Graphing LinesGraphing Lines - Advisory 207

1 624 Chapter 12 Graphing LinesGraphing LinesWhooping Crane PopulationYear1940196019802000 Cranes153679202 WildlifeEcologist Whatever happened to the Carolina parakeet and thepassenger pigeon, two species of birds that once inhabited theUnited States? They are now asextinct as Tyrannosaurus primary focus of wildlifeecologists is to keep other animals from becoming have been successful withthe whooping crane, the largestwild bird in North America. Thetable shows howthe whoopingcrane has come back from thebrink of Equations12-1 Graphing LinearEquations12-2 slope of a Line12-3 Using Slopes andInterceptsLABG raph Equations in slope -Intercept Form12-4 Point- slope Form12 BLinear Relationships12-5 Direct Variation12-6 Graphing Inequalities inTwo Variables12-7 Lines of Best FitExtSolving Systems ofEquations by GraphingKEYWORD:MT7 Ch12m807_c12_624_625 1/6/06 4:02 PM Page 624 Graphing Lines625 VocabularyChoose the best term from the list to complete each expression 4 3 is an example of a(n) __?

2 __expression. you divide both sides of the equation 2x 20 by 2, you are __?__. example of a(n) __?__is 3x 12. expression 7 6 can be rewritten as the __?__expression 7 ( 6). Complete these exercises to review skills you will need for this with 7 25 6. 32 53 7. 82 28 8. 126 9. 222 10. 12 9 Evaluate ExpressionsEvaluate each expression for the given value of the 2 for x 2 8 12 yfor y (x 1) for x 2 14. 3(y 2) yfor y 1 4 8 (a 3) 4 2k 4 1 1 5(m 2) 1 Determine whether each ordered pair is a solution to 12 x 3 (4, 1)22 82 , 2 23(0, 5)24.( 4, 5)25.(8, 1) 26.(2, 2) 27.( 2, 4) 28.(0, 1) Solve Inequalities in One VariableSolve and graph each 4 230. 3x 1 5additionequationinequality in one variablesolving for the variable subtractionm807_c12_624_625 1/6/06 4:02 PM Page 625 Previously, you located and named points on acoordinate plane using orderedpairs of integers.

3 Graphed data to demonstraterelationships between sets of can use the skillslearned in this chapter to predict the distance a carneeds to come to a completestop, given its speed. to estimate the maximumdistance a robotic vehicle cantravel during a given period of ConnectionsTo become familiar with some of thevocabulary terms in the chapter, consider thefollowing. You may refer to the chapter, theglossary, or a dictionary if you word linearmeans relating to a line . What do you think the graph of alooks like? word intercept can mean to interrupt a course or path. Where on a graph do you think you should look to find theof a line ? adjective directcan mean passing in a straight line . What do you suppose thegraph of an equation with looks like? boundaryis a limit. What do you thinkthe represents in a graph of a linear inequality?

4 Boundary linedirect variationy-interceptlinear equation626 Chapter 12 Key Vocabulary/Vocabularioboundary linel nea de lim teconstant of constante de proportionalityproporcionalidaddirect variationvariaci n directalinear equationecuaci n lineallinear inequalitydesigualdad linealpoint- slope formforma punto-pendienteslope-intercept formforma pendiente-intersecci nx-interceptintersecci n con el ejede las xy-interceptintersecci n con el ejede las yStudy Guide: PreviewYou will study locating and naming points on acoordinate plane using orderedpairs of rational numbers. generating differentrepresentations of data usingtables, graphs, and equations. Graphing linear equations usingslope and y-intercept. Graphing inequalities involvingtwo variables on a 1/6/06 4:02 PM Page 626 Writing Strategy: Use Your Own WordsExplaining a concept in your own words will help you better understand example, learning to solve two-step inequalities might seem difficult ifthe textbook does not use the same words that you would you work through each lesson, do the following: Identify the important concepts.

5 Use your own words to explain the concepts. Use examples to help clarify your Lines627 Rewrite each statement in your own terms can be grouped together because they have the samevariable raised to the same power. an equation contains fractions, consider multiplying both sides of theequation by the least common denominator (LCD) to clear the fractionsbefore you isolate the solve multi-step equations with variables on both sides, firstcombine like terms and then clear fractions. Then add or subtractvariable terms on both sides so that the variable occurs on only one sideof the equation. Then use properties of equality to isolate the ThisReading and Writing MathSolving a two-stepinequality uses the sameinverse operations as solvinga two-step or dividing an inequality by a negativenumber reverses theinequality Miguel ReadsWhat Miguel WritesSolve a two-step inequality like a two-step equation.

6 Use operations that undo each you multiply or divide bya negative number, switch theinequality symbol so that itfaces the opposite direction. 4y 8 Divide by 4 andy 2switch the symbol. m807_c12_627_627 1/6/06 4:02 PM Page 627 Learnto identify andgraph linear equationLight travels faster than s why you see lightningbefore you hear thunder. Thelinear equation d expressesthe approximate distance, d, inmiles of a thunderstorm for agiven number of seconds, s,between the lightning flash and the thunder is an equation whose solutions fall on a line on thecoordinate plane. All solutions of a particular linear equation fall onthe line , and all the points on the line are solutions of the find a solution that lies between two points (x1, y1) and (x2, y2),choose an x-value between x1and x2and find the corresponding y-value.

7 If an equation is linear, a constant change in the x-value corresponds to a constant change in the y-value. The graph shows an examplewhere each time the x-value increases by 3,the y-value increases by EquationsGraph each equation and tell whether it is 3x 4 The equation y 3x 4 is a linear equation because it is thegraph of a straight line , and each time xincreases by 1 unit, yincreases by 3 equation2233 EXAMPLE1O242 2 4xy 2x3x 4y(x, y)03(0) 4 4(0, 4)13(1) 4 1(1, 1)23(2) 42(2, 2)33(3) 45(3, 5)Read x1as xsubone or xone. 12-1 Graphing Linear Equations628 Chapter 12 Graphing Linesm807_c12_628_632 1/6/06 4:02 PM Page 628 Graph each equation and tell whether it is x2 The equation y x2is not alinear equation because its graph isnot a straight line . Also notice thatasxincreases by a constant of 1,the change in yis not constant.

8 Y 34x The equation y 34x is a linear equation because the points form a straight line . Each time the value of xincreases by 1, the value of ydecreases by 34 , or ydecreases by 3 each time xincreases by 3 For any value of x, y equation y 3 is a linear equation because the points form a straight line . As the value of xincreases, the value of yhas a constant change of 2 1012y 4 10 1 42 2 4xy 2 3 1 1 32121 2 1 2 1xyOxO22 2 2yx 3y(x, y) 2 3 3( 2, 3) 1 3 3( 1, 3)0 3 3(0, 3)1 3 3(1, 3)2 3 3(2, 3)x 34x y(x, y) 2 3( 42) 32 2, 32 1 3( 41) 34 1, 34 0 3(40) 0(0, 0)1 3(41) 34 1, 34 2 3(42) 32 2, 32 x x2y(x, y) 2 ( 2)2 4( 2, 4) 1 ( 1)2 1( 1, 1)0 (0)20(0, 0)1 (1)2 1(1, 1)2 (2)2 4(2, 4)12-1 Graphing Linear Equations629Be careful whengraphing eachordered pair. Double check eachpoint you 1/6/06 4:02 PM Page 629 Physical Science ApplicationThe equation d the approximatedistance, d, in miles of athunderstorm when sseconds pass between aflash of lightning and thesound of thunder.

9 About how far is thethunderstorm from eachstudent listed in the table?Draw a graph that represents the relationship between thetime between lightning and thunder and the distance of thestorm from the approximate distances areSandy, 1 mile; Diego, miles;Ted, mile; Cecilia, miles;and Massoud, miles. This is a linear equation because whensincreases by 10 seconds, dincreases by 2 miles. Time BetweenStudentFlash and Thunder (s)Sandy5 Diego9 Ted4 Cecilia11 Massoud82468101214161820sdd of a ThunderstormDistance (mi)2 EXAMPLET hink and an equation is linear if three ordered-pairsolutions lie on a straight line but a fourth does not. equations y 3x 2 and y 3x2. Withoutgraphing, explain why one of the equations is not linear. neither number in the ordered pair can be negative in Example 12 Graphing Linessd (s, d)5d (5) (5, 1)9d (9) (9, )4d (4) (4, )11d (11) (11, )8d (8) (8, )m807_c12_628_632 1/6/06 4:02 PM Page 63012-1 ExercisesExercisesGraph each equation and tell whether it is x ScienceTyrannosaurus rex was one of the largest meat-eaters that ever lived.

10 By 14 years of age, a T. r e xwas growing about 4 poundsevery day. If you found a T. r e xthat was 14 years old, the equation w 4d 5110 would represent the weight wof the animal ddays much would it weigh after 2 days? after days? after 5 days? Graph the equation and tell whether it is linear?Graph each equation and tell whether it is 14 x 13 x2 3x charter bus service charges a $125 transportation fee plus $ for each passenger. This is represented by the equation C 125, where Cis the total cost based on ppassengers. What isthe total cost of transportation for the following numbers of passengers:50, 100, 150, 200, and 250? Graph the equation and tell whether it is minute hand of a clock moves 110 degree every second. If you look atthe clock when the minute hand is 10 degrees past the 12, you can use theequation y 110 x 10 to find how many degrees past the 12 the minutehand is after xseconds.