1.2. GRAPHS OF RELATIONS
|
|
|
- Ellen Stanley
- 7 years ago
- Views:
Transcription
1 1.2. GRAPHS OF RELATIONS Graphs of relations as sets in coordinate plane Let us recall that a coordinate plane is formed by choosing two number lines (lines where points represent real numbers) which intersect at the right angle. Usually, one line is horizontal and the other vertical. The horizontal line is the x-axis and vertical line y-axis. The point of intersection of the two lines is called the origin. In a coordinate plane each point represents uniquely an ordered pair of real numbers (x, y) and each ordered pair of real numbers is represented by a point. If a point P represents the pair (x, y) then x is called x-coordinate and y is called y-coordinate of the point P. 1
2 DEFINITION. The graph of a relation R is the set of all points (x, y) in a coordinate plane such that x is related to y through the relation R EXAMPLE. The map of Australia below graphically represents a relation since any set in a coordinate plane does it. 2
3 Graphing relations A relation consisting of finitely many ordered pairs of numbers could be graphed by simple plotting of points EXAMPLE. The figure below shows the graph of the relation R = {(2, 5), (4, 3), (6, 1), (2, 7)}. For a relation given by an equation we plot enough points to reveal the essential behavior of the graph. 3
4 EXAMPLE. The figure below shows the graph of the relation 2x + 3y = 6. The graph could be sketched by plotting the points (0, 2) and (3, 0) and connecting the points with a straight line. 4
5 EXAMPLE. The figure below shows the graph of the relation x 2 + y 2 = 25. The graph could be sketched by plotting the points (5, 0), (4, 3), (3, 4), (0, 5), ( 3, 4), ( 4, 3), ( 5, 0), ( 4, 3), ( 3, 4), (0, 5), (3, 4), (4, 3) and connecting the points with a curve. 5
6 Identifying relations given graphically Any graph in a coordinate plane defines a relation in the following way: if a point representing the ordered pair (x, y) lies on the graph then x is related to y and if the point representing the pair (x, y) does not belong to the graph then x is not related to y. The next example illustrates the above definition. 6
7 EXAMPLE. In the figure below the graph of the letter Z in the coordinate plane represents a relation. The graph shows that the number 5 is related to the numbers 1, 3, 5 because the vertical line passing through the point representing 5 in the x- axis intersects the graph at the points which have the y-coordinates 1, 3, 5, respectively. In the same way we see that 3 is related to 2 and 6. The number 1 is not related to any number because the vertical line passing through the point 1 on the x-axis does not intersect the graph. 7
8 EXERCISES I. 1. Exercise. Describe the relation represented by the graph below. Go to answer 1 2. Exercise. Describe the relation represented by the graph below. 8
9 Go to answer 2 3. Exercise. Describe the relation represented by the graph below. 9
10 Go to answer 3 4. Exercise. Describe the relation represented by the graph below. 10
11 Go to answer 4 5. Exercise. Describe the relation represented by the shaded region shown below. 11
12 Go to answer 5 12
13 ANSWERS I. 1. Answer to Exercise 1. The graph represents the relation x = y. Go back 1 2. Answer to Exercise 2. The graph represents the relation x > y. Go back 2 3. Answer to Exercise 3. Since all points on the graph have the x- coordinate equal to 2 the graph represents the relation 2 is related to each number y. Go back 3 4. Answer to Exercise 4. Since all points on the graph have the y- coordinate equal to 3 the graph represents the relation each number x is related to the number 3. Go back 4 5. Answer to Exercise 5. The shade region consists of all points but the points on the line x = y. So the graph describes the relation x is related to y if x y. Go back 5 13
14 Finding intercepts of a graph Obviously, when we sketch a graph in a coordinate plane, the points of intersection of the graph with the axes are essential DEFINITION. The x-intercepts of of a graph are the points at which the graph intersects the x-axis and the y-intercepts of of a graph are the points at which the graph intersects the y-axis EXAMPLE. To find the x-intercepts, let y be zero and solve for x. For the relation given by the equation 2x + y = 4 substitution y = 0 gives us x = 2 so the x-intercept is (2, 0). 14
15 EXAMPLE. To find the y-intercepts, let x be zero and solve for y. For the relation given by the equation 2x + y = 4 substitution x = 0 gives us y = 4 so the y-intercept is (0, 4) EXERCISES II. 1. Exercise. Find the intercepts of the graph of the equation 16x2+9y2 =
16 Go to answer 1 16
17 ANSWERS II. 1. Answer to Exercise 1. The x-intercepts are (3, 0) and ( 3, 0). The y-intercepts are (4, 0) and ( 4, 0). Substituting y = 0 we obtain 16x2 + 9(02) = 144. It means that x2 = 9 and x = 3 or x = 3. The y-intercepts (0, 4) and (0, 4) are obtained in the same way by substituting x = 0 and solving the equation 16(0)2 + 9y2 = 144. Go back 1 17
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 y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 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
1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
Lecture 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
Chapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
Graphing 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
Solving 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
EQUATIONS 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
Elements 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 y-intercept in the equation of a line Comparing lines on
Linear 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 x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
The Point-Slope Form
7. The Point-Slope 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
Slope-Intercept 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 y-intercept. Determine
Lecture 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
5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
Chapter 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
3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
Part 1: Background - Graphing
Department of Physics and Geology Graphing Astronomy 1401 Equipment Needed Qty Computer with Data Studio Software 1 1.1 Graphing Part 1: Background - Graphing In science it is very important to find and
PLOTTING 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
Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES
Co-ordinate 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 co-ordinates. The Gradient of a Line. As
In this chapter, you will learn to use cost-volume-profit analysis.
2.0 Chapter Introduction In this chapter, you will learn to use cost-volume-profit analysis. Assumptions. When you acquire supplies or services, you normally expect to pay a smaller price per unit as the
Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate
12.5 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
CHAPTER 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 x-axis, and the vertical axis or
1.3 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
A synonym is a word that has the same or almost the same definition of
Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given
GRAPHING 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 - y-axis,
Section 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 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
Section 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
The Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
Performance. 13. Climbing Flight
Performance 13. Climbing Flight In order to increase altitude, we must add energy to the aircraft. We can do this by increasing the thrust or power available. If we do that, one of three things can happen:
CIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
To Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
Chapter 6: Break-Even & CVP Analysis
HOSP 1107 (Business Math) Learning Centre Chapter 6: Break-Even & CVP Analysis One of the main concerns in running a business is achieving a desired level of profitability. Cost-volume profit analysis
High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.
Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student
What are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
Section 9.1 Vectors in Two Dimensions
Section 9.1 Vectors in Two Dimensions Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in the first Figure below with an
(Least Squares Investigation)
(Least Squares Investigation) o Open a new sketch. Select Preferences under the Edit menu. Select the Text Tab at the top. Uncheck both boxes under the title Show Labels Automatically o Create two points
Unit 7: Radical Functions & Rational Exponents
Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving
www.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
Lecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
Exam 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
4.4 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
5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
x 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
EdExcel Decision Mathematics 1
EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation
Higher 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
6.2 Solving Nonlinear Equations
6.2. SOLVING NONLINEAR EQUATIONS 399 6.2 Solving Nonlinear Equations We begin by introducing a property that will be used extensively in this and future sections. The zero product property. If the product
STRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
Contents. 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..................................
Linear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
Circle 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
Graphing Quadratic Functions
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the
Graphing Equations. with Color Activity
Graphing Equations with Color Activity Students must re-write equations into slope intercept form and then graph them on a coordinate plane. 2011 Lindsay Perro Name Date Between The Lines Re-write each
Systems of Linear Equations and Inequalities
Systems of Linear Equations and Inequalities Recall that every linear equation in two variables can be identified with a line. When we group two such equations together, we know from geometry what can
Algebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
Section 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
Graphing - Slope-Intercept Form
2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,
Plot 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
Solving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
F B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below.
PHYSICS 176 UNIVERSITY PHYSICS LAB II Experiment 9 Magnetic Force on a Current Carrying Wire Equipment: Supplies: Unit. Electronic balance, Power supply, Ammeter, Lab stand Current Loop PC Boards, Magnet
Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross
CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal
Correlation key concepts:
CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)
3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS. Copyright Cengage Learning. All rights reserved.
3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.
Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
MA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
Vector 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
Mathematical goals. Starting points. Materials required. Time needed
Level C1 of challenge: D C1 Linking the properties and forms of quadratic of quadratic functions functions Mathematical goals Starting points Materials required Time needed To enable learners to: identif
with 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
PARABOLAS 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
Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
Week 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter
Pennsylvania System of School Assessment
Pennsylvania System of School Assessment The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read
Academic Support Center. Using the TI-83/84+ Graphing Calculator PART II
Academic Support Center Using the TI-83/84+ Graphing Calculator PART II Designed and Prepared by The Academic Support Center Revised June 2012 1 Using the Graphing Calculator (TI-83+ or TI-84+) Table of
Electrical Resonance
Electrical Resonance (R-L-C series circuit) APPARATUS 1. R-L-C Circuit board 2. Signal generator 3. Oscilloscope Tektronix TDS1002 with two sets of leads (see Introduction to the Oscilloscope ) INTRODUCTION
Make sure you look at the reminders or examples before each set of problems to jog your memory! Solve
Name Date Make sure you look at the reminders or examples before each set of problems to jog your memory! I. Solving Linear Equations 1. Eliminate parentheses. Combine like terms 3. Eliminate terms by
In following this handout, sketch appropriate graphs in the space provided.
Dr. McGahagan Graphs and microeconomics You will see a remarkable number of graphs on the blackboard and in the text in this course. You will see a fair number on examinations as well, and many exam questions,
What Does Your Quadratic Look Like? EXAMPLES
What Does Your Quadratic Look Like? EXAMPLES 1. An equation such as y = x 2 4x + 1 descries a type of function known as a quadratic function. Review with students that a function is a relation in which
CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
Graphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
Trigonometric 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
Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
SPIRIT 2.0 Lesson: A Point Of Intersection
SPIRIT 2.0 Lesson: A Point Of Intersection ================================Lesson Header============================= Lesson Title: A Point of Intersection Draft Date: 6/17/08 1st Author (Writer): Jenn
Section 8.4 - Composite and Inverse Functions
Math 127 - Section 8.4 - Page 1 Section 8.4 - Composite and Inverse Functions I. Composition of Functions A. If f and g are functions, then the composite function of f and g (written f g) is: (f g)( =
x 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
Fatigue Analysis of an Inline Skate Axel
FATIGUE ANALYSIS OF AN INLINE SKATE AXEL 57 Fatigue Analysis of an Inline Skate Axel Authors: Faculty Sponsor: Department: Garrett Hansen, Mike Woizeschke Dr. Shanzhong (Shawn) Duan Mechanical Engineering
Hydrostatic Force on a Submerged Surface
Experiment 3 Hydrostatic Force on a Submerged Surface Purpose The purpose of this experiment is to experimentally locate the center of pressure of a vertical, submerged, plane surface. The experimental
Geometry 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
Review of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
Writing the Equation of a Line in Slope-Intercept Form
Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing
For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false.
Level A11 of challenge: D A11 Mathematical goals Starting points Materials required Time needed Factorising cubics To enable learners to: associate x-intercepts with finding values of x such that f (x)
Volumes of Revolution
Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students
REVIEW 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.
Common Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
6.5 Factoring Special Forms
440 CHAPTER 6. FACTORING 6.5 Factoring Special Forms In this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial. Squaring a binomial.
Coordinate Plane, Slope, and Lines Long-Term 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 y-value change? b. From point
Area and Arc Length in Polar Coordinates
Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly
Elasticity. I. What is Elasticity?
Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in