Graph Linear Equations

Transcription

1 Lesson 4. Objectives Graph linear equations. Identif the slope and -intercept of linear equations. Graphing Linear Equations Suppose a baker s cookie recipe calls for a miture of nuts, raisins, and dried fruit. Some customers prefer lots of nuts; others would rather have more raisins. The recipe states that whichever wa ou mi them, when ou add the number of f cups c of raisins to the number of cups of nuts, and then add this to 4 cup of dried fruit, ou must have a total of cups. How can ou write an equation to model this situation? How would ou draw the graph? Graph Linear Equations The equation epresses a relationship between two variables and. In this relationship, the value of depends on the value that is substituted for. Thus, is called the independent variable and is called the dependent variable. If ou select as the value of the independent variable, then is the value of the dependent variable. Eample Graphing a Linear Equation Graph the equation. How does the graph compare to the graph of? The equation is solved when ou find the ordered pairs that make the equation a true statement. One solution of the equation is the ordered pair (, ). Because the solution of is a set of ordered pairs, a table of values or a graph can represent. In a table, the first column is usuall labeled as the independent variable. The second column is labeled as the dependent variable. Because ou cannot list all the ordered pairs that make the equation a true statement, three to five pairs are usuall enough From the table, it can be n that for each increase in, increases b. Therefore the slope of the equation is. 4 Chapter 4 Linear Equations

2 After ou complete the table, graph each pair of (, ) values as a point. Draw the line that passes through all the points. Then graph the parent function 5 on the same coordinate plane = = Eamine the figure that shows the graphs of and on the same pair of aes. Because the graphs are straight lines, and are called linear equations. Because the graph of the equation is determined b the table, ou can find the slope of from the table. Critical Thinking Eplain how to use the table to find the slope of the linear equation. Activit The Slope of a Line Use a table to graph each of the three equations on the same coordinate aes. Use four ordered pairs for each table. a. b. c. 5 Use the slope formula to determine the slope of each line. Compare the slope of each line to the coefficient of the independent variable. What do ou notice? Guess the slope of 6. Use a table to check our guess. Notice that the graph of each equation is a straight line that passes through the origin. Critical Thinking Let m represent the slope of a line that passes through the origin. What is the equation of the line? 4. Graphing Linear Equations 5

3 Eample Negative Slope Compare the slope of with the slope of. Then compare each graph to the parent function. Make a table of values and graph. First, choose four values for. Then, use the equation to find the values for. After ou complete the table, graph each pair of (, ) values as a point. Draw the line that passes through the points. Write the equation along the line. The steepness of these two lines is the same. However, the rise in opposite directions. Notice that the slopes are opposites. Graph the parent function. The graph of is steeper than the graph of. The graph of is steeper and rises in the opposite direction as = = 6 5 = Positive and Negative Slope A line that rises to the right has a positive slope. A line that rises to the left has a negative slope. Look at our table of values for. Notice that as increases in value, decreases. In this case, the graph of the equation has a negative slope. Critical Thinking In Eample, do ou need to graph all four ordered pairs to draw the line through them? What is the least number of points ou need in order to draw the line that contains the point? Name a reason wh ou would graph more points. 6 Chapter 4 Linear Equations

4 Activit The -intercept Make tables of values and draw the graphs of these four equations on the same coordinate plane. a. b. c. d. What is the slope of each line? Where does each line cross the -ais? 4 Where does the graph of 5 cross the -ais? Slope-Intercept Form of an Equation The -value of the point where the line crosses the -ais is called the -intercept. The equation 5 is written in slope-intercept form. The slope is, and the -intercept is (0, 5). Slope-Intercept Form of a Linear Equation m b is a linear equation. The slope is m and the -intercept is b. The line crosses the -ais at the point (0, b). Eample Graphing a Linear Equation Graph the equation 6 using slope-intercept form. Write the equation in slope-intercept form. Solve the equation for. 6 Given ( 6) Multiplication Propert of Equalit Distributive Propert and Simplif The slope is, and the line crosses the -ais at (0, ). Now draw the coor dinate aes. Graph the -intercept (0, ) as one point on the line. The slope is. You can find another point b -a moving units up and units left. Locate that point. Connect the two points with a line. = + (, 6) (0, ) 4. Graphing Linear Equations 7

5 Critical Thinking How can ou graph a line if ou are given the slope and a point on the line? How is this similar to graphing an equation that is in slopeintercept form? Plot the point and then use the slope to plot another point. The -intercept is also a point on the line. Standard Form of an Equation Another wa to write linear equations is in standard form. Standard Form of a Linear Equation A B C, where A, B, and C are real numbers and A and B are not both zero. Sometimes ou ma have to graph an equation that is in standard form. You can rewrite the equation in slope-intercept form. Then ou can graph the equation using the slope and -intercept. Eample 4 Writing Equations in Slope-Intercept Form Write the equation 4 in slope-intercept form. To write it in slope-intercept form, solve for. 4 4 Subtraction Propert of Equalit Division Propert and Simplif Therefore, the slope is and the -intercept is (0, ). Ongoing Assessment Write the equation in slope-intercept form. Graph the equation. ; for graph Eample 5 Graphing a Linear Equation Refer to the introductor paragraph about the baker's cookie recipe. Write an equation to model this situation. Then graph the equation. l First, translate the problem into sentence form. be The number of cups of raisins plus the number of cups of nuts plus of dried fruit must equal cups. 4 cup 8 Chapter 4 Linear Equations

6 l Second, let r represent the number of cups of raisins and n represent the number of nuts. cups N of nuts. Now ou can change the sentence to the equation: r n 4. To graph the equation, choose one of the variables as the independent variable. For eample, let n be the independent variable. Then r takes the position of, and n takes the position of. To write the equation in slope-intercept form, isolate r on the left side of the equation. r n 4 r n n 4 n r 4 4 n 4 r n 4 Subtraction Propert of Equalit Subtraction Propert of Equalit Simplif. B comparing the equation r n 4 with the slope-intercept form m b, ou can that the slope slo (m) is. So ou would epect this line to be higher on the left. What numbers do ou want to use on the r and n aes? Because the values for both will be small positive numbers, including fractions, make each unit. You can use the -intercept to locate one point and use the slope to find another point. Or ou can make a table of several values. With onl two ordered pairs, ou can draw a straight line, but with several additional values, ou can check our work. If all points are not on the same straight line, ou have an error. If this happens, go back and check our arithmetic and the wa ou made the graph. Graph our values and draw the line. The graph of the equation r n 4 is a line that lies in quadrants one, two, and four. But the situation in the slo problem makes sense onl in the first quadrant, where both r and n are positive numbers. As a result, the line ou graph to fit the problem is limited to the first quadrant. Use a solid line in the first quadrant to emphasize where the data in the problem appl. 4. Graphing Linear Equations 9

7 ASSESSMENT Think and Discuss CULTURAL CONNECTION Hpatia, the first woman mentioned in the histor of mathematics, wrote about the Hpatia, work of the a mathematician first woman known mentioned as Diophantus in the of Aleandria. histor of mathematics, Sometime in the wrote third about centur B.C.E., Diophantus wrote a tet the about work arithmetic. of a mathematician In his tet, known Diophantus as Diophantus worked with of Aleandria. equations that Sometime have more in the than third one centur whole number bce, Diophantus solution. The equations are called Diophantine wrote equations. a tet Consider about arithmetic. this In his problem. tet, Diophantus worked with equations that have more than one whole number solution. Lesson Assessment How can ou use Think an equation and Discuss to make a table? How can ou use a. table How to graph can ou an use equation? an equation to make a table? Eplain wh m is the. slope How of can the ou equation use a table m. to graph an equation? 4 Eplain wh the graph. Eplain of the linear wh m equation is the slope m of the bequation m. crosses the -ais at 4. (0, Eplain b). wh the graph of the linear equation m b crosses 5 How can ou graph the the equation -ais at (0, b). 9 using slopeintercept form? Practice and Problem Solving Practice and Problem Solving Identif the slope of the line represented b each equation. Identif the slope Find of the line -intercepts. represented b each ; (0, 0) equation? Find the - and -intercepts., (0, 0) , (0, 0) , (0,,; 0) (0, ) 8. 8.,, 0, + ; (0, ) ; (0,. ) 00. 0; (0, ) (, 0), (0, ) ( 5, 0), (0, ) none, (0, ) In a pet shop there are several Cultural kittens and birds. The Connection shopkeeper counted eactl 0 legs in the shop. The How equations man kittens are and called birds does Let represent the Diophantine the shopkeeper equations. have? number of child biccles and Consider Let represent this problem. the number of birds represent the number of and the number of kittens. The A linear to manufacturer equation uses go-carts. The linear equation 4 0 the models same this tpe situation. of wheel on 4 0 models this a child s There are biccle an infinite as on number a situation. of go-cart. solutions Inventor the equation. records But the There are an infinite number show solution onl must 0 of make this sense. tpe of How of solutions to the equation. man kittens and birds can the wheel shopkeeper on hand. have? How man But the solution must make child biccles and go-carts sense. How man biccles can be manufactured using and go-carts can the to the wheel inventor? manufacturer make? 0 Chapter 4 Linear Equations 4.4 Graphing Linear Equations

8 Use the graphs and tables a f to answer Eercises. a. b. c Use the the graphs and tables a f a f to to answers Eercises 4. 4 a. a. b. b. c. c d. e. f. 4 4 d. d. e. e. f. f Is the slope of each line positive, negative, or neither?.. Is Is the the slope of of each line line positive, negative, or or neither?. What is the slope of each line?.. What is is the the slope of of each line?. What is the -intercept of each line? What is is the the -intercept of of each line? Graph zero Graph each of each line the graph. given line given the slope the the slope and -intercept. and -intercept. Identif 4. slope the the zero ; of of -int. the the graph. 5. slope ; -int slope ; ; -int slope ; -int. 6. slope ; -int slope 0; -int slope ; 8. 0; ; -int slope 0; -int. 4 Graph Graph each line each given line given the slope the the slope and d a and a point a point on on the the on on line. the the line. 8. slope slope ; (0, ; ; ) (0, (0, ) slope 0. slope ; ( 4, ) 5 ; 5 ; ( 4, ) 0. slope.. slope ; (, ; ) (,.. slope 4; (, 6) 6) ; (, ). slope 4; (, 6) Graph Graph each each equation. equation. Compare Compare each each graph graph to the to to parent the the parent function function Graph each equation. Give the the slope and -intercept of of Graph each graph. equation. Give the slope and -intercept of for for graphs graphs each 6. graph m ; ; b 8 m ; ; b 6 6 m.. m ; 5 5 ; ; b b m 5; 5; b. m ; 5 ; b m. ; ; b 4.. An An asphalt compan has has found the the equation G 0 0L approimates the the number of of trucks of of gravel required for for surfacing three-lane cit cit streets. G represents the the number of of trucks of of gravel required and L represents the the length of of the the 4. Graphing Linear Equations

9 . An asphalt compan has found the equation G 0 0L approimates the number of trucks of gravel required for surfacing three-lane cit streets. G represents the number of trucks of gravel required and L represents the length of the street in kilometers. a. What are the slope and G-intercept of the equation? b. Complete a table for the street lengths and the number of trucks of gravel. Use street lengths between and 0 kilometers. c. Last week, the asphalt compan used 0 truckloads of gravel. How much of the street did the compan resurface?. You can tell roughl what the temperature (T ) is on a summer evening if ou count how man times (N) a cricket chirps in one minute. form The formula to find the temperature in degrees fahrenheit is T 4 N 40. a. Draw mula graph of the formula. Use our graph to find the temperature if ou count 00 chirps per minute. Find the number of chirps ou might hear if the temperature is 95 F. b. What are reasonable values for T and N in this problem? cm In an eperiment on plant growth, a certain species of plant is found to grow 0.05 centimeters per da. The plant measured two centimeters when the eperiment started. Let H represent the ending measurement of the plant. Let d represent the number of das during which the eperiment takes place. a. Make a table that models the growth of the plant. Write an equation from our table. b. What are the slope and -intercept for the equation? c. At the end of the eperiment, the plant was. cm tall. How man das did the eperiment last? Mied Review 4. Paul is going to use 0% of his savings to make a down pament on a car. He can make a down pament of $,500. How much does Paul have in his savings account? 5. At 7 a.m., Jared notes that it is 5 C. The weather forecaster reported that from 5 a.m. to 7 a.m., the temperature rose 5. What was the temperature at 5 a.m.? Chapter 4 Linear Equations