Graphing Linear Equations


 Homer Copeland
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1 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are now read to combine our work of the previous two sections. In Section 6.1 ou learned to write the solutions of equations in two variables as ordered pairs. Then, in Section 6.2, these ordered pairs were graphed in the plane. Putting these ideas together will let us graph certain equations. Eample 1 illustrates this approach. Eample 1 Graphing a Linear Equation Graph 2 4. NOTE We are going to find three solutions for the equation. We ll point out wh shortl. NOTE The table is just a convenient wa to displa the information. It is the same as writing (0, 2), (2, 1), and (4, 0). Step 1 Find some solutions for 2 4. To find solutions, we choose an convenient values for, sa 0, 2, and 4. Given these values for, we can substitute and then solve for the corresponding value for. So If 0, then 2, so (0, 2) is a solution. If 2, then 1, so (2, 1) is a solution. If 4, then 0, so (4, 0) is a solution. A hand wa to show this information is in a table such as this: Step 2 We now graph the solutions found in step (0, 2) (2, 1) (4, 0) What pattern do ou see? It appears that the three points lie on a straight line, and that is in fact the case. NOTE The arrows on the end of the line mean that the line etends indefinitel in either direction. Step 3 Draw a straight line through the three points graphed in step (0, 2) (2, 1) (4, 0) 495
2 496 CHAPTER 6 AN INTRODUCTION TO GRAPHING NOTE The graph is a picture of the solutions for the given equation. The line shown is the graph of the equation 2 4. It represents all of the ordered pairs that are solutions (an infinite number) for that equation. Ever ordered pair that is a solution will have its graph on this line. An point on the line will have coordinates that are a solution for the equation. Note: Wh did we suggest finding three solutions in step 1? Two points determine a line, so technicall ou need onl two. The third point that we find is a check to catch an possible errors. CHECK YOURSELF 1 Graph 2 6, using the steps shown in Eample 1. Let s summarize. An equation that can be written in the form A B C in which A, B, and C are real numbers and A and B cannot both be 0 is called a linear equation in two variables. The graph of this equation is a straight line. The steps of graphing follow. Step b Step: To Graph a Linear Equation Step 1 Find at least three solutions for the equation, and put our results in tabular form. Step 2 Graph the solutions found in step 1. Step 3 Draw a straight line through the points determined in step 2 to form the graph of the equation. Eample 2 NOTE Let 0, 1, and 2, and substitute to determine the corresponding values. Again the choices for are simpl convenient. Other values for would serve the same purpose. Graphing a Linear Equation Graph 3. Step Some solutions are
3 GRAPHING LINEAR EQUATIONS SECTION Step 2 Graph the points. (2, 6) (1, 3) (0, 0) NOTE Notice that connecting an two of these points produces the same line. Step 3 Draw a line through the points. 3 CHECK YOURSELF 2 Graph the equation 2 after completing the table of values Let s work through another eample of graphing a line from its equation. Eample 3 Graphing a Linear Equation Graph 2 3.
4 498 CHAPTER 6 AN INTRODUCTION TO GRAPHING Step 1 Some solutions are Step 2 Graph the points corresponding to these values. (0, 3) (1, 5) (2, 7) Step 3 Draw a line through the points. 2 3 CHECK YOURSELF 3 Graph the equation 3 2 after completing the table of values
5 GRAPHING LINEAR EQUATIONS SECTION In graphing equations, particularl when fractions are involved, a careful choice of values for can simplif the process. Consider Eample 4. Graphing a Linear Equation Graph Eample As before, we want to find solutions for the given equation b picking convenient values for. Note that in this case, choosing multiples of 2 will avoid fractional values for and make the plotting of those solutions much easier. For instance, here we might choose values of 2, 0, and 2 for. Step 1 If 2: 3 2 ( 2) If 0: NOTE Suppose we do not choose a multiple of 2, sa, 3. Then 3 2 (3) , 5 2 is still a valid solution, but we must graph a point with fractional coordinates. 3 2 (0) If 2: 3 2 (2) In tabular form, the solutions are Step 2 Graph the points determined above. ( 2, 5) (0, 2) (2, 1)
6 500 CHAPTER 6 AN INTRODUCTION TO GRAPHING Step 3 Draw a line through the points CHECK YOURSELF 4 Graph the equation 1 after completing the table of values Some special cases of linear equations are illustrated in Eamples 5 and 6. Eample 5 Graphing an Equation That Results in a Vertical Line Graph 3. The equation 3 is equivalent to 0 3. Let s look at some solutions. If 1: If 4: If 2: ( 2) In tabular form,
7 GRAPHING LINEAR EQUATIONS SECTION What do ou observe? The variable has the value 3, regardless of the value of. Look at the graph on the following page. 3 (3, 4) (3, 1) (3, 2) The graph of 3 is a vertical line crossing the ais at (3, 0). Note that graphing (or plotting) points in this case is not reall necessar. Simpl recognize that the graph of 3 must be a vertical line (parallel to the ais) that intercepts the ais at (3, 0). CHECK YOURSELF 5 Graph the equation 2. Eample 6 is a related eample involving a horizontal line. Eample 6 Graphing an Equation That Results in a Horizontal Line Graph 4. Because 4 is equivalent to 0 4, an value for paired with 4 for will form a solution. A table of values might be
8 502 CHAPTER 6 AN INTRODUCTION TO GRAPHING Here is the graph. ( 2, 4) (2, 4) (0, 4) This time the graph is a horizontal line that crosses the ais at (0, 4). Again the graphing of points is not required. The graph of 4 must be horizontal (parallel to the ais) and intercepts the ais at (0, 4). CHECK YOURSELF 6 Graph the equation 3. The following bo summarizes our work in the previous two eamples: Definitions: Vertical and Horizontal Lines 1. The graph of a is a vertical line crossing the ais at (a, 0). 2. The graph of b is a horizontal line crossing the ais at (0, b). NOTE With practice, all this can be done mentall, which is the big advantage of this method. To simplif the graphing of certain linear equations, some students prefer the intercept method of graphing. This method makes use of the fact that the solutions that are easiest to find are those with an coordinate or a coordinate of 0. For instance, let s graph the equation First, let 0 and solve for So (0, 4) is one solution. Now we let 0 and solve for.
9 GRAPHING LINEAR EQUATIONS SECTION A second solution is (3, 0). The two points corresponding to these solutions can now be used to graph the equation. NOTE Remember, onl two points are needed to graph a line. A third point is used onl as a check (0, 4) (3, 0) NOTE The intercepts are the points where the line cuts the and aes. The ordered pair (3, 0) is called the intercept, and the ordered pair (0, 4) is the intercept of the graph. Using these points to draw the graph gives the name to this method. Let s look at a second eample of graphing b the intercept method. Eample 7 Using the Intercept Method to Graph a Line Graph , using the intercept method. To find the intercept, let The intercept is (5, 0) To find the intercept, let The intercept is (0, 3) So (5, 0) and (0, 3) are solutions for the equation, and we can use the corresponding points to graph the equation. (5, 0) (0, 3)
10 504 CHAPTER 6 AN INTRODUCTION TO GRAPHING CHECK YOURSELF 7 Graph , using the intercept method. NOTE Finding the third checkpoint is alwas a good idea. This all looks quite eas, and for man equations it is. What are the drawbacks? For one, ou don t have a third checkpoint, and it is possible for errors to occur. You can, of course, still find a third point (other than the two intercepts) to be sure our graph is correct. A second difficult arises when the and intercepts are ver close to one another (or are actuall the same point the origin). For instance, if we have the equation , 0 0, 1 2. the intercepts are and It is hard to draw a line accuratel through these intercepts, so choose other solutions farther awa from the origin for our points. Let s summarize the steps of graphing b the intercept method for appropriate equations. Step b Step: Graphing a Line b the Intercept Method Step 1 To find the intercept: Let 0, then solve for. Step 2 To find the intercept: Let 0, then solve for. Step 3 Graph the and intercepts. Step 4 Draw a straight line through the intercepts. A third method of graphing linear equations involves solving the equation for. The reason we use this etra step is that it often will make finding solutions for the equation much easier. Let s look at an eample. NOTE Remember that solving for means that we want to leave isolated on the left. Eample 8 Graphing a Linear Equation Graph Rather than finding solutions for the equation in this form, we solve for Subtract Divide b 3. 3
11 GRAPHING LINEAR EQUATIONS SECTION or Now find our solutions b picking convenient values for. NOTE Again, to pick convenient values for, we suggest ou look at the equation carefull. Here, for instance, picking multiples of 3 for will make the work much easier. If 3: ( 3) So ( 3, 4) is a solution. If 0: So (0, 2) is a solution. If 3: So (3, 0) is a solution. We can now plot the points that correspond to these solutions and form the graph of the equation as before ( 3, 4) (0, 2) (3, 0) CHECK YOURSELF 8 Graph the equation Solve for to determine solutions.
12 506 CHAPTER 6 AN INTRODUCTION TO GRAPHING CHECK YOURSELF ANSWERS (3, 0) 2 6 (3, 0) (2, 2) (2, 2) (1, 4) (1, 4) (0, 4) (5, 0)
13 Name 6.3 Eercises Section Date Graph each of the following equations ANSWERS
14 ANSWERS
15 ANSWERS
16 ANSWERS
17 ANSWERS Graph each of the following equations, using the intercept method
18 ANSWERS Graph each of the following equations b first solving for
19 ANSWERS Write an equation that describes the following relationships between and. Then graph each relationship is twice. 38. is 3 times is 3 more than. 40. is 2 less than. 41. is 3 less than 3 times. 42. is 4 more than twice. 513
20 ANSWERS The difference of and the 44. The difference of twice and product of 4 and is 12. is Graph each pair of equations on the same aes. Give the coordinates of the point where the lines intersect Graph of winnings. The equation describes the amount of winnings a group earns for collecting plastic jugs in the reccling contest described in eercise 27 at the end of Section 6.2. Sketch the graph of the line on the coordinate sstem below. $600 $400 $ (Pounds) 514
21 ANSWERS 48. Minimum values. The contest sponsor will award a prize onl if the winning group in the contest collects 100 lb of jugs or more. Use our graph in eercise 47 to determine the minimum prize possible. 49. Fundraising. A high school class wants to raise some mone b reccling newspapers. The decide to rent a truck for a weekend and to collect the newspapers from homes in the neighborhood. The market price for reccled newsprint is currentl $11 per ton. The equation describes the amount of mone the class will make, in which is the amount of mone made in dollars, is the number of tons of newsprint collected, and 100 is the cost in dollars to rent the truck (a) (b) (c) (d) 50. (a) (b) (c) (d) (a) Using the aes below, draw a graph that represents the relationship between newsprint collected and mone earned. $400 $200 $ (Tons) (b) The truck is costing the class $100. How man tons of newspapers must the class collect to break even on this project? (c) If the class members collect 16 tons of newsprint, how much mone will the earn? (d) Si months later the price of newsprint is $17 dollars a ton, and the cost to rent the truck has risen to $125. Write the equation that describes the amount of mone the class might make at that time. 50. Production costs. The cost of producing a number of items is given b C m b, in which b is the fied cost and m is the variable cost (the cost of producing one more item). (a) If the fied cost is $40 and the variable cost is $10, write the cost equation. (b) Graph the cost equation. C (c) The revenue generated from the sale of items is given b R 50. Graph the revenue equation on the same set of aes as the cost equation. (d) How man items must be produced for the revenue to equal the cost (the breakeven point)? 515
22 ANSWERS 51. Graph each set of equations on the same coordinate sstem. Do the lines intersect? What are the intercepts? a. b. c. d. Getting Read for Section 6.4 [Section 1.4] Evaluate the following epressions ( 2) (a) (b) (c) (d) ( 4) 8 2 Answers
23
24 (3, 1) 47. Graph $600 $400 $ (Pounds) (a) Graph; (b) or L9 tons; 51. The lines do not intersect. 11 The intercepts are (0, 0), (0, 4), (c) $76; (d) and (0, 5). $400 $200 $ (Tons) a. 1 b. 2 c. d
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