUTIONS
MATH 41 FALL 009 HOMEWORK 3 SOLUTIONS H3P1 (i) We have the points A : (0, 0), B : (3, 0), and C : (x, y) We now from the distance formula that AC/BC = if and only if x + y (3 x) + y = which is equivalent
More informationThree-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.
More informationx and y, called the coordinates of the point.
P.1 The Cartesian Plane The Cartesian Plane The Cartesian Plane (also called the rectangular coordinate system) is the plane that allows you to represent ordered pairs of real numbers by points. It is
More information(b) Find, in terms of c and p, the coordinates of Q. (4) Do a diagram showing hyperbola, P, normal, and Q
Jan 2016 Recent IAL questions on parabolas and hyperbolas, "scaffolded" 6. The rectangular hyperbola H has equation xy = c 2, where c is a non-zero constant. The point P cp, c p, where p 0, lies on H.
More informationTable of contents. Jakayla Robbins & Beth Kelly (UK) Precalculus Notes Fall / 53
Table of contents The Cartesian Coordinate System - Pictures of Equations Your Personal Review Graphs of Equations with Two Variables Distance Equations of Circles Midpoints Quantifying the Steepness of
More information9.7 Extension: Writing and Graphing the Equations
www.ck12.org Chapter 9. Circles 9.7 Extension: Writing and Graphing the Equations of Circles Learning Objectives Graph a circle. Find the equation of a circle in the coordinate plane. Find the radius and
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More informationThe Chain Rule. Mathematics 11: Lecture 18. Dan Sloughter. Furman University. October 10, 2007
The Chain Rule Mathematics 11: Lecture 18 Dan Sloughter Furman University October 10, 2007 Dan Sloughter (Furman University) The Chain Rule October 10, 2007 1 / 15 Example Suppose that a pebble is dropped
More informationAntiderivatives. Mathematics 11: Lecture 30. Dan Sloughter. Furman University. November 7, 2007
Antiderivatives Mathematics 11: Lecture 30 Dan Sloughter Furman University November 7, 2007 Dan Sloughter (Furman University) Antiderivatives November 7, 2007 1 / 9 Definition Recall: Suppose F and f are
More informationIntermediate Level Learning Targets
Learning Target #1: Develop proficiency in analyzing, graphing and solving linear equations and inequalities. F1.1,,, B1. C1. 1.1 Students will be able to identify different types of relations and functions.
More informationUtah Math Standards for College Prep Mathematics
A Correlation of 8 th Edition 2016 To the A Correlation of, 8 th Edition to the Resource Title:, 8 th Edition Publisher: Pearson Education publishing as Prentice Hall ISBN: SE: 9780133941753/ 9780133969078/
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mathematics SKE, Strand J STRAND J: TRANSFORMATIONS, VECTORS and MATRICES J4 Matrices Text Contents * * * * Section J4. Matrices: Addition and Subtraction J4.2 Matrices: Multiplication J4.3 Inverse Matrices:
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More informationDistance and Midpoint Formula 7.1
Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units
More informationUMUC MATH-107 Final Exam Information
UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from
More information1! i 3$ (( )( x! 1+ i 3)
Math 4C Fall 2008 Final Exam (Name) (PID) (Section) Read each question carefully; answer each question completely. Show all work: no credit for unsupported answers. Attach additional sheets if necessary.
More informationMath 75 Mini-Mod Due Dates Spring 2016
Mini-Mod 1 Whole Numbers Due: 4/3 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing
More informationMath Conic Sections
Math 114 - Conic Sections Peter A. Perry University of Kentucky April 13, 2017 Bill of Fare Why Conic Sections? Parabolas Ellipses Hyperbolas Shifted Conics Goals of This Lecture By the end of this lecture,
More informationCollege Algebra & Trig w Apps
WTCS Repository 10-804-197 College Algebra & Trig w Apps Course Outcome Summary Course Information Description Total Credits 5.00 This course covers those skills needed for success in Calculus and many
More informationALGEBRA 2. Background Knowledge/Prior Skills Knows what operation properties hold for operations with matrices
ALGEBRA 2 Numbers and Operations Standard: 1 Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers, and number
More informationReview of Coordinate Systems
Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate
More informationUtah Secondary Mathematics Core Curriculum Precalculus
A Correlation of Trigonometry Lial To the Utah Secondary Mathematics Core Curriculum Precalculus Resource Title: Trigonometry Publisher: Pearson Education Inc Publishing as Prentice Hall ISBN (10 or 13
More informationThe Distance Formula. The Midpoint Formula
Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x
More informationSenior Math Circles February 18, 2009 Conics III
University of Waterloo Faculty of Mathematics Senior Math Circles February 18, 2009 Conics III Centre for Education in Mathematics and Computing Eccentricity of Conics Fix a point F called the focus, a
More informationImportant Math 125 Definitions/Formulas/Properties
Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient
More informationAlgebra I. Course Outline
Algebra I Course Outline I. The Language of Algebra A. Variables and Expressions B. Order of Operations C. Open Sentences D. Identity and Equality Properties E. The Distributive Property F. Commutative
More informationSuccessful completion of the core function transformations unit. Algebra manipulation skills with squares and square roots.
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses,
More informationPivotal Quantities. Mathematics 47: Lecture 16. Dan Sloughter. Furman University. March 30, 2006
Pivotal Quantities Mathematics 47: Lecture 16 Dan Sloughter Furman University March 30, 2006 Dan Sloughter (Furman University) Pivotal Quantities March 30, 2006 1 / 10 Pivotal quantities Definition Suppose
More informationSection 8.1 Vector and Parametric Equations of a Line in
Section 8.1 Vector and Parametric Equations of a Line in R 2 In this section, we begin with a discussion about how to find the vector and parametric equations of a line in R 2. To find the vector and parametric
More informationA video College Algebra course & 6 Enrichment videos
A video College Algebra course & 6 Enrichment videos Recorded at the University of Missouri Kansas City in 1998. All times are approximate. About 43 hours total. Available on YouTube at http://www.youtube.com/user/umkc
More informationMATH Spring 2010 Topics per Section
MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line
More information4 The Cartesian Coordinate System- Pictures of Equations
4 The Cartesian Coordinate System- Pictures of Equations Concepts: The Cartesian Coordinate System Graphs of Equations in Two Variables x-intercepts and y-intercepts Distance in Two Dimensions and the
More informationChapter 1 Coordinates, points and lines
Cambridge Universit Press 978--36-6000-7 Cambridge International AS and A Level Mathematics: Pure Mathematics Coursebook Hugh Neill, Douglas Quadling, Julian Gilbe Ecerpt Chapter Coordinates, points and
More informationSection 4.2 Polynomial Functions of Higher Degree
Section 4.2 Polynomial Functions of Higher Degree Polynomial Function P(x) P(x) = a degree 0 P(x) = ax +b (degree 1) Graph Horizontal line through (0,a) line with y intercept (0,b) and slope a P(x) = ax
More informationDescriptor Formula Standard
Grade/ Course Descriptor Standard 4 th -D Figures 4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional
More informationSome Highlights along a Path to Elliptic Curves
11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational
More informationSome Trigonometric Limits
Some Trigonometric Limits Mathematics 11: Lecture 7 Dan Sloughter Furman University September 20, 2007 Dan Sloughter (Furman University) Some Trigonometric Limits September 20, 2007 1 / 14 The squeeze
More informationAlgebra II Learning Targets
Chapter 0 Preparing for Advanced Algebra LT 0.1 Representing Functions Identify the domain and range of functions LT 0.2 FOIL Use the FOIL method to multiply binomials LT 0.3 Factoring Polynomials Use
More informationFundamentals of Engineering (FE) Exam Mathematics Review
Fundamentals of Engineering (FE) Exam Mathematics Review Dr. Garey Fox Professor and Buchanan Endowed Chair Biosystems and Agricultural Engineering October 16, 2014 Reference Material from FE Review Instructor
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationNonparametric Tests. Mathematics 47: Lecture 25. Dan Sloughter. Furman University. April 20, 2006
Nonparametric Tests Mathematics 47: Lecture 25 Dan Sloughter Furman University April 20, 2006 Dan Sloughter (Furman University) Nonparametric Tests April 20, 2006 1 / 14 The sign test Suppose X 1, X 2,...,
More informationPRACTICE TEST 1 Math Level IC
SOLID VOLUME OTHER REFERENCE DATA Right circular cone L = cl V = volume L = lateral area r = radius c = circumference of base h = height l = slant height Sphere S = 4 r 2 V = volume r = radius S = surface
More informationAlgebra II Curriculum Guide Dunmore School District Dunmore, PA
Algebra II Dunmore School District Dunmore, PA Algebra II Prerequisite: Successful completion of Geometry This course continues and reinforces the mathematical material learned in Algebra I. It deals with
More informationUnit 2 Quadratics. Mrs. Valentine Math 3
Unit 2 Quadratics Mrs. Valentine Math 3 2.1 Factoring and the Quadratic Formula Factoring ax 2 + bx + c when a = ±1 Reverse FOIL method Find factors of c that add up to b. Using the factors, write the
More informationConstructions with ruler and compass
Constructions with ruler and compass An application of algebra to geometry K. N. Raghavan http://www.imsc.res.in/ knr/ IMSc, Chennai July 2013 Cartesian geometry COORDINATE GEOMETRY FOLLOWING DESCARTES
More informationVectors and 2D Kinematics. AIT AP Physics C
Vectors and 2D Kinematics Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels
More informationDATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS
NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)
More informationNew Jersey Quality Single Accountability Continuum (NJQSAC) A-SSE 1-2; A-CED 1,4; A-REI 1-3, F-IF 1-5, 7a
ALGEBRA 2 HONORS Date: Unit 1, September 4-30 How do we use functions to solve real world problems? What is the meaning of the domain and range of a function? What is the difference between dependent variable
More informationPOINTS, LINES, DISTANCES
POINTS, LINES, DISTANCES NIKOS APOSTOLAKIS Examples/Exercises: (1) Find the equation of the line that passes through (4, 5), (4, ) () Find the equation of the line that passes through the points (1, ),
More informationPage 1
Pacing Chart Unit Week Day CCSS Standards Objective I Can Statements 121 CCSS.MATH.CONTENT.HSG.C.A.1 Prove that all circles are similar. Prove that all circles are similar. I can prove that all circles
More information1 Quadratic Functions
Unit 1 Quadratic Functions Lecture Notes Introductory Algebra Page 1 of 8 1 Quadratic Functions In this unit we will learn many of the algebraic techniques used to work with the quadratic function fx)
More information3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(4, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -9), find M 3. A( 2,0)
More informationPrep for College Algebra with Trigonometry
Prep for College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (246 topics +
More informationMATH-1420 Review Concepts (Haugen)
MATH-40 Review Concepts (Haugen) Unit : Equations, Inequalities, Functions, and Graphs Rational Expressions Determine the domain of a rational expression Simplify rational expressions -factor and then
More informationAlgebra II Honors Curriculum Guide Dunmore School District Dunmore, PA
Algebra II Honors Dunmore School District Dunmore, PA Algebra II Honors Prerequisite: Successful completion of Geometry Honors This course continues and reinforces the mathematical material learned in
More informationAlgebra Vocabulary. abscissa
abscissa The x-value of an ordered pair that describes the horizontal distance from the x-axis. It is always written as the first element in the ordered pair. 3 is the abscissa of the ordered pair (3,
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More informationConic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.
Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)
More informationCommutative laws for addition and multiplication: If a and b are arbitrary real numbers then
Appendix C Prerequisites C.1 Properties of Real Numbers Algebraic Laws Commutative laws for addition and multiplication: If a and b are arbitrary real numbers then a + b = b + a, (C.1) ab = ba. (C.2) Associative
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
Chapter 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. Overview We know that the square of a real number is always non-negative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square
More informationMATH HISTORY ACTIVITY
A. Fisher Acf 92 workbook TABLE OF CONTENTS: Math History Activity. p. 2 3 Simplify Expressions with Integers p. 4 Simplify Expressions with Fractions.. p. 5 Simplify Expressions with Decimals.. p. 6 Laws
More informationVector Algebra August 2013
Vector Algebra 12.1 12.2 28 August 2013 What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationCollege Algebra with Corequisite Support: Targeted Review
College Algebra with Corequisite Support: Targeted Review 978-1-63545-056-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable)
More informationCourse: Algebra 1-A Direct link to this page:http://www.floridastandards.org/courses/publicpreviewcourse5.aspx?ct=1
Course: 1200370 Algebra 1-A Direct link to this page:http://www.floridastandards.org/courses/publicpreviewcourse5.aspx?ct=1 BASIC INFORMATION Course Number: 1200370 Course Title: Algebra 1-A Course Abbreviated
More informationCourse Number 420 Title Algebra I Honors Grade 9 # of Days 60
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More information6A The language of polynomials. A Polynomial function follows the rule. Degree of a polynomial is the highest power of x with a non-zero coefficient.
Unit Mathematical Methods Chapter 6: Polynomials Objectives To add, subtract and multiply polynomials. To divide polynomials. To use the remainder theorem, factor theorem and rational-root theorem to identify
More informationCOURSE SYLLABUS Part I Course Title: MATH College Algebra Credit Hours: 4, (4 Lecture 0 Lab G) OTM-TMM001
COURSE SYLLABUS Part I Course Title: MATH 1340 - College Algebra Credit Hours: 4, (4 Lecture 0 Lab G) OTM-TMM001 Course Description: College Algebra in conjunction with MATH 1350, Pre-Calculus, provides
More informationSTEM-Prep Pathway SLOs
STEM-Prep Pathway SLOs Background: The STEM-Prep subgroup of the MMPT adopts a variation of the student learning outcomes for STEM from the courses Reasoning with Functions I and Reasoning with Functions
More informationAlgebra II Vocabulary Alphabetical Listing. Absolute Maximum: The highest point over the entire domain of a function or relation.
Algebra II Vocabulary Alphabetical Listing Absolute Maximum: The highest point over the entire domain of a function or relation. Absolute Minimum: The lowest point over the entire domain of a function
More informationPrecalculus Summer Assignment 2015
Precalculus Summer Assignment 2015 The following packet contains topics and definitions that you will be required to know in order to succeed in CP Pre-calculus this year. You are advised to be familiar
More informationAlgebra and Trigonometry
Algebra and Trigonometry 978-1-63545-098-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Jay Abramson, Arizona State
More informationPre Calculus Gary Community School Corporation Unit Planning Map
UNIT/TIME FRAME STANDARDS Functions and Graphs (6 weeks) PC.F.1: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities,
More information1.2 Graphs and Lines. Cartesian Coordinate System
1.2 Graphs and Lines Cartesian Coordinate System Note that there is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs (a, b) of real numbers. Graphs
More informationUNIT 3 CIRCLES AND VOLUME Lesson 1: Introducing Circles Instruction
Prerequisite Skills This lesson requires the use of the following skills: performing operations with fractions understanding slope, both algebraically and graphically understanding the relationship of
More informationIntroduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves
Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves 7.1 Ellipse An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r from two fixed
More informationCourse Outcome Summary
Course Information: Algebra 2 Description: Instruction Level: 10-12 Total Credits: 2.0 Prerequisites: Textbooks: Course Topics for this course include a review of Algebra 1 topics, solving equations, solving
More information(arrows denote positive direction)
12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate
More informationMath 1 packet for Coordinate Geometry part 1. Reviewing the basics. The coordinate plane
Math 1 packet for Coordinate Geometry part 1 Reviewing the basics The coordinate plane The coordinate plane (also called the Cartesian plane named after French mathematician Rene Descartes, who formalized
More information