Math 1310 Section 1.1 Points, Regions, Distance and Midpoints


 Ronald Casey
 1 years ago
 Views:
Transcription
1 Math 1310 Section 1.1 Points, Regions, Distance and Midpoints In this section, we ll review plotting points in the coordinate plane, then graph vertical lines, horizontal lines and some inequalities. We ll also review the Pythagorean Theorem, develop a formula for finding the distance between two points in the coordinate plane and one for finding the midpoint of a line segment with given endpoints. Here s the coordinate plane: Graphing Points It consists of two number lines that are perpendicular to one another. The horizontal number line is called the x axis and the vertical number line is called the y axis. The point where the two number lines intersect is called the origin. The number lines divide the plane into four regions, called quadrants. Math 1310 Class Notes Section 1.1, Page 1 of 1
2 The pair, ( x, y), is called an ordered pair. It corresponds to a single unique point in the coordinate plane. The first number is called the x coordinate, and the second number is called the y coordinate. The ordered pair (0, 0) names to the origin. The x coordinate tells us the horizontal distance a point is from the origin. The y coordinate tells us the vertical distance a point is from the origin. You ll move right or up for positive coordinates and left or down for negative coordinates. So, the point (3, 6) is three units to the right of the origin, and then 6 units up. The point (, 8) is two units to the left of the origin, and then 8 units down. The point (5, 4) is five units to the right of the origin, and then 4 units down. Example 1: State the ordered pairs associated with each point labeled on the graph: Math 1310 Class Notes Section 1.1, Page of 1
3 Example : Plot the following points in the coordinate plane: A (3, 5) B (6, 0) C (7, 1) D (0, 8) E (5, 3) F (8, ) Math 1310 Class Notes Section 1.1, Page 3 of 1
4 Graphing Regions in the Coordinate Plane The set of all points in the coordinate plane with y coordinate k is the horizontal line y = k. The set of all points in the coordinate plane with x coordinate k is the vertical line x = k. Example 3: Graph y = 3 in the coordinate plane. Example 4: Graph x = 7 in the coordinate plane. Math 1310 Class Notes Section 1.1, Page 4 of 1
5 The region y < k is the part of the coordinate plane below the line y = k, not including the line itself. The region y k is the part of the coordinate plane below the line y = k, including the line itself. The region y > k is the part of the coordinate plane above the line y = k, not including the line itself. The region y k is the part of the coordinate plane above the line y = k, including the line itself. The region x < k is the part of the coordinate plane to the left of the line x = k, not including the line itself. The region x k is the part of the coordinate plane to the left of the line x = k, including the line itself. The region x > k is the part of the coordinate plane to the right of the line x = k, not including the line itself. The region x k is the part of the coordinate plane to the right of the line x = k, including the line itself. For y > k or y < k, the line y = k is drawn as a dotted line. For x < k or x > k, the line x = k is drawn as a dotted line. For y k or y k, the line y = k is drawn as a solid line. For x k or x k, the line x = k is drawn as a solid line. In the exercises, you will see these written using set notation, e.g., ( x y) Example 4: Graph in the coordinate plane: y. { y k},. Math 1310 Class Notes Section 1.1, Page 5 of 1
6 Example 5: Graph in the coordinate plane: x < 3. Example 6: Graph in the coordinate plane: < x 5 Math 1310 Class Notes Section 1.1, Page 6 of 1
7 Example 7: Graph ( y) {, 3 y 8 } x. Example 8: Graph ( y) {, x < 4 and y 1 } x. Math 1310 Class Notes Section 1.1, Page 7 of 1
8 The Pythagorean Theorem In a previous course, you probably learned to work with the Pythagorean Theorem. This theorem states that, in a right triangle, with legs measuring a and b and with hypoteneuse measuring c, c = a + b. Given any two of these values, you can solve the equation to find the third. Example 9: In right triangle ABC, with right angle C, a = 0 and b = 15. Find c. Example 10: Find the length of the missing side of the right triangle Example 11: Find the length of the missing side of the right triangle Math 1310 Class Notes Section 1.1, Page 8 of 1
9 The Distance Formula We can use the Pythagorean Theorem to develop a formula for finding the distance between any two points in the coordinate plane. Suppose you have two ordered pairs, ( x, y ) and ( x y ). We can find a point C, so that we can draw a right triangle in the coordinate plane. Then we can find the lengths of the legs of the right triangle and use the Pythagorean Theorem to find the length of the hypoteneuse. This will give us the distance between the two points: 1 1, This gives us a formula that we can use to find the distance between any two points in the coordinate plane. Suppose (, y ) and ( x y ) x are points in the coordinate plane. Then the distance between the 1 1, two points is ( ) ( ) d = x. We call this the Distance Formula. x1 + y y1 Math 1310 Class Notes Section 1.1, Page 9 of 1
10 Example 1: Use the distance formula to find the distance between the points (3, ) and (4, 1). Example 13: Use the distance formula to find the distance between the points , and,. 6 4 Example 14: Use the distance formula to find the distance between the points, 3 and, 3 3. ( ) ( ) Math 1310 Class Notes Section 1.1, Page 10 of 1
11 The Midpoint Formula x 1 1, and ask for the coordinates of the point that is halfway between the two given points. We call this point the midpoint. Suppose we look at two points in the coordinate plane (, y ) and ( x y ) So the midpoint of a segment with two given endpoints, ( x ) and ( y ) x + x y1 + M =, 1 y 1, y 1. This is the Midpoint Formula. x is Example 15: Find the midpoint of the segment with endpoints at (4, 7) and (3, 1)., Math 1310 Class Notes Section 1.1, Page 11 of 1
12 Example 16: Find the midpoint of the segment with endpoints at , and,. 6 4 Example 17: Find the midpoint of the segment with endpoints at (.3, 0.7) and (5.8, .9). Example 18: The midpoint of a line segment is (3, 8). One endpoint is (4, ). Find the other endpoint. Math 1310 Class Notes Section 1.1, Page 1 of 1
13 Math 1310 Section 1. Lines Topics in this section include finding the slope of a line, writing an equation of the line, graphing a line in the coordinate plane, finding x and y intercepts, and writing an equation of a line that is parallel or perpendicular to a given line through a stated point. Slope The slope of a line represents the rate of change of the line. A steep line that rises from left to right will have a large positive slope, while a line that falls gradually from left to right will have a small negative slope. Example 1: Determine whether each of these lines has a positive slope, negative slope, slope 0 or undefined slope. Math 1310 Class Notes Section 1., Page 1 of 8
14 You can find the slope of a line that contains two given points ( x, y ) and ( x y ) y y1 using the slope formula: m =. x x 1 1 1, Example : Find the slope of the line containing the points (4, 3) and (, 1). by Example 3: Find the slope of the line containing the points 1 5, and, 3 4. Example 4: Find the slope of the line containing the points (7, 4) and (3, 4). Example 5: Find the slope of the line containing the points (, 5) and (, 7). Math 1310 Class Notes Section 1., Page of 8
15 You can also find the slope of a line from its graph. Example 6: Find the slope of the line: Example 8: Find the slope of the line: Math 1310 Class Notes Section 1., Page 3 of 8
16 Writing Equations of Lines One of the main objectives of this section is to write an equation of a line. You ll need to know the slope of the line and a point that lies on the line in order to write an equation of the line. You can be given this information in a number of ways. You can write an equation of the line using either the point slope form of the equation of the line, y y1 = m( x x1), or the slope intercept form of the equation of the line, y = mx + b. In both forms, m represents the slope. In the point slope form, ( x 1, y1) is a point on the line. In the slope intercept form, b represents the y intercept, the point where the graph of the line crosses the y axis.. Example 9: Write an equation of the line: Example 10: Write an equation of the line: Math 1310 Class Notes Section 1., Page 4 of 8
17 Example 11: Write an equation of the line that has slope 3 and y intercept 7. Example 1: Write an equation of the line that has slope 5 1 and passes through the point (10, 4). Example 13: Write an equation of the line that passes through the points (3, 8) and (4, 1). Example 14: Write an equation of the line that passes through the points (4, 0) and (4, 7). Math 1310 Class Notes Section 1., Page 5 of 8
18 Sometimes you will be given the x intercept and the y intercept. The x intercept is the point where the graph of the line crosses the x axis. Example 15: Write an equation of the line with x intercept 6 and y intercept 1. The form Ax + By = C is called the standard form of the equation of the line. You will need to be able to rewrite an equation given in standard form in slope intercept form. Example 16: Write the equation in slope intercept form. Then identify the slope and the y intercept: 4 x + y = 3 Graphing Lines You ll need to be able to graph lines in the coordinate plane. You can use the slope and the y intercept, or any two points, such as the x and y intercepts. Math 1310 Class Notes Section 1., Page 6 of 8
19 Example 17: Graph the line y = x + 1 using the slope and the y intercept. Example 18: Graph the line x 5y = 10 using the intercepts. Math 1310 Class Notes Section 1., Page 7 of 8
20 Example 19: Graph the line x + 3y = 1. Math 1310 Class Notes Section 1., Page 8 of 8
The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6
Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means
More informationGRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?
GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra  Linear Equations & Inequalities T37/H37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationSection summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2
Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationLines That Pass Through Regions
: Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationLesson 19: Equations for Tangent Lines to Circles
Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line
More informationSection 1.4 Graphs of Linear Inequalities
Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationSect The SlopeIntercept Form
Concepts # and # Sect.  The SlopeIntercept Form SlopeIntercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not
More informationLesson 19: Equations for Tangent Lines to Circles
Classwork Opening Exercise A circle of radius 5 passes through points ( 3, 3) and (3, 1). a. What is the special name for segment? b. How many circles can be drawn that meet the given criteria? Explain
More informationCoordinate Plane Project
Coordinate Plane Project C. Sormani, MTTI, Lehman College, CUNY MAT631, Fall 2009, Project XI BACKGROUND: Euclidean Axioms, Half Planes, Unique Perpendicular Lines, Congruent and Similar Triangle Theorems,
More informationWriting the Equation of a Line in SlopeIntercept Form
Writing the Equation of a Line in SlopeIntercept Form SlopeIntercept Form y = mx + b Example 1: Give the equation of the line in slopeintercept form a. With yintercept (0, 2) and slope 9 b. Passing
More informationCoordinate Plane, Slope, and Lines LongTerm Memory Review Review 1
Review. What does slope of a line mean?. How do you find the slope of a line? 4. Plot and label the points A (3, ) and B (, ). a. From point B to point A, by how much does the yvalue change? b. From point
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationLesson 19: Equations for Tangent Lines to Circles
Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
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 informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (1,3), (3,3), (2,3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the xcomponent of a point in the form (x,y). Range refers to the set of possible values of the ycomponent of a point in
More informationAll points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates?
Classwork Example 1: Extending the Axes Beyond Zero The point below represents zero on the number line. Draw a number line to the right starting at zero. Then, follow directions as provided by the teacher.
More informationGeometry Module 4 Unit 2 Practice Exam
Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More informationGRAPHING LINEAR EQUATIONS IN TWO VARIABLES
GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: SlopeIntercept Form: y = mx+ b In an equation
More informationCHAPTER 1 Linear Equations
CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or xaxis, and the vertical axis or
More informationPLOTTING DATA AND INTERPRETING GRAPHS
PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationMODERN APPLICATIONS OF PYTHAGORAS S THEOREM
UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationChapter 8 Graphs and Functions:
Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes
More informationx x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =
Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the
More information2.3 Writing Equations of Lines
. Writing Equations of Lines In this section ou will learn to use pointslope form to write an equation of a line use slopeintercept form to write an equation of a line graph linear equations using the
More informationLecture 9: Lines. m = y 2 y 1 x 2 x 1
Lecture 9: Lines If we have two distinct points in the Cartesian plane, there is a unique line which passes through the two points. We can construct it by joining the points with a straight edge and extending
More informationSolving Equations Involving Parallel and Perpendicular Lines Examples
Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines
More information2.1 Equations of Lines
Section 2.1 Equations of Lines 1 2.1 Equations of Lines The SlopeIntercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is
More informationTIME VALUE OF MONEY PROBLEM #8: NET PRESENT VALUE Professor Peter Harris Mathematics by Sharon Petrushka
TIME VALUE OF MONEY PROBLEM #8: NET PRESENT VALUE Professor Peter Harris Mathematics by Sharon Petrushka Introduction Creativity Unlimited Corporation is contemplating buying a machine for $100,000, which
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationSection 1.10 Lines. The Slope of a Line
Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through
More informationc sigma & CEMTL
c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,
More information1) (3) + (6) = 2) (2) + (5) = 3) (7) + (1) = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = 7) (5)(4) = 8) (3)(6) = 9) (1)(2) =
Extra Practice for Lesson Add or subtract. ) (3) + (6) = 2) (2) + (5) = 3) (7) + () = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = Multiply. 7) (5)(4) = 8) (3)(6) = 9) ()(2) = Division is
More informationSOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses
CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning
More informationCoordinate Geometry THE EQUATION OF STRAIGHT LINES
Coordinate Geometry THE EQUATION OF STRAIGHT LINES This section refers to the properties of straight lines and curves using rules found by the use of cartesian coordinates. The Gradient of a Line. As
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xyplane), so this section should serve as a review of it and its
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More information56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.
6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which
More informationPatterns, Equations, and Graphs. Section 19
Patterns, Equations, and Graphs Section 19 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
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 informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More informationThe measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures
8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 20132014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationThe Distance from a Point to a Line
: Student Outcomes Students are able to derive a distance formula and apply it. Lesson Notes In this lesson, students review the distance formula, the criteria for perpendicularity, and the creation of
More informationModule: Graphing Linear Equations_(10.1 10.5)
Module: Graphing Linear Equations_(10.1 10.5) Graph Linear Equations; Find the equation of a line. Plot ordered pairs on How is the Graph paper Definition of: The ability to the Rectangular Rectangular
More informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationStudents will understand 1. use numerical bases and the laws of exponents
Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?
More informationThe Cartesian Plane The Cartesian Plane. Performance Criteria 3. PreTest 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13
6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 PreTest 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit
More informationHomework from Section Find two positive numbers whose product is 100 and whose sum is a minimum.
Homework from Section 4.5 4.5.3. Find two positive numbers whose product is 100 and whose sum is a minimum. We want x and y so that xy = 100 and S = x + y is minimized. Since xy = 100, x = 0. Thus we have
More informationGraphing  SlopeIntercept Form
2.3 Graphing  SlopeIntercept Form Objective: Give the equation of a line with a known slope and yintercept. When graphing a line we found one method we could use is to make a table of values. However,
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationUnit 5: Coordinate Geometry Practice Test
Unit 5: Coordinate Geometry Practice Test Math 10 Common Name: Block: Please initial this box to indicate you carefully read over your test and checked your work for simple mistakes. What I can do in this
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
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 informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and yintercept in the equation of a line Comparing lines on
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More information4.4 Transforming Circles
Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different
More informationPRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71. Applications. F = mc + b.
PRIMARY CONTENT MODULE Algebra I Linear Equations & Inequalities T71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.
More informationMathematics Task Arcs
Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number
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 informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationWrite the Equation of the Line Review
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
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 informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationSlopeIntercept Equation. Example
1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the yintercept. Determine
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationPlot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.
Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationAlgebra 12. A. Identify and translate variables and expressions.
St. Mary's College High School Algebra 12 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used
More informationUnit 1: Integers and Fractions
Unit 1: Integers and Fractions No Calculators!!! Order Pages (All in CC7 Vol. 1) 31 Integers & Absolute Value 191194, 203206, 195198, 207210 32 Add Integers 33 Subtract Integers 215222 34 Multiply
More information