Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin.


 Kristopher Carr
 1 years ago
 Views:
Transcription
1 1. Introduction To describe position we need a fixed reference (start) point and a way to measure direction and distance. In Mathematics we use Cartesian coordinates, named after the Mathematician and Philosopher René Descartes ( ). The fixed reference point is called the Origin. The horizontal axis is labelled x. The vertical axis is labelled y. Coordinates are listed as (x, y). The origin is (0, 0) Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin. A is at (3, 2). From the origin, 3 to the right and 2 up. B is at (1, 5) C is at ( 4, 3). From the origin, 4 to the left and 3 down. D is at ( 6, 4) E is at (2, 5) Plot these F (5, 6) G ( 3, 2) H (4, 2) I ( 1, 1) J (0, 4) K ( 1, 0) Page 1 of 28
2 What do all these points have in common? Describe the line that would join the points. x = 2 Any point on this line has an xvalue of 2 They all meet the criterion x = 2 The line has the equation x = Page 2 of 28
3 Exercise 1a What are the equations of the following lines? Draw the lines x = 6, x = 2, x = 3, x = 5 What word could you use to describe the direction of these lines? So any vertical line has an equation of the form x = a number The yaxis has equation x = Page 3 of 28
4 What do all these points have in common? Describe the line that would join the points. y = 4 Any point on this line has a yvalue of 4. They all meet the criterion y = 4 The line has the equation y = Page 4 of 28
5 Exercise 1b What are the equations of the following lines? Draw the lines y = 6, y = 2, y = 3, y = 5 What word could you use to describe these lines? So any horizontal line has an equation of the form y = a number The xaxis has equation y = Page 5 of 28
6 2. Drawing a straight line from its equation Cartesian coordinates and straight lines booklet Think of a straight road that you are walking along. You might describe it as being flat (horizontal), or having an upward slope as you go up a hill. In mathematics we describe lines by using an equation. This is the rule that links the x and y values of any point on the line. Consider the line y = 2x (remember that 2x = 2 x) To draw a line we need a minimum of two points. We will use three points for confirmation. Let s set up a table: xvalue Why 0, 1, and 2? Because they are yvalue nice, easy numbers. We substitute these xvalues into the equation y = 2x to work out the related y value. y = 2x x = 0 y = 2 0 = 0 x = 1 y = 2 1 = 2 x = 2 y = 2 2 = 4 xvalue yvalue So our coordinates are (0, 0) (1, 2) (2, 4) Page 6 of 28
7 Now we plot these and draw the line through them. Cartesian coordinates and straight lines booklet Any point on this line will meet the criterion y= 2x y = 2x Check this for yourself: x = 3 y = 6 x = 2 y = 4 y= 3x Page 7 of 28
8 How about the line y = 3x 1? y = 3x 1 x = 0 y = 1 x = 1 y = 2 x = 2 y = 5 xvalue yvalue Remember BIDMAS? We Multiply before Subtracting = 0 1 = = 3 1 = = 6 1 = 5 So our coordinates are (0, 1) (1, 2) (2, 5) Page 8 of 28
9 Exercise 2 a. Draw the line y = 3x 5 y = 3x 5 x = 0 y = x = 1 y = x = 2 y = xvalue yvalue The coordinates are (0, ) (1, ) (2, ) Page 9 of 28
10 b. Draw the line y = x + 4 y = x + 4 x = 0 x = 1 x = 2 xvalue yvalue The coordinates are (0, ) (1, ) (2, ) Page 10 of 28
11 c. Draw the line y = 6 5x y = 6 5x x = 0 x = 1 x = 2 xvalue yvalue Page 11 of 28
12 d. Draw the line y = ½x + 3 y = ½x + 3 x = 0 x = 2 x = 4 xvalue yvalue Page 12 of 28
13 e. Draw the line x + y = 6 x + y = 6 x = = 6 x = 1 1+? = 6 x = 2 2+? = 6 xvalue yvalue This equation may look a bit different from previous ones but don t be put off. If in doubt, substitute values and have a look. What number plus 2 gives 6? Page 13 of 28
14 f. Draw the line y = 2x + 3 y = 2x + 3 x = 0 y = x = 1 y = 2(1) + 3 = x = 2 y = xvalue yvalue Page 14 of 28
15 3. Gradient The gradient of a line is the ratio of the vertical change in distance (height or rise) to the distance travelled horizontally (run). In road signs this is often expressed as a percentage. In mathematics we usually express a gradient as a fraction. gradient = vertical change (y) horizontal change (x) = y x The Greek letter delta, Δ is often used to denote a change in a quantity. gradient = vertical change (y) horizontal change (x) = y x Page 15 of 28
16 Let s compare these two graphs: Cartesian coordinates and straight lines booklet y= x + 3 What effect does a negative coefficient of x have on a line? y= 2x Page 16 of 28
17 Let s compare these two graphs: Cartesian coordinates and straight lines booklet y= 2x 3 What do these two equations have in common? What word could you use to describe the direction of these lines? Hint: railway tracks! y= 2x Page 17 of 28
18 Let s compare these two graphs: Cartesian coordinates and straight lines booklet If we pick any point on the line then move 4 right and 3 up, we return to the line. gradient = vertical (y) horizontal (x) = 3 4 gradient = vertical (y) horizontal(x) = Page 18 of 28
19 gradient = vertical (y) horizontal (x) = 4 1 = 4 Exercise 3a From the starting points on this grid, mark on two further points for each line, counting boxes, then draw the lines with these gradients: gradient A 3 3 up 1 right B up 2 right C down 3 right Page 19 of 28
20 If we are given the coordinates of two points we can calculate the gradient of the line segment joining them. Label the points (x 1, y 1 ) and (x 2, y 2 ). gradient = Δy Δx = y 2 y 1 x 2 x 1 Example (Check this calculation by counting along and up) (x 1, y 1 ) ( 1, 2 ) (x 2, y 2 ) ( 5, 4 ) gradient = y 2 y 1 x 2 x 1 = = 2 4 = Page 20 of 28
21 Does it matter which way we label (x 1, y 1 ) and (x 2, y 2 ). Let s check by reversing this last example: Cartesian coordinates and straight lines booklet (x 2, y 2 ) ( 1, 2 ) (x 1, y 1 ) ( 5, 4 ) gradient = y 2 y 1 x 2 x 1 = = 2 4 = 1 2 The gradient is the same. Exercise 3b Work out the gradient of the lines between these points: A (3, 2) and B (7, 6) C ( 3, 3) and D (9, 7) E ( 5, 2) and F( 3, 8) G (11, 2) and H (3, 2) Page 21 of 28
22 4. Working out an equation of a straight line Cartesian coordinates and straight lines booklet This is often expressed as y = mx + c It s important to identify the things that are different and the things that are the same. What about the equations we ve looked at? y = 3x 5 y = 6 5x x + y = 6 y = x + 4 y = 1 2 x + 3 y = 2x + 3 The coefficient of x and the number on its own can vary but these all fit the pattern of the straight line equation. Note: y = 2x is really y = 2x + 0 y = 3 is really y = 0x + 3 Memorise: y = mx + c m = gradient = Δy Δx = y 2 y 1 x 2 x 1 c = y intercept (where the line crosses the yaxis) Page 22 of 28
23 Example What is the equation of this line? Choose points on the line that are nice and easy to read. m = y 2 y 1 x 2 x 1 = 3 ( 3) 0 ( 4) = 6 4 = 3 2 The line crosses the yaxis at y = 3, therefore c = 3. y = 3 2 x Page 23 of 28
24 Example What is the equation of this line? Choose points on the line that are nice and easy to read. m = y 2 y 1 x 2 x 1 = = 4 6 = 2 3 The line crosses the yaxis at y = 1 therefore c = 1. y = 2 3 x Page 24 of 28
25 Exercise 4 Work out the equations of these lines: a. b. c Page 25 of 28
26 5. Quick method to draw a straight line Cartesian coordinates and straight lines booklet If asked to draw a line, given the equation, a quick way is to mark the yintercept and then count along and up/down to get a second point. Example: Draw the line y = 5x + 4 Exercise 5 a. Draw the line y = 2x Page 26 of 28
27 b. y = 4x 3 c. y = 4 3 x 1 d. y = 2 x + 4 c. y = 3x Page 27 of 28
28 Solutions Exercise 1a x = 4 x = 0 Exercise 1b y = 0 y = 3 Exercise 2 a. (0, 5), (1, 2), (2, 1) b. (0, 4), (1, 5), (2, 6) c. (0, 6), (1, 1), (2, 4) d. (0, 3), (1, 3.5), (2, 4) e. (0, 6), (1, 5), (2, 4) f. (0, 3), (1, 1), (2, 1) Exercise 3b Gradient AB = 1 Gradient CD = 1/3 Gradient EF = 5 Gradient GH = 1/2 Exercise 4 a. y = 2x + 3 b. y = 1 2 x + 2 c. y = 3x Page 28 of 28
The 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 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 informationGraphing Linear Equations in Two Variables
Math 123 Section 3.2  Graphing Linear Equations Using Intercepts  Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
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 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 informationTemperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.
Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is
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 informationWhy should we learn this? One realworld 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 realworld connection is to find the rate
More informationChapter 1 Linear Equations and Graphs
Chapter 1 Linear Equations and Graphs Section 1.1  Linear Equations and Inequalities Objectives: The student will be able to solve linear equations. The student will be able to solve linear inequalities.
More informationExample 1. Rise 4. Run 6. 2 3 Our Solution
. Graphing  Slope Objective: Find the slope of a line given a graph or two points. As we graph lines, we will want to be able to identify different properties of the lines we graph. One of the most important
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 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 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 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 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 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 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 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 informationA synonym is a word that has the same or almost the same definition of
SlopeIntercept Form Determining the Rate of Change and yintercept Learning Goals In this lesson, you will: Graph lines using the slope and yintercept. Calculate the yintercept of a line when given
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 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 informationLesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1
Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve
More informationLesson Plan Mine Shaft Grade 8 Slope
CCSSM: Grade 8 DOMAIN: Expressions and Equations Cluster: Understand the connections between proportional relationships, lines, and linear equations. Standard: 8.EE.5: Graph proportional relationships,
More informationYear 12 Pure Mathematics. C1 Coordinate Geometry 1. Edexcel Examination Board (UK)
Year 1 Pure Mathematics C1 Coordinate Geometry 1 Edexcel Examination Board (UK) Book used with this handout is Heinemann Modular Mathematics for Edexcel AS and ALevel, Core Mathematics 1 (004 edition).
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 informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationM 1310 4.1 Polynomial Functions 1
M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More information4.4 Concavity and Curve Sketching
Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at
More informationThe 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 informationGRAPH OF A RATIONAL FUNCTION
GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find
More informationGraphing 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 yvalues for a graph. 1. How are the x and yvalues related? What pattern do you see? To enter the
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 informationDetermine If An Equation Represents a Function
Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The
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 informationSection 2.1 Intercepts; Symmetry; Graphing Key Equations
Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the xaxis. 2. The xcoordinate of a point
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
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 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 informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationReview 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 decisionmaking tools
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 informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationBecause the slope is, a slope of 5 would mean that for every 1cm increase in diameter, the circumference would increase by 5cm.
Measurement Lab You will be graphing circumference (cm) vs. diameter (cm) for several different circular objects, and finding the slope of the line of best fit using the CapStone program. Write out or
More informationGraphing Rational Functions
Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function
More informationBasic Graphing Functions for the TI83 and TI84
Basic Graphing Functions for the TI83 and TI84 The primary benefits of the TI83 and TI84 are the abilities to graph functions and to identify properties those functions possess. The purpose of this
More informationOverview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: SlopeIntercept Form
Name Date Linear Functions: SlopeIntercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,
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 informationLesson 4: Solving and Graphing Linear Equations
Lesson 4: Solving and Graphing Linear Equations Selected Content Standards Benchmarks Addressed: A2M Modeling and developing methods for solving equations and inequalities (e.g., using charts, graphs,
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 informationAnswer Key Building Polynomial Functions
Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,
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 information2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.
2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results
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 informationEcon 101: Principles of Microeconomics Fall 2012
Problem 1: Use the following graph to answer the questions. a. From the graph, which good has the price change? Did the price go down or up? What is the fraction of the new price relative to the original
More informationTHE SPRING CONSTANT. Apparatus: A spiral spring, a set of weights, a weight hanger, a balance, a stop watch, and a twometer
THE SPRING CONSTANT Objective: To determine the spring constant of a spiral spring by Hooe s law and by its period of oscillatory motion in response to a weight. Apparatus: A spiral spring, a set of weights,
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 informationPart 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
More informationSimple Regression Theory I 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY I 1 Simple Regression Theory I 2010 Samuel L. Baker Regression analysis lets you use data to explain and predict. A simple regression line drawn through data points In Assignment
More informationSection 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
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 informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More informationMath 113 Review for Exam I
Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationBuilding Polynomial Functions
Building Polynomial Functions NAME 1. What is the equation of the linear function shown to the right? 2. How did you find it? 3. The slope yintercept form of a linear function is y = mx + b. If you ve
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 informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationRational Functions 5.2 & 5.3
Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at
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 informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
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 informationWhat 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
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 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 informationClimbing Stairs. Goals. Launch 4.1 4.1
4.1 Climbing Stairs Goals Introduce students to the concept of slope as the ratio of vertical change to between two points on a line or ratio of rise over run Use slope to sketch a graph of a line with
More information1.2 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 yintercepts of graphs of equations. Use symmetry to sketch graphs
More informationEdExcel 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
More informationUpdates to Graphing with Excel
Updates to Graphing with Excel NCC has recently upgraded to a new version of the Microsoft Office suite of programs. As such, many of the directions in the Biology Student Handbook for how to graph with
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationLinear Equations. 5 Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber
Linear Equations 5 Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI 83
More informationWARM UP EXERCSE. 13 Linear Functions & Straight lines
WARM UP EXERCSE A company makes and sells inline skates. The pricedemand function is p (x) = 190 0.013(x 10) 2. Describe how the graph of function p can be obtained from one of the library functions.
More informationTI83/84 Plus Graphing Calculator Worksheet #2
TI83/8 Plus Graphing Calculator Worksheet #2 The graphing calculator is set in the following, MODE, and Y, settings. Resetting your calculator brings it back to these original settings. MODE Y Note that
More informationActivity 6 Graphing Linear Equations
Activity 6 Graphing Linear Equations TEACHER NOTES Topic Area: Algebra NCTM Standard: Represent and analyze mathematical situations and structures using algebraic symbols Objective: The student will be
More informationYou might be surprised to know that the word Tshirt wasn t really used until
Hot Shirts Using Tables, Graphs, and Equations, Part 2 Learning Goals In this lesson, you will: Use different methods to represent a problem situation. Estimate values of expressions that involve decimals.
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 informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationWarm Up. Write an equation given the slope and yintercept. Write an equation of the line shown.
Warm Up Write an equation given the slope and yintercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and yintercept From the graph, you can see that the slope is
More informationSituation: Dividing Linear Expressions
Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product
More informationCreating Multiple Baseline (MB) SingleSubject Design Graphs in Microsoft Excel 2007
1 Creating Multiple Baseline (MB) SingleSubject Design Graphs in Microsoft Excel 2007 2 Step 1: Set up the Variables Multiple Baseline Design (MB) Xaxis 1st AB 2nd AB 3rd AB Use the top cells to Label
More informationMotion Graphs. Plotting distance against time can tell you a lot about motion. Let's look at the axes:
Motion Graphs 1 Name Motion Graphs Describing the motion of an object is occasionally hard to do with words. Sometimes graphs help make motion easier to picture, and therefore understand. Remember: Motion
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1100 Overview: This unit models realworld situations by using one and twovariable linear equations. This unit will further expand upon pervious
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 informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
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 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 informationExperiment 3 ~ Ohm's Law, Measurement of Voltage, Current and Resistance
Experiment 3 ~ Ohm's Law, Measurement of Voltage, Current and Resistance Objective: In this experiment you will learn to use the multimeter to measure voltage, current and resistance. Equipment: Bread
More informationGraphing Motion. Every Picture Tells A Story
Graphing Motion Every Picture Tells A Story Read and interpret motion graphs Construct and draw motion graphs Determine speed, velocity and accleration from motion graphs If you make a graph by hand it
More information