Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometry (3ed, Addison Wesley, 2007) 58 Chapter 1 Graphs, Functions, and Models
|
|
- Regina Fleming
- 6 years ago
- Views:
Transcription
1 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 8 Chapter Graphs, Functions, and Models.. Introduction Polnomial to Functions Graphing and Modeling Plot points. Determine whether an ordered pair is a solution of an equation. Graph equations. Find the distance between two points in the plane and find the midpoint of a segment. Find an equation of a circle with a given center and radius, and given an equation of a circle, find the center and the radius. Graph equations of circles. Graphs Graphs provide a means of displaing, interpreting, and analzing data in a visual format. It is not uncommon to open a newspaper or magazine and encounter graphs. Eamples of bar, circle, and line graphs are shown below. Americans Without Health Insurance More than 7 million Americans do not have health insurance. Percentage of uninsured b age % % 8% 7% 6% Source: U.S. Bureau of the Census Land Mines in Afghanistan Where land mines are found in Afghanistan: Agricultural land Roads 6% Residential areas 7% 4% % % Irrigation sstems 6% Other Grazing land Sources: United Nations; Landmine Monitor; howstuffworks.com Indianapolis Life 00 Festival Mini-Marathon Number of participants Year II I (, +) (+, +) (, ) (0, 0) III IV (, ) (+, ) Man real-world situations can be modeled, or described mathematicall, using equations in which two variables appear. We use a plane to graph a pair of numbers. To locate points on a plane, we use two perpendicular number lines, called aes, which intersect at 0, 0. We call this point the origin. The horizontal ais is called the -ais, and the vertical ais is called the -ais.(other variables, such as a and b, can also be used.) The aes divide the plane into four regions, called quadrants, denoted b Roman numerals and numbered counterclockwise from the upper right. Arrows show the positive direction of each ais. Each point, in the plane is described b an ordered pair. The first number,, indicates the point s horizontal location with respect to the -ais, and the second number,, indicates the point s vertical location with
2 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 9 respect to the -ais. We call the first coordinate, -coordinate, or abscissa. We call the second coordinate, -coordinate, or ordinate. Such a representation is called the Cartesian coordinate sstem in honor of the French mathematician and philosopher René Descartes (96 60). In the first quadrant, both coordinates of a point are positive. In the second quadrant, the first coordinate is negative and the second is positive. In the third quadrant, both coordinates are negative, and in the fourth quadrant, the first coordinate is positive and the second is negative. (, ) (0, 4) 4 (, 4) (4, ) (, 0) (0, 0) 4 4 ( 4, ) (, 4) 4 EXAMPLE Graph and label the points,, 4,,, 4, 4,,, 4, 0, 4,, 0,and 0, 0. Solution To graph or plot,,we note that the -coordinate,, tells us to move from the origin units to the left of the -ais. Then we move units up from the -ais.* To graph the other points, we proceed in a similar manner. (See the graph at left.) Note that the point 4, is different from the point, 4. Solutions of Equations Equations in two variables, like 8, have solutions, that are ordered pairs such that when the first coordinate is substituted for and the second coordinate is substituted for, the result is a true equation. The first coordinate in an ordered pair generall represents the variable that occurs first alphabeticall. EXAMPLE Determine whether each ordered pair is a solution of 8. a), 7 b), 4 Solution We substitute the ordered pair into the equation and determine whether the resulting equation is true. a) 8 7? 8 8 We substitute for and 7 for (alphabetical order). FALSE The equation 8 is false, so, 7 is not a solution. b) 8 4? We substitute for and 4 for. TRUE The equation 8 8 is true, so, 4 is a solution. *We first saw notation such as, in Section R.. There the notation represented an open interval. Here the notation represents an ordered pair. The contet in which the notation appears usuall makes the meaning clear.
3 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 60 Chapter Graphs, Functions, and Models We can also perform these substitutions on a graphing calculator. When we substitute for and 7 for,we get. Since 8,, 7 is not a solution of the equation. When we substitute for and 4 for, we get 8, so, 4 is a solution. ( ) Graphs of Equations The equation considered in Eample actuall has an infinite number of solutions. Since we cannot list all the solutions, we will make a drawing, called a graph, that represents them. Shown at left are some suggestions for drawing graphs. To Graph an Equation To graph an equation is to make a drawing that represents the solutions of that equation. Graphs of equations of the tpe A B C are straight lines. Man such equations can be graphed convenientl using intercepts. The -intercept of the graph of an equation is the point at which the graph crosses the -ais. The -intercept is the point at which the graph crosses the -ais. We know from geometr that onl one line can be drawn through two given points. Thus, if we know the intercepts, we can graph the line. To ensure that a computational error has not been made, it is a good idea to calculate and plot a third point as a check. - and -Intercepts An -intercept is a point a,0. To find a,let 0 and solve for. A -intercept is a point 0, b. To find b,let 0 and solve for. EXAMPLE Graph: 8. Solution The graph is a line. To find ordered pairs that are solutions of this equation, we can replace either or with an number and then solve for the other variable. In this case, it is convenient to find the intercepts of the graph. For instance, if is replaced with 0, then Dividing b
4 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 6 Suggestions for Drawing Graphs. Calculate solutions and list the ordered pairs in a table.. Use graph paper.. Draw aes and label them with the variables. 4. Use arrows on the aes to indicate positive directions.. Scale the aes; that is, label the tick marks on the aes. Consider the ordered pairs found in step () when choosing the scale. 6. Plot the ordered pairs, look for patterns, and complete the graph. Label the graph with the equation being graphed. Thus, 0, 6 is a solution. It is the -intercept of the graph. If is replaced with 0, then Dividing b Thus, 9, 0 is a solution. It is the -intercept of the graph. We find a third solution as a check. If is replaced with, then Subtracting 8. Dividing b Thus,, 8 is a solution. We list the solutions in a table and then plot the points. Note that the points appear to lie on a straight line., 0 6 0, , 0 8, 8 -intercept (0, 6) intercept 4 Were we to graph additional solutions of 8, the would be on the same straight line. Thus, to complete the graph, we use a straightedge to draw a line, as shown in the figure. This line represents all solutions of the equation. Ever point on the line represents a solution; ever solution is represented b a point on the line. When graphing some equations, it is easier to first solve for and then find ordered pairs. We can use the addition and multiplication principles to solve for. 4 8 (, h) (9, 0) GCM EXAMPLE 4 Graph:. Solution We first solve for : Subtracting on both sides. Multipling b on both sides
5 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 6 Chapter Graphs, Functions, and Models B choosing multiples of for, we can avoid fraction values when calculating.for eample, if we choose for,we get. The following table lists a few more points. We plot the points and draw the graph.,, 0 0,, 4 (0, ) (, ) 4 4 (, ) 4, or Plot Plot Plot \Y (/)X \Y \Y \Y4 \Y \Y6 \Y7 In the equation, the value of depends on the value chosen for, so is said to be the independent variable and the dependent variable. We can graph an equation on a graphing calculator. Man calculators require an equation to be entered in the form. In such a case, if the equation is not initiall given in this form, it must be solved for before it is entered. For the equation in Eample 4, we enter on the equation-editor, or, screen in the form, as shown in the window at left. Net, we determine the portion of the -plane that will appear on the calculator s screen. That portion of the plane is called the viewing window. The notation used in this tet to denote a window setting consists of four numbers L,R,B,T, which represent the Left and Right endpoints of the -ais and the Bottom and Top endpoints of the -ais, respectivel. The window with the settings,,, is the standard viewing window. On some graphing calculators, the standard window can be selected quickl using the ZSTANDARD feature from the ZOOM menu. WINDOW Xmin Xma Xscl Ymin Yma Yscl Xres Xmin and Xma are used to set the left and right endpoints of the -ais, respectivel; Ymin and Yma are used to set the bottom and top endpoints of the -ais. The settings Xscl and Yscl give the scales for the aes. For eample, Xscl and Yscl means that there is unit between tick marks on each of the aes.
6 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 6 After entering the equation and choosing a viewing window, we can then draw the graph shown at left. GCM EXAMPLE Graph: 9. Solution Note that this is not of the form A B C, thus it is not linear. We make a table of values, plot enough points to obtain an idea of the shape of the curve, and connect them with a smooth curve. It is important to scale the aes to include most of the ordered pairs listed in the table. Here it is appropriate to use a larger scale on the -ais than on the -ais. 9, 4, 4, 0 0, 6, 6 4 4,,, 4, Select values for. Compute values for. Plot Plot Plot \Y X 9X \Y \Y \Y4 \Y \Y6 \Y7 Figure TABLE SETUP TblStart Tbl Indpnt: Auto Ask Depend: Auto Ask Figure A graphing calculator can be used to create a table of ordered pairs that are solutions of an equation. For the equation in Eample, 9,we first enter the equation on the equation-editor screen (see Fig. ). Then we set up a table in AUTO mode b designating a value for TBLSTART and a value for TBL. The calculator will produce a table starting with the value of TBLSTART and continuing b adding TBL to suppl succeeding -values. For the equation 9, we let TBLSTART and TBL (see Fig. ). We can scroll up and down in the table to find values other than those shown in Fig.. We can also graph this equation on the graphing calculator, as shown in Fig X 0 X Y Figure Figure 4
7 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) Section. Introduction to Graphing 7. Eercise Set Graph and label the given points b hand.. 4, 0,,,, 4, 0,,,., 4, 4,,, 0,, 4, 4, 0.,,,,,,,, 0, 4. 4, 0, 4,,,,, 0,, Epress the data pictured in the graph as ordered pairs, letting the first coordinate represent the ear and the second coordinate the amount.. 6. Consumption of Bottled Water (U.S.) Number of gallons (in billions) Gift Card Sales (U.S.) Sales (in billions) $ $9 999 Source: Bain and Co. 4.6 $ $ 00.4 $ $4 00 estimated 6.4 Use substitution to determine whether the given ordered pairs are solutions of the given equation. 7.,, 0, ; Yes; no 8.,,, ; Yes; no 9.,,, 4 ; 6 4 Yes; no..,.6,, 0 ; 9 No; es., 0,, 4 ; a b No; es. 0,,, ; m 4n 6 Yes; es. 0.7,.7,, ; No; es 4., 4, 4, ; 70 No; es Use a graphing calculator to create a table of values with TBLSTART and TBL. Then graph the equation b hand In Eercises 7 40, use a graphing calculator to match the equation with one of the graphs (a) (d), which follow. a) b) Source: Beverage Marketing Corp. Answers to Eercises 6 and 6 can be found on p. IA-.
8 Beecher J.A, Penna J.A., Bittinger M.L. Algebra and Trigonometr (ed, Addison Wesle, 007) 7 Chapter Graphs, Functions, and Models c) 7. (b) 8. 6 (d) 9. (a) (c) Use a graphing calculator to graph the equation in the standard window Graph the equation in the standard window and in the given window. Determine which window better shows the shape of the graph and the - and -intercepts.. 6 4, 4, 4, 4 Standard window d) 6. 4,,, 0,with Xscl and Yscl,,, ,, 0., 0.,with Xscl 0. and Yscl 0.,, 0., ,,, Standard window Find the distance between the pair of points. Give an eact answer and, where appropriate, an approimation to three decimal places. 4, , 6 and, 9 60., 7 and,, , and 9, 6. 4, 7 and, 4, , , and., , and,, 4 6.,4 and, ,. and 8., , and 6,0 4 6, , and 0, 7 7, , 0 and a, b 70. r, s and r, s a b r s 7. The points, and 9, 4 are the endpoints of the diameter of a circle. Find the length of the radius of the circle The point 0, is on a circle that has center,. Find the length of the diameter of the circle. The converse of the Pthagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pthagorean theorem to determine whether the set of points could be vertices of a right triangle. 7. 4,, 6,,and 8, Yes 74.,,,,and 6, 9 Yes 7. 4,, 0,,and, 4 No 76. The points, 4,,,,, and 0, 7 are vertices of a quadrilateral. Show that the quadrilateral is a rectangle. (Hint: Show that the quadrilateral s opposite sides are the same length and that the two diagonals are the same length.) Find the midpoint of the segment having the given endpoints , 9 and, 4, , and 9, 8, ,.8 and.8, , ,.7 and 4.8, 0..,. 8. 6, and 6, 8 6, 8., and, 0, 0 8. and, 6, 4, and, 4 9, 4 4, , and,4, 7 86., and, 7 0, 87. Graph the rectangle described in Eercise 76. Then determine the coordinates of the midpoint of each Answers to Eercises 4 4 and 76 can be found on pp. IA- and IA-.
1.2 Visualizing and Graphing Data
6360_ch01pp001-075.qd 10/16/08 4:8 PM Page 1 1 CHAPTER 1 Introduction to Functions and Graphs 9. Volume of a Cone The volume V of a cone is given b V = 1 3 pr h, where r is its radius and h is its height.
More informationGraphs and Functions
CHAPTER Graphs and Functions. Graphing Equations. Introduction to Functions. Graphing Linear Functions. The Slope of a Line. Equations of Lines Integrated Review Linear Equations in Two Variables.6 Graphing
More informationLESSON 3.1 INTRODUCTION TO GRAPHING
LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137 OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The -plane b. The -ais and -ais c. The origin d. Ordered
More informationAppendix C: Review of Graphs, Equations, and Inequalities
Appendi C: Review of Graphs, Equations, and Inequalities C. What ou should learn Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points
More information2.1 The ReCTAngUlAR COORdInATe SySTemS And graphs
7 CHAPTER equations ANd inequalities learning ObjeCTIveS In this section ou will: Plot ordered pairs in a Cartesian coordinate sstem. Graph equations b plotting points. Graph equations with a graphing
More informationEquations and Inequalities
Equations and Inequalities Figure CHAPTER OUTLINE. The Rectangular Coordinate Sstems and Graphs. Linear Equations in One Variable. Models and Applications. Comple Numbers. Quadratic Equations.6 Other Tpes
More informationThe Graph of an Equation
60_0P0.qd //0 :6 PM Page CHAPTER P Preparation for Calculus Archive Photos Section P. RENÉ DESCARTES (96 60) Descartes made man contributions to philosoph, science, and mathematics. The idea of representing
More information+ b. From this we can derive the following equations:
A. GEOMETRY REVIEW Pythagorean Theorem (A. p. 58) Hypotenuse c Leg a 9º Leg b The Pythagorean Theorem is a statement about right triangles. A right triangle is one that contains a right angle, that is,
More informationConnecticut Common Core Algebra 1 Curriculum. Professional Development Materials. Unit 4 Linear Functions
Connecticut Common Core Algebra Curriculum Professional Development Materials Unit 4 Linear Functions Contents Activit 4.. What Makes a Function Linear? Activit 4.3. What is Slope? Activit 4.3. Horizontal
More informationSection 9.3: Functions and their Graphs
Section 9.: Functions and their Graphs Graphs provide a wa of displaing, interpreting, and analzing data in a visual format. In man problems, we will consider two variables. Therefore, we will need to
More informationscience. In this course we investigate problems both algebraically and graphically.
Section. Graphs. Graphs Much of algebra is concerned with solving equations. Man algebraic techniques have been developed to provide insights into various sorts of equations and those techniques are essential
More informationP.5 The Cartesian Plane
7_0P0.qp //07 8: AM Page 8 8 Chapter P Prerequisites P. The Cartesian Plane The Cartesian Plane Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real
More informationPartial Fraction Decomposition
Section 7. Partial Fractions 53 Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the eamples that follow. Note
More informationEach point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
More informationUNIT 4 MODELING AND ANALYZING EXPONENTIAL FUNCTIONS Lesson 1: Creating Exponential Equations
Guided Practice Eample 1 If a pendulum swings to 9% of its previous height on each swing and starts out at a height of 6 cm, what is the equation that models this scenario? What is its graph? 1. Read the
More informationLines and Their Slopes
8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation
More informationOn Your Own. Applications. Unit 3. b. AC = 8 units y BD = 16 units C(5, 2)
Applications 1 a. If students use uniform and y scales or ZSquare, they should have a shape that looks like a kite. Using the distance formula, DC = AD = 1 and BC = BA = 0. b. AC = 8 units y BD = 16 units
More informationL13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11, page 1 L13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11
L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 1 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11 1.1 Rectangular Coordinate System: Suppose we know the sum of
More information3.5 Rational Functions
0 Chapter Polnomial and Rational Functions Rational Functions For a rational function, find the domain and graph the function, identifing all of the asmptotes Solve applied problems involving rational
More informationPATTERNS AND ALGEBRA. He opened mathematics to many discoveries and exciting applications.
PATTERNS AND ALGEBRA The famous French philosopher and mathematician René Descartes (596 65) made a great contribution to mathematics in 67 when he published a book linking algebra and geometr for the
More information7.5. Systems of Inequalities. The Graph of an Inequality. What you should learn. Why you should learn it
0_0705.qd /5/05 9:5 AM Page 5 Section 7.5 7.5 Sstems of Inequalities 5 Sstems of Inequalities What ou should learn Sketch the graphs of inequalities in two variables. Solve sstems of inequalities. Use
More informationDeveloped in Consultation with Tennessee Educators
Developed in Consultation with Tennessee Educators Table of Contents Letter to the Student........................................ Test-Taking Checklist........................................ Tennessee
More informationChpt 1. Functions and Graphs. 1.1 Graphs and Graphing Utilities 1 /19
Chpt 1 Functions and Graphs 1.1 Graphs and Graphing Utilities 1 /19 Chpt 1 Homework 1.1 14, 18, 22, 24, 28, 42, 46, 52, 54, 56, 78, 79, 80, 82 2 /19 Objectives Functions and Graphs Plot points in the rectangular
More informationGraphs, Linear Equations, and Functions
Graphs, Linear Equations, and Functions. The Rectangular R. Coordinate Fractions Sstem bjectives. Interpret a line graph.. Plot ordered pairs.. Find ordered pairs that satisf a given equation. 4. Graph
More information1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
More informationMath 3 Coordinate Geometry part 1 Unit November 3, 2016
Reviewing the basics The number line A number line is a visual representation of all real numbers. Each of the images below are examples of number lines. The top left one includes only positive whole numbers,
More informationCalculator Basics TI-83, TI-83 +, TI-84. Index Page
Calculator Basics TI-83, TI-83 +, TI-84 Index Page Getting Started Page 1 Graphing Page 2 Evaluating Functions page 4 Minimum and Maximum Values Page 5 Table of Values Page 6 Graphing Scatter Plots Page
More informationChapter P Prerequisites
Section P. Real Numbers Chapter P Prerequisites Section P. Real Numbers Quick Review P.. {,,,,, 6}. {,, 0,,,,,, 6}. {,, }. {,,, }. (a) 87.7 (b).7 6. (a) 0.6 (b) 0.0 7. ( ) -( )+ ; (.) -(.)+.7 8. ( ) +(
More informationProblem 1: The relationship of height, in cm. and basketball players, names is a relation:
Chapter - Functions and Graphs Chapter.1 - Functions, Relations and Ordered Pairs Relations A relation is a set of ordered pairs. Domain of a relation is the set consisting of all the first elements of
More informationRational Numbers: Graphing: The Coordinate Plane
Rational Numbers: Graphing: The Coordinate Plane A special kind of plane used in mathematics is the coordinate plane, sometimes called the Cartesian plane after its inventor, René Descartes. It is one
More informationThe x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.
We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section
More information(0, 4) Figure 12. x + 3. d = c. = b. Figure 13
80 CHAPTER EQUATIONS AND INEQUALITIES Plot both points, and draw a line passing through them as in Figure. Tr It # _, 0 Figure Find the intercepts of the equation and sketch the graph: = _ +. (0, (This
More informationLet us start with a quick revision of algebra and how to work with variables. Now let us look at the simpler topic, namely Substitution.
Section F Algebra & Equations Objective In this section, we will be dealing with solving equations for the variable involved, manipulating formulae to get one variable as the subject of the formula and
More informationAppendix 2: PREPARATION & INTERPRETATION OF GRAPHS
Appendi 2: PREPARATION & INTERPRETATION OF GRAPHS All of you should have had some eperience in plotting graphs. Some of you may have done this in the distant past. Some may have done it only in math courses
More informationGlossary alternate interior angles absolute value function Example alternate exterior angles Example angle of rotation Example
Glossar A absolute value function An absolute value function is a function that can be written in the form, where is an number or epression. alternate eterior angles alternate interior angles Alternate
More informationSection 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations origin (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate) x axis Rectangular or
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More information3.6 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions
76 CHAPTER Graphs and Functions Find the equation of each line. Write the equation in the form = a, = b, or = m + b. For Eercises through 7, write the equation in the form f = m + b.. Through (, 6) and
More information3.1. 3x 4y = 12 3(0) 4y = 12. 3x 4y = 12 3x 4(0) = y = x 0 = 12. 4y = 12 y = 3. 3x = 12 x = 4. The Rectangular Coordinate System
3. The Rectangular Coordinate System Interpret a line graph. Objectives Interpret a line graph. Plot ordered pairs. 3 Find ordered pairs that satisfy a given equation. 4 Graph lines. 5 Find x- and y-intercepts.
More informationGRAPHICS OUTPUT PRIMITIVES
CHAPTER 3 GRAPHICS OUTPUT PRIMITIVES LINE DRAWING ALGORITHMS DDA Line Algorithm Bresenham Line Algorithm Midpoint Circle Algorithm Midpoint Ellipse Algorithm CG - Chapter-3 LINE DRAWING Line drawing is
More informationMath 2 Coordinate Geometry Part 1 Slope & Transformations
Math 2 Coordinate Geometry Part 1 Slope & Transformations 1 MATH 1 REVIEW: THE NUMBER LINE A number line is a visual representation of all real numbers. Each of the images below are examples of number
More information1.2 Functions and Graphs
Section.2 Functions and Graphs 3.2 Functions and Graphs You will be able to use the language, notation, and graphical representation of functions to epress relationships between variable quantities. Function,
More informationTransformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations
Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting,
More informationThree-Dimensional Coordinates
CHAPTER Three-Dimensional Coordinates Three-dimensional movies superimpose two slightl different images, letting viewers with polaried eeglasses perceive depth (the third dimension) on a two-dimensional
More informationConcept Explanation Examples Distance formula The distance between (x 1, y 1 ) and (x 2, y 2 ) is d = 2(x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2.
660_ch0pp00-075.qd 0/6/08 4:8 PM Page 4 4 CHAPTER Introduction to Functions and Graphs continued from previous page Concept Eplanation Eamples Distance formula The distance between (, y ) and (, y ) is
More informationDetermine Whether Two Functions Are Equivalent. Determine whether the functions in each pair are equivalent by. and g (x) 5 x 2
.1 Functions and Equivalent Algebraic Epressions On September, 1999, the Mars Climate Orbiter crashed on its first da of orbit. Two scientific groups used different measurement sstems (Imperial and metric)
More informationGraphing Equations. The Rectangular Coordinate System
3.1 Graphing Equations The Rectangular Coordinate Sstem Ordered pair two numbers associated with a point on a graph. The first number gives the horizontal location of the point. The second gives the vertical
More information2.1. Rectangular Coordinates and Graphs. 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions. Graphs and Functions
2 Graphs and Functions 2 Graphs and Functions 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions Sections 2.1 2.4 2008 Pearson Addison-Wesley. All rights reserved Copyright
More informationPre-Algebra Notes Unit 8: Graphs and Functions
Pre-Algebra Notes Unit 8: Graphs and Functions The Coordinate Plane A coordinate plane is formed b the intersection of a horizontal number line called the -ais and a vertical number line called the -ais.
More informationConnecting Algebra and Geometry with Polygons
Connecting Algebra and Geometr with Polgons 15 Circles are reall important! Once ou know our wa around a circle, ou can use this knowledge to figure out a lot of other things! 15.1 Name That Triangle!
More informationUsing Polar Coordinates. Graphing and converting polar and rectangular coordinates
Using Polar Coordinates Graphing and converting polar and rectangular coordinates Butterflies are among the most celebrated of all insects. It s hard not to notice their beautiful colors and graceful flight.
More informationMATH STUDENT BOOK. 10th Grade Unit 9
MATH STUDENT BOOK 10th Grade Unit 9 Unit 9 Coordinate Geometr MATH 1009 Coordinate Geometr INTRODUCTION 3 1. ORDERED PAIRS 5 POINTS IN A PLANE 5 SYMMETRY 11 GRAPHS OF ALGEBRAIC CONDITIONS 19 SELF TEST
More information2.8 Distance and Midpoint Formulas; Circles
Section.8 Distance and Midpoint Formulas; Circles 9 Eercises 89 90 are based on the following cartoon. B.C. b permission of Johnn Hart and Creators Sndicate, Inc. 89. Assuming that there is no such thing
More informationSection 3.1 Graphing Using the Rectangular Coordinate System
Objectives Section 3.1 Graphing Using the Rectangular Coordinate System n Construct a rectangular coordinate system n Plot ordered pairs and determine the coordinates of a point n Graph paired data n Read
More informationUsing Characteristics of a Quadratic Function to Describe Its Graph. The graphs of quadratic functions can be described using key characteristics:
Chapter Summar Ke Terms standard form of a quadratic function (.1) factored form of a quadratic function (.1) verte form of a quadratic function (.1) concavit of a parabola (.1) reference points (.) transformation
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationGraphing Calculator Graphing with the TI-86
Graphing Calculator Graphing with the TI-86 I. Introduction The TI-86 has fift kes, man of which perform multiple functions when used in combination. Each ke has a smbol printed on its face. When a ke
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
More informationUnit 4 Trigonometry. Study Notes 1 Right Triangle Trigonometry (Section 8.1)
Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric
More informationPolar Functions Polar coordinates
548 Chapter 1 Parametric, Vector, and Polar Functions 1. What ou ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed b Polar Curves A Small Polar Galler... and wh Polar
More informationMATH College Algebra Review for Test 1
MATH 34 - College Algebra Review for Test Section.2. For the relation {(,4), (,2), (5, )}, (a) what is the domain and (b) what is the range? 2. (a) For the table of data shown in the table at the right,
More informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
More informationYear 6 Mathematics Overview
Year 6 Mathematics Overview Term Strand National Curriculum 2014 Objectives Focus Sequence Autumn 1 Number and Place Value read, write, order and compare numbers up to 10 000 000 and determine the value
More informationNumber- Algebra. Problem solving Statistics Investigations
Place Value Addition, Subtraction, Multiplication and Division Fractions Position and Direction Decimals Percentages Algebra Converting units Perimeter, Area and Volume Ratio Properties of Shapes Problem
More information= = The number system. Module. Glossary Math Tools... 33
- > + > < - %. < + a = - = = b in. F - - Module The number sstem Lesson Rational and Irrational Numbers........ 8.NS. Lesson ompare and Order Numbers......... 8 8.NS., 8.NS. Lesson Estimate the Value of
More information6 Using Technology Wisely
6 Using Technology Wisely Concepts: Advantages and Disadvantages of Graphing Calculators How Do Calculators Sketch Graphs? When Do Calculators Produce Incorrect Graphs? The Greatest Integer Function Graphing
More information9. f(x) = x f(x) = x g(x) = 2x g(x) = 5 2x. 13. h(x) = 1 3x. 14. h(x) = 2x f(x) = x x. 16.
Section 4.2 Absolute Value 367 4.2 Eercises For each of the functions in Eercises 1-8, as in Eamples 7 and 8 in the narrative, mark the critical value on a number line, then mark the sign of the epression
More informationNumber Mulitplication and Number and Place Value Addition and Subtraction Division
Number Mulitplication and Number and Place Value Addition and Subtraction Division read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to
More informationACTIVITY: Representing Data by a Linear Equation
9.2 Lines of Fit How can ou use data to predict an event? ACTIVITY: Representing Data b a Linear Equation Work with a partner. You have been working on a science project for 8 months. Each month, ou measured
More information2.3 Polynomial Functions of Higher Degree with Modeling
SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 185 2.3 Polnomial Functions of Higher Degree with Modeling What ou ll learn about Graphs of Polnomial Functions End Behavior of Polnomial
More informationNew Swannington Primary School 2014 Year 6
Number Number and Place Value Number Addition and subtraction, Multiplication and division Number fractions inc decimals & % Ratio & Proportion Algebra read, write, order and compare numbers up to 0 000
More informationLesson 11 Skills Maintenance. Activity 1. Model. The addition problem is = 4. The subtraction problem is 5 9 = 4.
Lesson Skills Maintenance Lesson Planner Vocabular Development -coordinate -coordinate point of origin Skills Maintenance ddition and Subtraction of Positive and Negative Integers Problem Solving: We look
More informationMaths. Formative Assessment/key piece of work prior to end of unit: Term Autumn 1
Term Autumn 1 3 weeks Negative numbers Multiples and factors Common factors Prime numbers Ordering decimal numbers Rounding Square numbers and square roots Prime factor decomposition LCM and HCF Square
More informationBarrhead High School Mathematics Department. National 4 Mathematics. Learning Intentions & Success Criteria: Assessing My Progress
Barrhead High School Mathematics Department National 4 Mathematics Learning Intentions & Success Criteria: Assessing My Progress Expressions and Formulae Topic Learning Intention Success Criteria I understand
More informationSECTION 8.2 the hyperbola Wake created from shock wave. Portion of a hyperbola
SECTION 8. the hperola 6 9 7 learning OjeCTIveS In this section, ou will: Locate a hperola s vertices and foci. Write equations of hperolas in standard form. Graph hperolas centered at the origin. Graph
More information3.1 Functions. The relation {(2, 7), (3, 8), (3, 9), (4, 10)} is not a function because, when x is 3, y can equal 8 or 9.
3. Functions Cubic packages with edge lengths of cm, 7 cm, and 8 cm have volumes of 3 or cm 3, 7 3 or 33 cm 3, and 8 3 or 5 cm 3. These values can be written as a relation, which is a set of ordered pairs,
More informationSLOPE A MEASURE OF STEEPNESS through 2.1.4
SLOPE A MEASURE OF STEEPNESS 2.1.2 through 2.1.4 Students used the equation = m + b to graph lines and describe patterns in previous courses. Lesson 2.1.1 is a review. When the equation of a line is written
More informationSECTION 3-4 Rational Functions
20 3 Polnomial and Rational Functions 0. Shipping. A shipping bo is reinforced with steel bands in all three directions (see the figure). A total of 20. feet of steel tape is to be used, with 6 inches
More informationWhat s the Point? # 2 - Geo Fashion
What s the Point? # 2 - Geo Fashion Graph the points and connect them with line segments. Do not connect points with DNC between them. Start (-4,1) (-5,5) (-2,2) (-4,1) DNC (2,-4) (3,-3) (4,-3) (5,-4)
More information9 CARTESIAN SYSTEM OF COORDINATES You must have searched for our seat in a cinema hall, a stadium, or a train. For eample, seat H-4 means the fourth seat in the H th row. In other words, H and 4 are the
More informationY6 MATHEMATICS TERMLY PATHWAY NUMBER MEASURE GEOMETRY STATISTICS
Autumn Number & Place value read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to a required degree of accuracy use negative numbers in
More informationYear 6 Maths Long Term Plan
Week & Focus 1 Number and Place Value Unit 1 2 Subtraction Value Unit 1 3 Subtraction Unit 3 4 Subtraction Unit 5 5 Unit 2 6 Division Unit 4 7 Fractions Unit 2 Autumn Term Objectives read, write, order
More informationSTRAND G: Relations, Functions and Graphs
UNIT G Using Graphs to Solve Equations: Tet STRAND G: Relations, Functions and Graphs G Using Graphs to Solve Equations Tet Contents * * Section G. Solution of Simultaneous Equations b Graphs G. Graphs
More informationSAMPLE. Interpreting linear relationships. Syllabus topic AM2 Interpreting linear relationships. Distance travelled. Time (h)
C H A P T E R 5 Interpreting linear relationships Sllabus topic AM Interpreting linear relationships Graphing linear functions from everda situations Calculating the gradient and vertical intercept Using
More informationYou should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
More informationReteaching Golden Ratio
Name Date Class Golden Ratio INV 11 You have investigated fractals. Now ou will investigate the golden ratio. The Golden Ratio in Line Segments The golden ratio is the irrational number 1 5. c On the line
More informationMATHEMATICS 105 Plane Trigonometry
Chapter I THE TRIGONOMETRIC FUNCTIONS MATHEMATICS 105 Plane Trigonometry INTRODUCTION The word trigonometry literally means triangle measurement. It is concerned with the measurement of the parts, sides,
More information4.1 The Coordinate Plane
4. The Coordinate Plane Goal Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin -ais -ais Ordered pair -coordinate -coordinate Quadrant Scatter plot Copright McDougal Littell, Chapter
More information8.6 Three-Dimensional Cartesian Coordinate System
SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 69 What ou ll learn about Three-Dimensional Cartesian Coordinates Distance and Midpoint Formulas Equation of a Sphere Planes and Other Surfaces
More informationWRITING AND GRAPHING LINEAR EQUATIONS ON A FLAT SURFACE #1313
WRITING AND GRAPHING LINEAR EQUATIONS ON A FLAT SURFACE #11 SLOPE is a number that indicates the steepness (or flatness) of a line, as well as its direction (up or down) left to right. SLOPE is determined
More informationCoordinate geometry Straight lines are an important part of our environment. We play sport on courts with parallel and perpendicular lines, and
Number and Algebra Coordinate geometr Straight lines are an important part of our environment. We pla sport on courts with parallel and perpendicular lines, and skscrapers would not be standing without
More informationAlgebra I Summer Math Packet
01 Algebra I Summer Math Packet DHondtT Grosse Pointe Public Schools 5/0/01 Evaluate the power. 1.. 4. when = Write algebraic epressions and algebraic equations. Use as the variable. 4. 5. 6. the quotient
More informationFunctions and Their Graphs
Functions and Their Graphs. Rectangular Coordinates. Graphs of Equations. Linear Equations in Two Variables. Functions.5 Analzing Graphs of Functions. A Librar of Parent Functions.7 Transformations of
More informationPreparation for Calculus
P Preparation for Calculus This chapter reviews several concepts that will help ou prepare for our stud of calculus. These concepts include sketching the graphs of equations and functions, and fitting
More informationu u 1 u (c) Distributive property of multiplication over subtraction
ADDITIONAL ANSWERS 89 Additional Answers Eercises P.. ; All real numbers less than or equal to 4 0 4 6. ; All real numbers greater than or equal to and less than 4 0 4 6 7. ; All real numbers less than
More informationSolving inequalities Expanding brackets and simplifying the result Graphing quadratic functions in simple cases Interpreting real-life graphs, e.g.
Grade C Descriptors Estimation and division by a number less than 1 Calculating compound interest - no rounding necessary Using a calculator in complex situations Multiplication and division by a number
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;
More information3 CHAPTER. Coordinate Geometry
3 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius Cartesian Plane Ordered pair A pair of numbers a and b instead in a specific order with a at the first place and b
More informationMaths Year 11 Mock Revision list
Maths Year 11 Mock Revision list F = Foundation Tier = Foundation and igher Tier = igher Tier Number Tier Topic know and use the word integer and the equality and inequality symbols use fractions, decimals
More information