Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Linear algebra wikipedia , lookup
Eigenvalues and eigenvectors wikipedia , lookup
Quartic function wikipedia , lookup
System of polynomial equations wikipedia , lookup
Cubic function wikipedia , lookup
Quadratic equation wikipedia , lookup
Elementary algebra wikipedia , lookup
History of algebra wikipedia , lookup
Median graph wikipedia , lookup
System of linear equations wikipedia , lookup
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line by the intercept method 4. Graph a line that passes through the origin 5. Determine domain and range 6. Graph horizontal and vertical lines In previous algebra classes you have solved equations in one variable such as 3x 2 5x 4. Solving such an equation required finding the value of the variable, in this case x, that made the equation a true statement. In this case, that value is x 3, because 3(3) 2 5(3) 4 This is a true statement because each side of the equation is equal to 11; no other value for x makes this statement true. The solution can be written in three different ways. We can write x 3, xx 3 which is read “the set of all x such that x equals 3,” or simply 3, which is the set containing the number 3. What if we have an equation in two variables, such as 3x y 6? The solution set is defined in a similar manner. Definitions: Solution Set for an Equation in Two Variables The solution set for an equation in two variables is the set containing all ordered pairs of real numbers (x, y) that will make the equation a true statement. The solution set for an equation in two variables is a set of ordered pairs. Typically, there will be an infinite number of ordered pairs that make an equation a true statement. We can find some of these ordered pairs by substituting a value for x, then solving the remaining equation for y. We will use that technique in Example 1. Example 1 Finding Ordered Pair Solutions (a) 3x y 6 We will pick three values for x, set up a table for ordered pairs, and then determine the related value for y. x 1 0 1 190 y © 2001 McGraw-Hill Companies Find three ordered pairs that are solutions for each equation. THE GRAPH OF A LINEAR EQUATION SECTION 4.1 Substituting 1 for x, we get 3(1) y 6 3 y 6 y9 The ordered pair (1, 9) is a solution to the equation 3x y 6. Substituting 0 for x, we get 3(0) y 6 0y6 y6 The ordered pair (0, 6) is a solution to the equation 3x y 6. Substituting 1 for x, we get 3(1) y 6 3y6 y3 NOTE To indicate the set of all solutions to the equation, we write {(x, y) 3x y 6} The ordered pair (1, 3) is a solution to the equation 3x y 6. Completing the table gives us the following: x y 1 0 1 9 6 3 (b) 2x y 1 Let’s try a different set of values for x. We will use the following table. x y 5 0 5 Substituting 5 for x, we get 2(5) y 1 © 2001 McGraw-Hill Companies 10 y 1 y 11 y 11 The ordered pair (5, 11) is a solution to the equation 2x y 1. Substituting 0 for x, we get 2(0) y 1 0y1 y 1 y 1 191 192 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS NOTE Again, the set of all solutions is {(x, y) 2x y 1} The ordered pair (0, 1) is a solution to the equation 2x y 1. Substituting 5 for x, we get 2(5) y 1 10 y 1 y 9 y9 The ordered pair (5, 9) is a solution to the equation 2x y 1. Completing the table gives us the following: x y 5 0 5 11 1 9 CHECK YOURSELF 1 Find three ordered pairs that are solutions for each equation. (a) 2x y 6 (b) 3x y 2 The graph of the solution set of an equation in two variables, usually called the graph of the equation, is the set of all points with coordinates (x, y) that satisfy the equation. In this chapter, we are primarily interested in a particular kind of equation in x and y and the graph of that equation. The equations we refer to involve x and y to the first power, and they are called linear equations. NOTE Why can A and B not both be zero? First, recall that, although x and y are variables, A, B, and C are constants. With that in mind, look at the equation if A and B are both zero. (0)x (0)y C 00C Definitions: Linear Equations An equation of the form Ax By C in which A and B cannot both be zero, is called the standard form for a line. Its graph is always a line. 0C 00 This would be a true statement regardless of the values of x and y. Its graph would be every point in the plane. NOTE Because two points determine a line, technically two points are all that are needed to graph the equation. You may want to locate at least one other point as a check of your work. Example 2 Graphing by Plotting Points Graph the equation xy5 This is a linear equation in two variables. To draw its graph, we can begin by assigning values to x and finding the corresponding values for y. For instance, if x 1, we have 1y5 y4 Therefore, (1, 4) satisfies the equation and is on the graph of x y 5. © 2001 McGraw-Hill Companies Because zero must be a constant, we are left with the statement THE GRAPH OF A LINEAR EQUATION SECTION 4.1 193 Similarly, (2, 3), (3, 2), and (4, 1) are in the graph. Often these results are recorded in a table of values, as shown below. We then plot the points determined and draw a line through those points. NOTE If you first rewrite an equation so that y is isolated on the left side, it can be easily entered and graphed with a graphing calculator. In this case, graph the equation y x 5 xy5 y x y 1 2 3 4 4 3 2 1 (1, 4) (2, 3) (3, 2) (4, 1) x Every point on the graph of the equation x y 5 has coordinates that satisfy the equation, and every point with coordinates that satisfy the equation lies on the line. CHECK YOURSELF 2 Graph the equation 2x y 6. NOTE An algorithm is a sequence of steps that solve a problem. The following algorithm summarizes our first approach to graphing a linear equation in two variables. Step by Step: To Graph a Linear Equation Step 1 Find at least three solutions for the equation, and write your results in a table of values. Step 2 Graph the points associated with the ordered pairs found in step 1. Step 3 Draw a line through the points plotted above to form the graph of the equation. Two particular points are often used in graphing an equation because they are very easy to find. The x intercept of a line is the point at which the line crosses the x axis. If the x intercept exists, it can be found by setting y 0 in the equation and solving for x. The y intercept is the point at which the line crosses the y axis. If the y intercept exists, it is found by letting x 0 and solving for y. © 2001 McGraw-Hill Companies Example 3 Graphing by the Intercept Method Use the intercepts to graph the equation NOTE Solving for y, we get 1 y x3 2 To graph this result on your calculator, you can enter Y1 (1 2)x 3 using the x, T, u, n key for x. x 2y 6 To find the x intercept, let y 0. x206 x6 The x intercept is (6, 0). CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS To find the y intercept, let x 0. 0 2y 6 2y 6 y 3 The y intercept is (0, 3). Graphing the intercepts and drawing the line through those intercepts, we have the desired graph. y x 2y 6 (6, 0) x (0, 3) CHECK YOURSELF 3 Graph, using the intercept method. 4x 3y 12 The following algorithm summarizes the steps of graphing a line by the intercept method. Step by Step: Graphing by the Intercept Method Step 1 Step 2 Step 3 Step 4 Find the x intercept. Let y 0, and solve for x. Find the y intercept. Let x 0, and solve for y. Plot the two intercepts determined in steps 1 and 2. Draw a line through the intercepts. y y intercept (x 0) x x intercept (y 0) When can the intercept method not be used? Some lines have only one intercept. For instance, the graph of x 2y 0 passes through the origin. In this case, other points must be used to graph the equation. © 2001 McGraw-Hill Companies 194 THE GRAPH OF A LINEAR EQUATION SECTION 4.1 195 Example 4 Graphing a Line That Passes Through the Origin NOTE Graph the equation 1 y x 2 Note that the line passes through the origin. Graph x 2y 0. Letting y 0 gives x200 x0 Thus (0, 0) is a solution, and the line has only one intercept. We continue by choosing any other convenient values for x. If x 2: 2 2y 0 2y 2 y 1 So (2, 1) is a solution. You can easily verify that (4, 2) is also a solution. Again, plotting the points and drawing the line through those points, we have the desired graph. y x 2y 0 x (2, 1) (0, 0) (4, 2) CHECK YOURSELF 4 Graph the equation x 3y 0. In Section 3.1, we defined the terms domain and range. Recall that the domain of a relation is the set of all the first elements in the ordered pairs. The range is the set of all the second elements. Recall that a line is the graph of a set of ordered pairs. In Example 5, we will examine the domain and range for the graph of a line. Example 5 Finding the Domain and Range Find the domain and range for the relation described by the equation © 2001 McGraw-Hill Companies xy5 We can analyze the domain and range either graphically or algebraically. First, we will look at a graphical analysis. From Example 2, let’s look at the graph of the equation. y (1, 4) (2, 3) (3, 2) (4, 1) x CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS The graph continues forever at both ends. For every value of x, there is an associated point on the line. Therefore, the domain (D) is the set of all real numbers. In set notation, we write D xx R This is read, “The domain is the set of every x that is a real number.” To find the range (R), we look at the graph to see what values are associated with y. Note that every y is associated with some point. The range is written as R yy R This is read, “The range is the set of every y that is a real number.” Let’s find the domain and range for the same relation by using an algebraic analysis. Look at the following equation. xy5 To determine the domain, we need to find every value of x that allows us to solve for y. That combination will result in an ordered pair (x, y). The set of all those x values is the domain of the relation. We can find a value for y for any real value of x. For example, if x 5, 5 y 5 y 10 The ordered pair (5, 10) is part of the relation. As in our graphical analysis, the domain is D xx R By a similar argument, we can substitute any value for y and solve the equation for x. The range is R yy R CHECK YOURSELF 5 Find the domain and range for the relation described by the following equation. xy4 Two types of linear equations are worthy of special attention. Their graphs are lines that are parallel to the x or y axis, and the equations are special cases of the general form Ax By C in which either A 0 or B 0. Rules and Properties: Vertical or Horizontal Lines 1. A line with an equation of the form yk is horizontal (parallel to the x axis). 2. A line with an equation of the form xh is vertical (parallel to the y axis). © 2001 McGraw-Hill Companies 196 THE GRAPH OF A LINEAR EQUATION SECTION 4.1 197 Example 6 illustrates both cases. Example 6 Graphing Horizontal and Vertical Lines NOTE Because part (a) is a function, it can be graphed on your calculator. Part (b) is not a function and cannot be graphed on your calculator. (a) Graph the line with equation y3 You can think of the equation in the equivalent form 0xy3 Note that any ordered pair of the form (__, 3) will satisfy the equation. Because x is multiplied by 0, y will always be equal to 3. For instance, (2, 3) and (3, 3) are on the graph. The graph, a horizontal line, is shown below. y y3 x The domain for a horizontal line is every real number. The range is a single y value. We write D xx R and R 3 (b) Graph the line with equation x 2 In this case, you can think of the equation in the equivalent form x 0 y 2 NOTE Notice that D 2 Now any ordered pair of the form (2, __) will satisfy the equation. Examples are (2, 1) and (2, 3). The graph, a vertical line, is shown below. and y R y y R x 2 © 2001 McGraw-Hill Companies x CHECK YOURSELF 6 Graph each equation and state the domain and range. (a) y 3 (b) x 5 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS CHECK YOURSELF ANSWERS 1. (a) Answers will vary, but could include (0, 6); (b) Answers will vary, but could y include (0, 2). 2. x y 0 1 2 6 4 2 x (2, 2) (1, 4) (0, 6) 3. 4. y 5. D xx R and R y y R y x 3y 0 (0, 4) 4x 3y 12 (3, 1) (0, 0) (6, 2) x (3, 0) x 6. (a) (b) y y x5 x x (5, 0) (0, 3) D xx R R 3 y 3 D 5 R y y R © 2001 McGraw-Hill Companies 198 Name 4.1 Exercises Section Date In exercises 1 to 8, find three ordered pairs that are solutions to the given equations. 1. 2x y 5 2. 3x y 7 3. 7x y 8 4. 5x y 3 5. 4x 5y 20 6. 2x 3y 6 7. 3x y 0 8. 2x y 0 ANSWERS 1. In exercises 9 to 26, graph each of the equations. 2. 9. x y 6 10. x y 6 11. y x 2 y y y 3. 4. 5. x x x 6. 7. 8. 9. 12. y x 5 13. y x 1 y y 14. y 2x 2 10. y 11. 12. x x x 13. 14. 15. 16. 17. 15. y 2x 1 16. y 3x 1 y y y © 2001 McGraw-Hill Companies 1 17. y x 3 2 x x x 199 ANSWERS 18. 18. y 2x 4 19. 19. y x 3 20. y 2x 4 y y y 20. 21. x x x 22. 23. 24. 25. 26. 21. x 2y 0 22. x 2y 0 y 23. x 4 y y x 24. x 4 25. y 4 26. y 6 y y y x x © 2001 McGraw-Hill Companies x x x 200 ANSWERS In exercises 27 to 38, find the x and y intercepts and then graph each equation. 27. x 2y 4 28. x 3y 6 y 27. 29. 2x y 6 28. y y 29. x x x 30. 31. 32. 33. 30. 3x 2y 12 31. 2x 5y 10 y y 33. 5x 6y 0 34. 2x 7y 0 y x 35. x 4y 8 0 y y x x © 2001 McGraw-Hill Companies x 35. y x x 34. 32. 2x 3y 6 201 ANSWERS 36. 2x y 6 0 36. 37. 8x 4y y 37. 38. 6x 7y y y 38. x 39. x x 40. 41. 42. 43. In exercises 39 to 46, find the domain and range of each of the relations. 44. 39. 3x 2y 4 40. 5x 4y 20 41. 6x 2y 18 42. x 5y 8 43. x 4 44. 2x 10 0 45. y 3 46. 3y 12 0 45. 46. 47. 48. 49. 50. 51. For exercises 47 to 54, select a window that allows you to see both the x and y intercepts on your calculator. If that is not possible, explain why not. 52. 53. 47. x y 40 48. x y 80 49. 2x 3y 900 50. 5x 8y 800 51. y 5x 90 52. y 3x 450 53. y 30x 54. y 200 202 © 2001 McGraw-Hill Companies 54. ANSWERS Two distinct lines in the plane either are parallel or they intersect. In exercises 55 to 58, graph each pair of equations on the same set of axes, and find the point of intersection, where possible. 55. x y 6, x y 4 56. y x 3, y x 1 y 56. 57. y x 55. 58. x 59. 60. 57. y 2x, y x 1 58. 2x y 3, 2x y 5 61. y y 62. 63. x x 64. 59. Graph y x and y 2x on the same set of axes. What do you observe? 60. Graph y 2x 1 and y 2x 1 on the same set of axes. What do you observe? 61. Graph y 2x and y 2x 1 on the same set of axes. What do you observe? 62. Graph y 3x 1 and y 3x 1 on the same set of axes. What do you © 2001 McGraw-Hill Companies observe? 1 2 63. Graph y 2x and y x on the same set of axes. What do you observe? 1 3 you observe? 64. Graph y x 7 and y 3x 2 on the same set of axes. What do 3 203 ANSWERS 65. Use your graphing utility to graph each of the following equations. 66. 65. y 3 66. y 2 67. 67. y 3x 1 68. y 2x 2 69. Write an equation whose graph will have no x intercept but will have a y intercept at (0, 68. 6). 70. Write an equation whose graph will have no y intercept but will have an x intercept 69. at (5, 0). 70. Answers 3. (0, 8), (1, 1), (1, 15) 1. (0, 5), (1, 3), (1, 7) 7. (0, 0), (1, 3), (1, 3) 9. x y 6 11. y x 2 13. y x 1 y y y x x 1 2 15. y 2x 1 17. y x 3 19. y x 3 y y 21. x 2y 0 y x x 23. x 4 x x 25. y 4 y y 204 x y x x © 2001 McGraw-Hill Companies 24 5. (0, 4), (5, 0), 1, 5 27. x 2y 4; y intercept (0, 2); 29. 2x y 6; y intercept (0, 6); x intercept (4, 0) x intercept (3, 0) y y x 31. 2x 5y 10; y intercept (0, 2); x 33. 5x 6y 0; x intercept (5, 0) intercepts: (0, 0) y y x x 35. x 4y 8 0; y intercept (0, 2); 37. 8x 4y; x intercept (8, 0) intercepts: (0, 0) y y © 2001 McGraw-Hill Companies x 39. 41. 43. 45. 47. 49. 51. 53. x D: xx R; R: yy R D: xx R; R: yy R D: 4; R: yy R D: xx R; R: 3 X max 40, Y max 40 X max 450, Y max 300 X min 18, Y max 90 Any viewing window that shows the origin 205 55. Intersection: (5, 1) 57. Intersection: (1, 2) y y x x 59. The line corresponding to 61. The two lines appear to be parallel. y 2x is steeper than that corresponding to y x. y y x x 63. The lines appear to be 65. y 3 4 perpendicular. 2 y 4 2 2 4 2 4 x y 3x1 67. 69. y 6 206 2 2 4 © 2001 McGraw-Hill Companies 4