Chapter 8 Graphs and Functions:


 Dwain Reeves
 2 years ago
 Views:
Transcription
1 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 to locate points on it. See diagram below The vertical axis is the y axis and the horizontal axis is the x axis. Any point in this plane has a set of numbers called an ordered pair, which is a set of coordinates that locate a point in the plane. In the diagram above the top right point has coordinates (x=5,y=7) i.e. 5 units on the x axis and 7 units on the y axis. It is in quadrant 1. However because we always write the x coordinate first we just write (5,7) instead of (x=5,y=7). The top right quadrant is quadrant 1, then going anticlockwise we get the other 3 quadrants. For example the bottom left point has coordinates (5,6) i.e. 5 units on the x axis and 6 units on the y axis. It is in quadrant 3. E8.1 Give the coordinates of the point in the (i) second quadrant (ii) fourth quadrant A8.1 The point in the (i) second quadrant is (5,4). (ii) fourth quadrant is (3,5). 8.2 The point (3,2) is located 3 units to the right and 2 units below the origin. It is in the 4 th (bottom right) quadrant. E8.2 Complete the Table below on the location of a point relative to the origin. The first row has been done for you. Location Coordinates Quadrant 3 to the right and 2 (3,2) 4 th below origin 87
2 (5, 2) At the origin Junction of all 4 (, 4) Boundary Between 1 st 4 to the left and 2 above. (2,0) Plot these points on the axes above. and 2 nd. (5, ) Boundary between 1 st and 4 th A8.2 Location Coordinates Quadrant 3 to the right and 2 (3,2) 4 th directly below origin 5 to the left, 2 below (5, 2) 2 nd At the origin (0,0) Junction of all 4 4 above the origin (0, 4) Boundary Between 1 st and 2 nd. 2 to the left (2,0) Between 2 nd And 3 rd. 4 to the left and 2 (4,2) 2 nd above. 5 to the right (5,0 ) Boundary between 1 st and 4 th Equations with infinite solutions 8.3 We know that x + 3 = 5 is an equation that we can solve by inspection. There is only one solution, x = 2. Now consider the equation, x + y =10. It has two unknowns and we can produce many solutions by inspection such as x = 5 and y = 5, x = 0 and y = 10, x = 1 and y = 9. Also solutions with negative values such as x = 4 and y = 14 etc. Why does the first equation have one solution and the second equation have an infinite number? The reason is that the first equation has just one unknown, x whereas the second equation has two unknowns x and y and yet we have just one piece of information. E8.3 How many solutions do the following equations have? 88
3 (i) 2x + 7 = 9. Identify it. (ii) 5x + 6y = 10 Can you identify some of them? A8.3 (i) 2x + 7 = 9. Has 1 solution, x = 1. (ii) 5x + 2y = 10. Has an infinite number of solutions e.g. (2,0), (0,5), (1,2.5) etc. Can you identify some of them? 8.4 Since 1 equation with two unknowns has an infinite number of solutions we can produce a graph by plotting the set of solutions on a set of Cartesian axes. Here are some solutions to the equation: y = 2x + 3 (0,3) because y = 3 when x = 0. (2,7) because y = 7 when x = 2. (3, 3) because when x = 3, y = 3. E8.4 Complete the following coordinates for some of the solutions of the equation: y = 2x + 3 (2, ) ; (3, ) ; (1, ) ; (1, ) ; (, 4) ; (, 3) ; (0, ) Plot the points on the axes given below A8.4 Complete the following coordinates for some of the solutions of the equation: y = 2x + 3 (2,1) ; (3,3) ; (1,1) ; (1,5) ; (0.5, 4) ; (3, 3) ; (0,3) 89
4 Linear graphs 8.5 What is the graph of the entire solution set? It is a straight line as shown below: Graph of y =2x + 3 E8.5 Examine the graph and identify the points where the graph crosses the y axis and the x axis. Graph crosses y axis at (0,3) and cross the x axis at (1.5,0) In graphing the equation, y = 2x + 3 let s see what the significance of the coefficient of 2 and and the constant term 3 is. Note the 2 is the coefficient of the x term (linear term) and the 3 is the constant term. First consider the constant term, 3. Notice that when x = 0 then y = 3. Hence (0,3) is a point on the graph. Which quadrant is it in? It is on the boundary between the 1 st and 2 nd quadrants i.e. it lies on the y axis. We call this point the y intercept. So we say either the y intercept is 3 or we say the y intercept is the point (0,3). We say that in the graph y = mx + c, c is the y intercept. E8.6 What is the y intercept of the following graphs? (i) y = 2x + 4 (ii) y = 2x 3 (iii) y = 2x? A8.6 The y intercept of (i) y = 2x + 4 is 4 i.e. the point (0,4). (ii) y = 2x 3 is 3 i.e. the point (0,3). (iii) y = 2x is 0 i.e. the point (0,0). 8.7 Now what is the significance of the coefficient of x of the RHS (2 in this case) in the graph of y = 2x + 3? 90
5 Let us look at two random points on the graph, say point A at (1,5) and point B at (3,9) We can by looking at the x coordinates of the two points determine that B is 2 units to the right of A. We write x = x value of B x value of A. = 3 1 =2 We can by looking at the y coordinates of the two points determine that B is 4 units above A. We write y = y value of B y value of A = 9 5 = 4 So in going from A to B the graph is climbing 4 units up for every 2 units to the right. So we say the graph has a slope or gradient of 4 in 2 which we write as y 4 gradient 2. x 2 So the coefficient of the x term in the equation y = 2x + 3 is the gradient. Summarise: constant term = 3 is the y intercept coefficient of x term = the slope or gradient of the line given by y gradient x E8.7 Complete the Table below: The first row has been done for you. A8.7 Equation of line Slope Y intercept y = 2x y = 3x y = 4x 04 y = 5 2x y = Equation of line Slope Y intercept y = 2x y = 4x y = 3x y = 2x y = 4x 4 0 y =
6 y = 5 2x 2 5 y = y = 5x What is the equation of the line that passes through the points (0,3) and has a slope of 2? We know that in general the equation is of the form: y = mx + c, where m is the slope (coefficient of x term) and y intercept is c (the constant term). Since the slope is 2 (i.e. the coefficient of x is 2) y = mx + c becomes y = 2x + c. Since the graph passes through (0,3) we know the y intercept is 3; So c is 3 and y = 2x + c becomes y = 2x + 3. So the equation is y = 2x + 3 OR y = 3 2x. E8.8 What is the equation of the line that passes through the point i. (0, 1) and has a slope of 2 ii. (0, 2) and has a slope of 3 iii. (0,0) and has a slope of 1 A8.8 i. y = 2x 1 ii. y = 3x + 2 iii. y = x 8.9 What is the equation of the line that passes through the points (0,3) and (2,5)? We know that in general the equation is of the form: y = mx + c, where m is the slope (coefficient of x term) and y intercept is c (the constant term). Since the graph passes through (0,3), which is on the y axis, we know the y intercept is 3; so c is 3, so y = mx + c becomes y = mx + 3. The point (2,5) is 2 units to the right (2 0) and 2 units above (5 3) the point (0,3). y gradient 1. x So the slope is 2 in 2 i.e. m=1 92
7 so y = mx + 3 becomes y = x + 3. So the equation of the line is y = x + 3 E8.9 What is the equation of the line that passes through the points (0,1) and (1,4)? A8.9 (0,1) tells us that the y intercept is 1 so we can write y = mx 1. (0,1) and (1,4) tells us that the gradient is 5. So y = 5x Now consider the line that passes through the points (2,4) and (3,5). We can t determine the slope or the y intercept at sight easily: So we proceed as follows: gradient y x So m= 9/5 in the equation y = mx + c. 9x So write: y  c. 5 Now remember (3,5) is (x=3,y=5). So substitute in the equation above to get: 9(3) c 5 9(3) c x 9x 2 So our line y  c. becomes y E8.10 Find the equation of the line that passes through (3, 6) and (2,1) A8.10 gradient y x So m= 1 in the equation y = mx + c. 93
8 So write: y  x c. Now remember (2,1) is (x=1,y=1). So substitute in the equation above to get: 21 c c 3 So our line y  x c. becomes y 3 x Consider the line y = 2x + 3 again. It has a slope of 2 and the y intercept is 3. What is the x intercept? We know that at the x intercept y=0. If we substitute y=0 in y = 2x + 3 we get 0 = 2x We can solve this equation to get x So the x intercept is  and the point of interception on the x axis is 2 3 (  2, 0) Check by seeing the graph in box 8.5. E8.11 For the graph y = 4 3x what is i. the slope ii. y intercept iii. x intercept? A8.11 y = 4 3x i. slope = 3 ii. y intercept = 4 iii. 0 = 4 3x x intercept = 4/3 Graphs of Quadratic functions 8.12 A quadratic function is function of the form : y = ax 2 + bx + c where a,b,c are constants. There are three terms in the function the first is a term in x 2 (called the quadratic term), the second is a term in x (called a linear term) and the third is a constant term. This is a 2 nd degree function, whereas linear functions are first degree functions. The simplest quadratic is y = x 2 i.e. no linear term and no constant term. We can complete the ordered pairs using the x values given below: (5,25), (4,16), (3,9), (2,4), (1,1), (0,0), (1,1), (2,4), (3,9), (4,16), (5,25). Note the x values range from 5 to +5 but the y values are all positive 94
9 because the square of a number is always +ve. E8.12 Complete the following points on this quadratic: (,25), (,0), (,25) so that the points are distinct (i.e. separate) A8.12 ( 5,25), (0,0), (5,25) 8.13 The graph is plotted below E8.13 What is the y intercept? What is the y intercept? the x intercept? What is the axis of symmetry? What is the minimum point? What is the minimum y value? A8.13 y intercept is 0, the x intercept is 0, the axis of symmetry is x = 0, the minimum point is (0,0), the minimum value is Now suppose we want to sketch (rather than draw) the quadratic, y = x 2 2x 15 We can find the y intercept by putting x = 0 in y = x 2 2x 15 to get y = 15. So the graph crosses the y axis at (0,15). Now to find the x intercepts we put y = 0 to get: 0 = x 2 2x 15. Write this as x 2 2x 15 = 0 and solve viz. (x 5)(x + 3) = 0 So x = 3 and x = 5. The graph crosses the x axis at (3,0) and (5,0). The axis of symmetry is a vertical line bisecting the line joining (3,0) and (5,0). 95
10 How do we find it? The middle value of any two numbers is its arithmetic average. (3 5) x The average of x = 3 and x = 5 is thus 2 x 1 The axis of symmetry is x = 1. Now we get the minimum y value using the fact that minimum y value is on the axis of symmetry, x = 1. So putting x=1 in y = x 2 2x 15 we get y min = 1 2 2(1) 15 = 17. So the minimum point of the graph is (1,17). We can now sketch our graph by putting all this information together: Point where graph crosses y axis. (0,15) Two axis points where graph (3,0) and (5,0) crosses x axis. Minimum point on the axis of (1,17) symmetry The sketch is confirmed by this drawing E8.14 Sketch the graph of the quadratic function y = x 2 2x 8 1. Find the y intercept by putting x = 0 in y = x 2 2x 8 to get y =. 2. Find the x intercepts by putting put y = 0 to get:. = x 2 2x 8 and solve viz. (x )(x ) = 0 So x =.. and x =.. 96
11 The two points of interception on the x axis are (,.) and (, ) The axis of symmetry is a vertical line bisecting the line joining (., ) and (, ) The middle x value is its arithmetic average. The average of x = and x = is thus x =.. The axis of symmetry is x =... The minimum y value is on the axis of symmetry, x =.. So putting x=. in y = x 2 2x 8 we get y min =. So the minimum point of the graph is (.,.). We can now sketch our graph by putting all this information together: Point of interception on y axis (, ) Two points of interception on x (, ) and (, ) axis Minimum point on the axis of (, ) symmetry Sketch: A8.14 Sketch the graph of the quadratic function y = x 2 2x 8 1. Find the y intercept by putting x = 0 in y = x 2 2x 8 to get y = 8. So the y intercept is the point (0,8). 2. Find the x intercepts by putting put y = 0 to get: 0 = x 2 2x 8 and solve viz. (x 4)(x + 2) = 0 So x = 4 and x = 2 are the x intercepts. 97
12 The two points of interception on the x axis are (4,0) and (2,0). The axis of symmetry is a vertical line bisecting the line joining (4,0) and (2,0). The middle x value is its arithmetic average. The average of x = 4 and x = 2 is thus x = (4 2)/2 = 1 i.e. the axis of symmetry is the vertical line, x = 1 The minimum y value is on the axis of symmetry, x = 1. So putting x = 1 in y = x 2 2x 8 we get y min = (1) 2 2(1) 8 = = 9. So the minimum point of the graph is (1,9). We can now sketch our graph by putting all this information together: Point of interception on y axis (0,8) Two points of interception on x (2,0) and (0,4) axis Minimum point on the axis of (1,9) symmetry Sketch: 98
13 99
14 Exercise 8 1. Find the equation of the straight line with a gradient of 2 passing thru the point ( 1,1). 2. A line passes thru the points (0,3)and (3,0). Find Deltax, Deltay, the slope, and the equation of the line. 3. A variable P changes from 20 to 30. This causes a dependent variable Q to change from 100 to 25. Find DeltaP, deltaq, the slope, the proportional change in P, proportional change in Q and the ratio of the proportional change in P to the proportional change in Q. 4. (i) (ii) (iii) (iv) Sketch the graph of the straight line y = 2 3x what is the gradient? what is the y intercept? what is the x intercept? 5 Sketch the quadratic y = x 2 2x 8 100
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 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 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 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 informationLines and Linear Equations. Slopes
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 informationOrdered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.
Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value
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 informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
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 informationMATH 105: Finite Mathematics 11: Rectangular Coordinates, Lines
MATH 105: Finite Mathematics 11: Rectangular Coordinates, Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Rectangular Coordinate System 2 Graphing Lines 3 The Equation of
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 information2.7. The straight line. Introduction. Prerequisites. Learning Outcomes. Learning Style
The straight line 2.7 Introduction Probably the most important function and graph that you will use are those associated with the straight line. A large number of relationships between engineering variables
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 informationIn this section, we ll review plotting points, slope of a line and different forms of an equation of a line.
Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:
More informationKey Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function)
Outcome R3 Quadratic Functions McGrawHill 3.1, 3.2 Key Terms: Quadratic function Parabola Vertex (of a parabola) Minimum value Maximum value Axis of symmetry Vertex form (of a quadratic function) Standard
More informationLinear Equations and Graphs
2.12.4 Linear Equations and Graphs Coordinate Plane Quadrants  The xaxis and yaxis form 4 "areas" known as quadrants. 1. I  The first quadrant has positive x and positive y points. 2. II  The second
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 informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More informationChapter 12. The Straight Line
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
More informationHelpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize
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 informationPreCalculus III Linear Functions and Quadratic Functions
Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3
More informationPositive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin.
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
More informationSection 2.3. Learning Objectives. Graphing Quadratic Functions
Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic
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 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 informationClass 9 Coordinate Geometry
ID : in9coordinategeometry [1] Class 9 Coordinate Geometry For more such worksheets visit www.edugain.com Answer t he quest ions (1) Find the coordinates of the point shown in the picture. (2) Find
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 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 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 informationPreAP Algebra 2 Unit 3 Lesson 1 Quadratic Functions
Unit 3 Lesson 1 Quadratic Functions Objectives: The students will be able to Identify and sketch the quadratic parent function Identify characteristics including vertex, axis of symmetry, xintercept,
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 information10.1 NotesGraphing Quadratics
Name: Period: 10.1 NotesGraphing Quadratics Section 1: Identifying the vertex (minimum/maximum), the axis of symmetry, and the roots (zeros): State the maximum or minimum point (vertex), the axis of symmetry,
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 informationMathematics Chapter 8 and 10 Test Summary 10M2
Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus
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 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 information3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes
Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
More informationAlgebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )
Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.11.4, 1.61.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order
More informationElementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination
Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used
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 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 informationREVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011
REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working
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 informationEquations of Lines Derivations
Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated
More informationBeginning of the Semester ToDo List
Beginning of the Semester ToDo List Set up your account at https://casa.uh.edu/ Read the Math 13xx Departmental Course Policies Take Course Policies Quiz until your score is 100%. You can find it on the
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationCorrelation 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)
More informationSlopeIntercept Form of a Linear Equation Examples
SlopeIntercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the yintercept of AB. Suppose you want to find an equation
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationwith "a", "b" and "c" representing real numbers, and "a" is not equal to zero.
3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,
More informationIntroduction to Modular Arithmetic, the rings Z 6 and Z 7
Introduction to Modular Arithmetic, the rings Z 6 and Z 7 The main objective of this discussion is to learn modular arithmetic. We do this by building two systems using modular arithmetic and then by solving
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 informationPacket: Lines (Part 1) Standards covered:
Packet: Lines (Part 1) Standards covered: *(2)MA.912.A.3.8 Graph a line given any of the following information: a table of values, the x and y intercepts, two points, the slope and a point, the equation
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationSection 3.4 The Slope Intercept Form: y = mx + b
SlopeIntercept Form: y = mx + b, where m is the slope and b is the yintercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear
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 informationMathematics Common Core Cluster. Mathematics Common Core Standard. Domain
Mathematics Common Core Domain Mathematics Common Core Cluster Mathematics Common Core Standard Number System Know that there are numbers that are not rational, and approximate them by rational numbers.
More informationSection 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.
Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:
More informationFunctions and straight line graphs. Jackie Nicholas
Mathematics Learning Centre Functions and straight line graphs Jackie Nicholas c 004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Functions and Straight Line Graphs Functions
More informationThe Parabola and the Circle
The Parabola and the Circle The following are several terms and definitions to aid in the understanding of parabolas. 1.) Parabola  A parabola is the set of all points (h, k) that are equidistant from
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 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 informationTitle: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under
Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Activity and follow all steps for each problem exactly
More information3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes
Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
More informationLinear Approximations ACADEMIC RESOURCE CENTER
Linear Approximations ACADEMIC RESOURCE CENTER Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations
More information9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots.
9.1 Solving Quadratic Equations by Finding Square Roots 1. Evaluate and approximate square roots. 2. Solve a quadratic equation by finding square roots. Key Terms Square Root Radicand Perfect Squares Irrational
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 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 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 informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationGraphing Quadratic Functions
Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x
More informationAlgebra. Indiana Standards 1 ST 6 WEEKS
Chapter 1 Lessons Indiana Standards  11 Variables and Expressions  12 Order of Operations and Evaluating Expressions  13 Real Numbers and the Number Line  14 Properties of Real Numbers  15 Adding
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 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 informationIntroduction to Finite Systems: Z 6 and Z 7
Introduction to : Z 6 and Z 7 The main objective of this discussion is to learn more about solving linear and quadratic equations. The reader is no doubt familiar with techniques for solving these equations
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 informationChapter 10: Analytic Geometry
10.1 Parabolas Chapter 10: Analytic Geometry We ve looked at parabolas before when talking about the graphs of quadratic functions. In this section, parabolas are discussed from a geometrical viewpoint.
More informationLINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH
59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More informationTEKS 2A.7.A Quadratic and square root functions: connect between the y = ax 2 + bx + c and the y = a (x  h) 2 + k symbolic representations.
Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum
More informationGradient  Activity 1 Gradient from the origin.
Name: Class: p 31 Maths Helper Plus Resource Set 1. Copyright 2002 Bruce A. Vaughan, Teachers Choice Software Gradient  Activity 1 Gradient from the origin. 1) On the graph below, there is a line ruled
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 informationPortable Assisted Study Sequence ALGEBRA IIA
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
More informationSection 2.2 Equations of Lines
Section 2.2 Equations of 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
More informationChapter 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
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 information2015 Junior Certificate Higher Level Official Sample Paper 1
2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list
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 informationPreAP Algebra 2 Lesson 25 Graphing linear inequalities & systems of inequalities
Lesson 25 Graphing linear inequalities & systems of inequalities Objectives: The students will be able to  graph linear functions in slopeintercept and standard form, as well as vertical and horizontal
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 information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationThe xintercepts of the graph are the xvalues for the points where the graph intersects the xaxis. A parabola may have one, two, or no xintercepts.
Chapter 101 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax
More informationSection 3.2. Graphing linear equations
Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:
More informationLesson 17: Graphing the Logarithm Function
Lesson 17 Name Date Lesson 17: Graphing the Logarithm Function Exit Ticket Graph the function () = log () without using a calculator, and identify its key features. Lesson 17: Graphing the Logarithm Function
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 informationChapter 8. Quadratic Equations and Functions
Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property
More informationMath 181 Spring 2007 HW 1 Corrected
Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the xaxis (horizontal axis)
More informationNorwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction
1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K7. Students must demonstrate
More informationChapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms
Contents 3 Algebra 3 3.1 Rational expressions................................ 3 3.1.1 BAa1: Reduce to lowest terms...................... 3 3.1. BAa: Add, subtract, multiply, and divide............... 5
More information