MATH STUDENT BOOK 9th Grade Unit 8
Unit 8 Graphing Math 908 Graphing INTRODUCTION 3. USING TWO VARIABLES 5 EQUATIONS 5 THE REAL NUMBER PLANE TRANSLATIONS 5 SELF TEST. APPLYING GRAPHING TECHNIQUES 5 LINES 5 INEQUALITIES 54 ABSOLUTE VALUES 6 SELF TEST 7 3. WRITING EQUATIONS OF LINES 77 GIVEN TWO POINTS 77 GIVEN ONE POINT AND THE SLOPE 84 GIVEN THE GRAPH 87 GIVEN A RELATED LINE 90 SELF TEST 3 95 GLOSSARY 00 LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. Section
Graphing Unit 8 Authors: Arthur C. Landre, M.A.Ed. Robert L. Zenor, M.A., M.S. Editor-In-Chief: Richard W. Wheeler, M.A.Ed. Editor: Robin Hintze Kreutzberg, M.B.A. Consulting Editor: Rudolph Moore, Ph.D. Revision Editor: Alan Christopherson, M.S. Westover Studios Design Team: Phillip Pettet, Creative Lead Teresa Davis, DTP Lead Nick Castro Andi Graham Jerr Wingo 804 N. nd Ave. E. Rock Rapids, IA 546-759 MCMXCVI b Alpha Omega Publications, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, Inc. All trademarks and/or service marks referenced in this material are the propert of their respective owners. Alpha Omega Publications, Inc. makes no claim of ownership to an trademarks and/ or service marks other than their own and their affiliates, and makes no claim of affiliation to an companies whose trademarks ma be listed in this material, other than their own. Section
Unit 8 Graphing Graphing INTRODUCTION In this LIFEPAC ou will continue our stud in the mathematical sstem of algebra b learning about graphing. After seeing how two variables are used, ou will learn the various techniques for showing the solutions to open sentences on the real-number plane. Finall, ou will learn to write the equations of lines drawn in this plane. Objectives Read these objectives. The objectives tell ou what ou will be able to do when ou have successfull completed this LIFEPAC. When ou have finished this LIFEPAC, ou should be able to:. Find ordered-pair solutions for two-variable equations.. Locate points on the real-number plane. 3. Translate verbal statements to equations. 4. Draw the graphs for linear equations, linear inequalities, and open sentences involving absolute values. 5. Find the equations of lines from given information. Section 3
Unit 8 Graphing. USING TWO VARIABLES In this first section ou will learn the introductor concepts and definitions needed for graphing: solving two-variable equations, plotting points on the realnumber plane, and translating verbal sentences to equations. OBJECTIVES. Find ordered-pair solutions for two-variable equations.. Locate points on the real-number plane. 3. Translate verbal statements to equations. EQUATIONS You have alread learned to find numerical answers for equations having one variable, such as: + = 5, 3m + = 7, and 4 t =. For eample, ou know that the equation, + = 5, has eactl one integral solution, = 3; but what about the equation, + = 5? Is = 3 still a solution? Is 3 the onl value for that will give a solution? Let us investigate further. The equation, + = 5, indicates that the sum of and is five. If is 3, then the sum of 3 and must be five; therefore must be. Thus, = 3 is a solution onl when =. Now 3 and are certainl not the onl two numbers having a sum of five; and 4, 5 and 0, and -3 and 8 are just three eamples of other pairs of integers with sums of five. Certain pairs of rational numbers (such as 3 3 and.6) and irrational numbers (such as 3 8 and 3 3) also have a sum of five. In fact, infinitel man real number solutions eist to the equation + = 5. When an equation contains two variables, ou must look for a relationship between those variables rather than just for the value(s) of a single variable. The solutions for such equations will be pairs of numbers that make true sentences. These solutions are called ordered pairs since the numbers are written in the alphabetical order of the two variables. Model : Find several ordered pair solutions for + = 5 Solution: The ordered pairs are written as (, ) since is before in the alphabet: (3, ), (, 4), (5, 0), (-3, 8), (3 3,.6), ( 3 8, 3 3). Section 5
Graphing Unit 8 You should notice that if the numbers are reversed in the solutions of Model, the resulting ordered pairs will also be solutions of + = 5. This situation is not alwas true. Model : Find three solutions for a b =. Solution: You ma use an real values ou wish for one of the variables (usuall the one nearer the beginning of the alphabet). Substitute each of the chosen values in the equation and solve for the remaining variable. Suppose we choose 5,, and 0 for a: a = 5: 5 b = 0 b = b = 9 a = : b = b = b = 0 a = 0: 0 b = 0 b = b = - Three ordered-pair solutions of a b = are (5, 9), (, 0), and (0, -). The are in the order (a, b). These ordered pairs cannot be reversed and still be solutions for a b =. (9, 5) is not a solution since 9 5. (0, ) is not a solution since 0. (-, 0) is not a solution since (-) 0. You must be ver careful to put our pairs of numbers in the correct order when writing solutions to two-variable equations. The ordered pairs that make an equation true are said to satisf that equation. 6 Section
Unit 8 Graphing Complete the following activities. Find the value of for the given value of in each of the following equations. Then write the ordered pairs.. + = 0. = 8 (, ) (, ) 0 4 0 8 6 Complete the ordered pair solutions for each of the following equations..3 + = 6 A = {(, ), (0, ), (-, ) }.4 3 + = 5 B = {(0, ), (3, ), (6, ) }.5 = 3 C = {(-, ), (0, ), (, ) } Find three ordered pair solutions for each of the following equations..6 = a. b. c..7 + = - a. b. c..8 = 7 a. b. c..9 + = 0 a. b. c..0 3 + = 0 a. b. c. Sometimes ou ma wish to find solutions b first changing the form of the original equation so that the variable nearer the end of the alphabet is written alone on one side of the equation. Model : Solve = 7 for. Solution: = 7 = 7 + or = + 7 Section 7
Graphing Unit 8 Model : Solve a + b = 5 for b. Solution: a + b = 5 b = 5 a b = 5 a or b = -a + 5 Model 3: Solve m n = for n. Solution: [ m n ] = [ ] m n = -n = -m + -[ -n ] = -[ -m + ] n = m Solve each of the following equations for the variable indicated.. 3 + = for :. + = -6 for :.3 3a + b = 6 for b:.4 r 3 3s = 0 for s:.5 5 = for : VOCABULARY domain for a two variable equation, the domain is the set of numbers to be used for the first (alphabetical) variable. REMEMBER? The elements of a set are listed between braces: { }. The smbol element of the set. Sets are often named with capital letters. means is an 8 Section
Unit 8 Graphing Model : Find the ordered pairs that satisf the equation 3m + n = 7 when the domain of m is {-5, 0, 3 3 }. Solution: First solve for n: 3m + n = 7 n = 7 3m n = 7 3m T hen complete the table: 3 m -5 0 3 or 3 7 3m 7 3 ( 5 ) 7 + 5 7 3 0 7 0 n 3-7 7 3 3 7-4 The ordered pairs are (-5, ), (0, 3 ), and (3 3, -). Model : F ind the ordered pairs that satisf the equation 4s t =. when s {-, 0.3, }. Solution: Solve for t : 4s t =. Complete the table: t =. 4s t = -. + 4s or t = 4s. s - 0.3 4s. 4(-). 4( 0.3). 4( ). -8... 4. t -9. 0.8 t no values 0 -.8 or.8 (since t cannot be negative) (0.3, 0), (, -.8), and (,.8) are solutions of 4s t =.. Section 9
Graphing Unit 8 a. Solve each of the following equations for ; b. find the ordered pairs that satisf the equation for the given domain..6 = {-, 0, } a. = b. (-, ), (0, ), (, ).7 + = 0 {, 4, 8 } a. = b. 3.8 5 + = - {0, 5, 0} a. = b..9 7 + = 0 {0.5.5.5} a. = b. -.0 + 3 = 5 { 3, 3, 0} a. = b. Find three ordered pair solutions for each of the following equations b selecting three convenient elements of the domain. 3. = 4. + 3 = 5.3 7 = 7.4 + = 6.5 + = 0 Section
Unit 8 Graphing THE REAL NUMBER PLANE In Mathematics LIFEPAC 907, ou learned to graph the solution points of one-variable equations on the real-number line. In this LIFEPAC ou will be graphing the solution points of two-variable equations on the real-number plane. First, however, ou need to learn the terminolog and procedures of graphing. Two reference lines or aes are drawn in the plane, one horizontall and one verticall, meeting at a common zero point called the origin. Each ais is a number line for one of the two variables. Since and are the letters used most frequentl, the horizontal ais is known as the -ais and the vertical ais is known as the -ais. ais origin ais On the -ais, positive numbers are to the right of the origin, and negative numbers are to the left of the origin. On the -ais, positive numbers are above the origin, and negative numbers are below the origin. The aes separate the plane into four regions called quadrants, which are labeled with Roman numerals as indicated in the diagram. An ordered pair of numbers in the form (, ) is used to locate an point in the plane. The value of indicates the horizontal direction and distance of the point from the origin, and the value of indicates the vertical direction and distance of the point from the origin. Of course, the ordered pair (0, 0) represents the origin itself. Suppose we wish to locate the point corresponding to the ordered pair (4, -) on the plane at the right. (NOTE: A complete grid of intersecting lines is used so that points ma be found more easil and accuratel.) The first number indicates that the point is four units to the right of the origin; the second number indicates that the point is two units below the origin. Thus, to find the point (4, -), begin at the origin and move four units right then two units down. You arrive at point J in Quadrant IV; a heav dot is used to show (or plot) the point on the plane. II -4-3 - - III - - -3-4 4 3 I 3 4 IV > > > > J > > IV Section
> > > > The order of the numbers written in the pair, as well as the order of movements from the origin, is ver important. To see this fact, notice that point K corresponding to the ordered pair (-, 4) is in Quadrant II and certainl is not the same point as J. > > Graphing Unit 8 II K J Model : Plot the point corresponding to the ordered pair (-5, -) and describe its location. Solution: Begin at the origin and move five units left then one unit down. This point is located in Quadrant III. (-5, -) III Model : Plot the point for (0, -3) and describe its location. Solution: The first value of 0 indicates that no horizontal movement is to be made. Thus, the point corresponding to (0, -3) is three units below the origin on the -ais. (0, -3) This point is located between Quadrants III and IV. III IV Section
Unit 8 Graphing Model 3: Describe the locations and name the ordered pairs corresponding to points L, M, and N in the diagram. N Solution: Point L beginning at the origin, move 3 units right then unit up; thus, the ordered pair is (3, ). M L Point M beginning at the origin, move units left then 0 units verticall; thus, the ordered pair is (-, 0). Point N beginning at the origin, move 0 units horizontall then 4 units up; thus, the ordered pair is (0, 4). Plot and label the point corresponding to each given ordered pair; then describe its location..6 (4, 3).7 (3, 4).8 (-, 5).9 (-5, 3).30 (-, -).3 (-6, -3).3 (, -3).33 (3, -3).34 (0, 4).35 (4, 0).36 (0, -) Section 3
Graphing Unit 8 SELF TEST For each of the following points, describe its location on a grid (each answer, 3 points)..0 (6, ).0 (-3, 5).03 (0, ).04 (0, 0).05 (-6, -6) Name the ordered pair corresponding to each point on the graph (each answer, 3 points)..06 Point A.07 Point B.08 Point C C A B.09 Point D D E.00 Point E For each of the following sentences, write a translation using and (each answer, 3 points)..0 The ordinate is one-half the abscissa..0 The abscissa less the ordinate is one..03 The product of the abscissa and ordinate is ten. Section
Unit 8 Graphing For each of the following equations, solve for (each answer, 3 points)..04 + = 6.05 + 6 = 0.06 + 3 = 0 For each of the following equations, a. solve for ; b. find three ordered pairs; and c. graph the points (a. 3 points; b. 3 points; c. 4 points)..07 T he ordinate is twice the abscissa. c. a. b..08 + = 0 c. a. b. Section 3
Graphing Unit 8.09 + = and {-, 0, } a. b. c..00 3 = - c. a. b. 7 88 SCORE TEACHER initials date 4 Section
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