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1 points of a diameter. ▼ SO LUTI O N (a) Using the equation of a circle with r 3, h 2, and k 5, we obtain The graph is shown in Figure 8 _2 2 x (2, _5) FIGURE 8 (x-2)™+(y+5)™=9 (b) We first observe that the center is the midpoint of the diameter PQ, so by the Midpoint Formula the center is, 1 1 2 The radius r is the distance from P to the center, so by the Distance Formula 1 8 2 22 2 53 (1, 8) (3, 1) x Q(5, _6) (x-3)™+(y-1)™=53 FIGURE 9 SE CTI ON 2. 2 | Graphs of Equations in Two Variables 151 Therefore, the equation of the circle is The graph is shown in Figure 9 53 ✎ Practice what you’ve learned: Do Exercises 55 and 59. ▲ Let’s expand the equation of the circle in the preceding example. 2 1 2 x 3 y 1 2 53 x 2 6x 9 y 2 2y 1 53 x 2 6x y 2 2y 43 2 1 Standard form Expand the squares Subtract 10 to get expanded form Completing the square is used in many contexts in algebra. In Section 1.3 we used completing the square to solve quadratic equations. Suppose we are given the equation of a circle in expanded form. Then to find its center and radius, we must put the equation back in standard form. That means that we must reverse the steps in the preceding calculation, and to do that, we need to know what to add to an expression such as x 2 6x to make it a perfect square—that is, we need to complete the square, as in the next example. Identifying an Equation of a Circle E X AM P L E 7 | Show that the equation x 2 y 2 2x 6y 7 0 represents a circle, and find the center and radius of the circle. We must add the same numbers to each side to maintain equality. ▼ SO LUTI O N We first group the x-terms and y-terms. Then we complete the square within each grouping. That is, we complete the square for x 2 2x by adding 2 1, and we complete |
the square for y 2 6y by adding 2x x 2 2x y2 6y y2 6y Group terms Complete the square by adding 1 and 9 to each side Factor and simplify r 13 Comparing this equation with the standard equation of a circle, we see that h 1, k 3, and 13 radius. ✎ Practice what you’ve learned: Do Exercise 65., so the given equation represents a circle with center 1, 3 1 2 and ▲ ■ Symmetry Figure 10 shows the graph of y x 2. Notice that the part of the graph to the left of the y-axis is the mirror image of the part to the right of the y-axis. The reason is that if the point, and these points are reflections of each other about the y-axis. In this situation we say that the graph is symmetric with respect to the y-axis. Similarly, we say that a graph is symmetric with respect to the x-axis. A graph is symmetric with if whenever the point respect to the origin if whenever 2 is on the graph, so is is on the graph, then so is is on the graph, then so is x, y x, y x, y x, y x, y x_x, y) y 1 0 y=≈ (x, y) 1 x FIGURE 10 152 CHAPTER 2 | Coordinates and Graphs DEFINITION OF SYMMETRY Type of symmetry Symmetry with respect to the x-axis How to test for symmetry The equation is unchanged when y is replaced by y Symmetry with respect to the y-axis The equation is unchanged when x is replaced by x What the graph looks like (figures in this section) y 0 (x, y) x (x, _y) (Figures 6, 11, 12) (_x, y) y 0 (x, y) x (Figures 2, 3, 4, 6, 10, 12) Geometric meaning Graph is unchanged when reflected in the x-axis Graph is unchanged when reflected in the y-axis Symmetry with respect to the origin The equation is unchanged when x is replaced by x and y by y y 0 (x, y) x Graph is unchanged when rotated 180 about the origin (_x, _y) (Figures 6, 12) The remaining examples in this section show how symmetry helps us |
to sketch the graphs of equations. E X AM P L E 8 | Using Symmetry to Sketch a Graph Test the equation x y 2 for symmetry, and sketch the graph. ▼ SO LUTI O N If y is replaced by y in the equation x y 2, we get y x 1 x y 2 2 2 Replace y by –y Simplify so the equation is unchanged. Therefore, the graph is symmetric about the x-axis. But changing x to x gives the equation x y 2, which is not the same as the original equation, so the graph is not symmetric about the y-axis. We use the symmetry about the x-axis to sketch the graph by first plotting points just for y 0 and then reflecting the graph in the x-axis, as shown in Figure 11 on the next page. SE CTI ON 2. 2 | Graphs of Equations in Two Variables 153 y 0 1 2 3 y 4 (1, 1) (0, 0) x y2 0 1 4 9 x, y 1 0, 0 1, 1 4, 2 99, 3) (4, 2) 4 x FIGURE 11 x=¥ ✎ Practice what you’ve learned: Do Exercises 77 and 83. ▲ E X AM P L E 9 | Testing an Equation for Symmetry Test the equation y x 3 9x for symmetry. ▼ SO LUTI O N If we replace x by x and y by y in the equation, we get 9x y x 3 9x Replace x by –x and y by –y Simplify Multiply by –1 so the equation is unchanged. This means that the graph is symmetric with respect to the origin. ✎ Practice what you’ve learned: Do Exercise 79. ▲ E X AM P L E 10 | A Circle That Has All Three Types of Symmetry Test the equation of the circle x 2 y 2 4 for symmetry. The equation x 2 y2 4 remains unchanged when x is replaced by x ▼ SO LUTI O N and y is replaced by y, since 2 y 2, so the circle exhibits all three and types of symmetry. It is symmetric with respect to the x-axis, the y-axis, and the origin, as shown in Figure 12_x, y) (x, y) 0 2 x (_x, _y) (x |
, _y) ≈+¥=4 FIGURE 12 ✎ Practice what you’ve learned: Do Exercise 81. ▲ 154 CHAPTER 2 | Coordinates and Graphs 2. ▼ CONCE PTS 1. If the point (2, 3) is on the graph of an equation in x and y, then the equation is satisfied when we replace x by ✎ and y by of the equation 2y x 1?. Is the point 2, 3 1 2 on the graph 2. (a) To find the x-intercept(s) of the graph of an equation, we set x-intercept of 2y x 1 is equal to 0 and solve for.. So the (b) To find the y-intercept(s) of the graph of an equation, we equal to 0 and solve for. So the set y-intercept of 2y x 1 is x 1 3. The graph of the equation with center (, 2. y 2 2 1 2 9 is a circle. 2 2 1 and radius 4. (a) If a graph is symmetric with respect to the x-axis and a, b) is on the graph, then 1 graph. 1, is also on the 2 (b) If a graph is symmetric with respect to the y-axis and is on the graph, then a, b 2 1 graph., 1 is also on the 2 (c) If a graph is symmetric about the origin and a, b is on the graph, then, 1 2 2 is also on the graph. 1 ▼ SKI LLS 5–10 ■ Determine whether the given points are on the graph of the equation. 5. y 3x 2; 1, 1 0, 2,, 1 3, 1 1 y 2x 1; 6. 7. x 2y 1 0; 1; 2 1 8. 1 9. x 2 xy y 2 4; x y 1 1 2 A 2 1, 0, 1 B 0, 1, 2 3, 2 2 1, 1 1 1, 1 1 2 0, 0 1 2 1, 0, 1 1, 1 2B, 2, A 2 0, 2, 1 2 1, 1, 2B A 1, 2, 2 1 2 1 2 10. x 2 y 2 1; 0, 1, 2 a 1 1 12, 1 12 b, a |
2 2, 2 1 13 2, 1 2 b ✎ ✎ 11–36 ■ Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts. 11. y x 13. y x 4 15. 2x y 6 17. y 1 x 2 19. 4y x 2 21. y x 2 9 23. xy 2 12. y 2x 14. y 3x 3 16. x y 3 18. y x 2 2 20. 8y x 3 22. y 9 x 2 24. x y 2 4 26. x 2 y 2 9 25. 27. y 1x y 24 x 2 28. y 24 x 2 29. y x 0 0 y 4 31. 33. x y 3 35. y x 4 x 0 0 0 30. x y 0 4 x y 32. 34. y x 3 1 36. y 16 x 4 0 0 37–40 ■ An equation and its graph are given. Find the x- and y-intercepts. 37. y 4x x 2 38 39. x 4 y 2 xy 16 40. x 2 y 3 x 2y 2 64 41–48 ■ Find the x- and y-intercepts of the graph of the equation. 41. y x 3 43. y x 2 9 42. y x 2 5x 6 44. y 2xy 2x 1 ✎ 45. x 2 y 2 4 47. xy 5 y 1x 1 46. 48. x 2 xy y 1 ✎ ✎ 49–54 ■ Find the center and radius of the circle, and sketch its graph. 49. x 2 y 2 9 51. (x 3)2 y 2 16 53. (x 3)2 (y 4)2 25 50. x 2 y 2 5 52. x 2 (y 2)2 4 54. (x 1)2 (y 2)2 36 55–62 ■ Find an equation of the circle that satisfies the given conditions. ✎ 55. Center 1 56. Center 2, 1 ; 2 1, 4 radius 3 ; radius 8 2 57. Center at the origin; passes through 1 ✎ 58. Center 1, 5 1 2 ; passes through 59. Endpoints of a diameter are P 1 60. Endpoints of a diameter are 7, 3 2 tangent to the x-axis 61. Center P ; 1 1 2 4, 7 1 4 |
, 6 2 1 1, 1 1, 3 2 Q and 2 and Q 5, 9 2 7, 5 1 1 2 62. Circle lies in the first quadrant, tangent to both x-and y-axes; radius 5 85. Symmetric with respect 86. Symmetric with respect to the origin to the origin SE CTI ON 2. 2 | Graphs of Equations in Two Variables 155 63–64 ■ Find the equation of the circle shown in the figure. 63. 64. y 2 y 2 _2 0 2 x _2 0 2 x 65–72 ■ Show that the equation represents a circle, and find the center and radius of the circle. ✎ 65. x 2 y 2 2x 4y 1 0 66. x 2 y 2 2x 2y 2 67. x 2 y 2 4x 10y 13 0 68. x 2 y 2 6y 2 0 69. x 2 y 2 x 0 70. x 2 y 2 2x y 1 0 71. 72 2y 1 8 16 0 73–76 ■ Sketch the graph of the equation. 73. x 2 y 2 4x 10y 21 74. 4x 2 4y 2 2x 0 75. x 2 y 2 6x 12y 45 0 76. x 2 y 2 16x 12y 200 0 77–82 ■ Test the equation for symmetry. 77. y x 4 x 2 79. y x 3 10x 81. x 4y 4 x 2y 2 1 78. x y 4 y 2 80. y x 2 0 82. x 2y 2 xy 1 0 x 83–86 ■ Complete the graph using the given symmetry property. 83. Symmetric with respect 84. Symmetric with respect to the y-axis to the x-axis y 0 y= 1 1+≈ x y 0 ¥-≈=1 x ✎ ✎ ✎ ✎ y 0 y= x 1+≈ x y 0 y= 1 x£ x 0 2 x 5 1 87. 88. x, y 87–90 ■ Sketch the region given by the set 2x x 6 2 9 6 2 4 x, y x, y x, y 89. 90 91. Find the area of the region that lies outside the circle x 2 y 2 4 but inside the circle 2 y 2 4y 12 0 x 92. Sketch the region in the coordinate plane that |
satisfies both the. What is the area of this y x inequalities x 2 y 2 9 and region? 0 0 ▼ APPLICATIONS 93. U.S. Inflation Rates The graph shows the annual inflation rate in the United States from 1975 to 2003. (a) Estimate the inflation rates in 1980, 1991, and 1999 to the nearest percent. (b) For which years in this period did the inflation rate exceed 6%? (c) Did the inflation rate generally increase or decrease in the years from 1980 to 1985? What about from 1987 to 1992? (d) Estimate the highest and lowest inflation rates in this time period to the nearest percent. 16 14 12 10 1970 1975 1980 1985 1990 1995 2000 2005 Year 94. Orbit of a Satellite A satellite is in orbit around the moon. A coordinate plane containing the orbit is set up with the center of the moon at the origin, as shown in the graph on the next page, with distances measured in megameters (Mm). The equation of the satellite’s orbit is 156 CHAPTER 2 | Coordinates and Graphs 2 1 x 3 25 2 y 2 16 1 (a) From the graph, determine the closest and the farthest that the satellite gets to the center of the moon. (b) There are two points in the orbit with y-coordinates 2. Find the x-coordinates of these points, and determine their distances to the center of the moon. y 2 2 x ▼ DISCOVE RY • DISCUSSION • WRITI NG 95. Circle, Point, or Empty Set? Complete the squares in the general equation x 2 ax y 2 by c 0 and simplify the result as much as possible. Under what conditions on the coefficients a, b, and c does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius. 96. Do the Circles Intersect? (a) Find the radius of each circle in the pair and the distance between their centers; then use this information to determine whether the circles intersect. (i; 2 16 (ii) (iii; 2 y 14 ; 2 25 (b) How can you tell, just by knowing the radii of two circles and the distance between their centers |
, whether the circles intersect? Write a short paragraph describing how you would decide this, and draw graphs to illustrate your answer. 97. Making a Graph Symmetric The graph shown in the figure is not symmetric about the x-axis, the y-axis, or the origin. Add more line segments to the graph so that it exhibits the indicated symmetry. In each case, add as little as possible. (a) Symmetry about the x-axis (b) Symmetry about the y-axis (c) Symmetry about the origin y 1 0 1 x Graphing Calculators: Solving Equations and Inequalities Graphically 2.3 LEARNING OBJECTIVES After completing this section, you will be able to: ■ Use a graphing calculator to graph equations ■ Solve equations graphically ■ Solve inequalities graphically In Chapter 1 we solved equations and inequalities algebraically. In the preceding section we learned how to sketch the graph of an equation in a coordinate plane. In this section we use graphs to solve equations and inequalities. To do this, we must first draw a graph using a graphing device. So we begin by giving a few guidelines to help us use graphing devices effectively. SE CTI ON 2. 3 | Graphing Calculators: Solving Equations and Inequalities Graphically 157 ■ Using a Graphing Calculator A graphing calculator or computer displays a rectangular portion of the graph of an equation in a display window or viewing screen, which we call a viewing rectangle. The default screen often gives an incomplete or misleading picture, so it is important to choose the viewing rectangle with care. If we choose the x-values to range from a minimum value of Xmin a to a maximum value of Xmax b and the y-values to range from a minimum value of Ymin c to a maximum value of Ymax d, then the displayed portion of the graph lies in the rectangle a, b c, d x, y a x b, c y d 2 as shown in Figure 1. We refer to this as the 51 4 4 3 3 0 by x, y a, b The graphing device draws the graph of an equation much as you would. It plots points for a certain number of values of x, equally spaced between a and b. If of the form the equation is not defined for an x-value or if the corresponding y-value lies outside the viewing |
rectangle, the device ignores this value and moves on to the next x-value. The machine connects each point to the preceding plotted point to form a representation of the graph of the equation. c, d 2 1 3 4 4 3 6 viewing rectangle. (a, d) y=d (b, d) x=a x=b (a, c) y=c (b, c) FIGURE 1 The viewing rectangle a, b c, d by 3 4 3 4 E X AM P L E 1 | Choosing an Appropriate Viewing Rectangle Graph the equation y x 2 3 in an appropriate viewing rectangle. ▼ SO LUTI O N viewing rectangle Let’s experiment with different viewing rectangles. We start with the 2, 2, so we set 2, 2 by 3 4 3 4 Xmin 2 Xmax 2 Ymin 2 Ymax 2 The resulting graph in Figure 2(a) is blank! This is because x 2 0, so x 2 3 3 for all x. Thus, the graph lies entirely above the viewing rectangle, so this viewing rectangle is not appropriate. If we enlarge the viewing rectangle to, as in Figure 2(b), we begin to see a portion of the graph. 4, 4 4, 4 by 3 4 4 3 Now let’s try the viewing rectangle 10, 10. The graph in Figure 2(c) seems to give a more complete view of the graph. If we enlarge the viewing rectangle even further, as in Figure 2(d), the graph doesn’t show clearly that the y-intercept is 3. by gives an appropriate representation of So the viewing rectangle 10, 10 5, 30 5, 30 by 3 3 4 4 3 4 3 4 the graph. 2 4 _2 2 _4 4 _2 (a) _4 (b) FIGURE 2 Graphs of y x2 3 30 _5 (c) _10 10 _50 ✎ Practice what you’ve learned: Do Exercise 5. 1000 _100 (d) 50 ▲ 158 CHAPTER 2 | Coordinates and Graphs by 1, 3 2.5, 1.5 E X AM P L E 2 | Two Graphs on the Same Screen Graph the equations y 3x 2 6x 1 and y 0.23x 2.25 together in the viewing rectangle. Do the graphs intersect in this viewing rectangle? 3 ▼ SO LUTI O N Figure 3(a) shows the essential features |
of both graphs. One is a parabola, and the other is a line. It looks as if the graphs intersect near the point. However, if we zoom in on the area around this point as shown in Figure 3(b), we see that although the graphs almost touch, they do not actually intersect. 1, 2 2 1 3 4 4 1.5 _1.85 _1 3 _2.5 (a) FIGURE 3 0.75 _2.25 (b) 1.25 ✎ Practice what you’ve learned: Do Exercise 23. ▲ You can see from Examples 1 and 2 that the choice of a viewing rectangle makes a big difference in the appearance of a graph. If you want an overview of the essential features of a graph, you must choose a relatively large viewing rectangle to obtain a global view of the graph. If you want to investigate the details of a graph, you must zoom in to a small viewing rectangle that shows just the feature of interest. Most graphing calculators can only graph equations in which y is isolated on one side of the equal sign. The next example shows how to graph equations that don’t have this property. E X AM P L E 3 | Graphing a Circle Graph the circle x 2 y 2 1. ▼ SO LUTI O N We first solve for y, to isolate it on one side of the equal sign. y 2 1 x 2 Subtract x2 y 21 x 2 Take square roots Therefore, the circle is described by the graphs of two equations: y 21 x 2 and y 21 x 2 The first equation represents the top half of the circle (because y 0), and the second represents the bottom half of the circle (because y 0). If we graph the first equation in the viewing rectangle, we get the semicircle shown in Figure 4(a). The graph of the second equation is the semicircle in Figure 4(b). Graphing these semicircles together on the same viewing screen, we get the full circle in Figure 4(c). 2, 2 2, 2 by 4 4 3 3 The graph in Figure 4(c) looks somewhat flattened. Most graphing calculators allow you to set the scales on the axes so that circles really look like circles. On the TI-82 and TI-83, from the ZSquare to set the scales appropriately. (On the TI |
-86 the command is Zsq.) menu, choose ZOO SE CTI ON 2. 3 | Graphing Calculators: Solving Equations and Inequalities Graphically 159 _2 2 _2 (a) 2 _2 2 _2 (b) 2 _2 FIGURE 4 Graphing the equation x 2 y 2 1 ✎ Practice what you’ve learned: Do Exercise 27. 2 _2 (c) 2 ▲ ■ Solving Equations Graphically In Chapter 1 we learned how to solve equations. To solve an equation such as 3x 5 0 we used the algebraic method. This means that we used the rules of algebra to isolate x on one side of the equation. We view x as an unknown, and we use the rules of algebra to hunt it down. Here are the steps in the solution: 3x 5 0 3x 5 x 5 3 Add 5 Divide by 3 So the solution is x 5 3. We can also solve this equation by the graphical method. In this method we view x as a variable and sketch the graph of the equation y 3x 5 y 1 0 y=3x-5 1 2 x FIGURE 5 Different values for x give different values for y. Our goal is to find the value of x for which y 0. From the graph in Figure 5 we see that y 0 when x 1.7. Thus, the solution is x 1.7. Note that from the graph we obtain an approximate solution. We summarize these methods in the box on the following page. Pierre de Fermat (1601–1665) was a French lawyer who became interested in mathematics at the age of 30. Because of his job as a magistrate, Fermat had little time to write complete proofs of his discoveries and often wrote them in the margin of whatever book he was reading at the time. After his death, his copy of Diophantus’ Arithmetica (see page 49) was found to contain a particularly tantalizing comment. Where Diophantus discusses the solutions of x 2 y 2 z 2 Ófor example, x 3, y 4, and z 5Ô, Fermat states in the margin that for n 3 there are no natural number solutions to the equation x n y n z n. In other words, it’s impossible for a cube to equal the sum of two cubes, a fourth power to equal the sum of two fourth powers, and so on. Fermat |
writes “I have discovered a truly wonderful proof for this but the margin is too small to contain it.” All the other margin comments in Fermat’s copy of Arithmetica have been proved. This one, however, remained unproved, and it came to be known as “Fermat’s Last Theorem.” In 1994, Andrew Wiles of Princeton University announced a proof of Fermat’s Last Theorem, an astounding 350 years after it was conjectured. His proof is one of the most widely reported mathematical results in the popular press 160 CHAPTER 2 | Coordinates and Graphs SOLVING AN EQUATION Algebraic method Graphical method Use the rules of algebra to isolate the unknown x on one side of the equation. Example: 2x 6 x 3x 6 x 2 Add x Divide by 3 The solution is x 2. Move all terms to one side and set equal to y. Sketch the graph to find the value of x where y 0. 2x 6 x Example: 0 6 3x Set y 6 3x and graph. y 2 0 y=6-3x 1 2 x From the graph the solution is x 2. The advantage of the algebraic method is that it gives exact answers. Also, the process of unraveling the equation to arrive at the answer helps us to understand the algebraic structure of the equation. On the other hand, for many equations it is difficult or impossible to isolate x. The graphical method gives a numerical approximation to the answer. This is an advantage when a numerical answer is desired. (For example, an engineer might find an answer expressed as x 2.6 more immediately useful than.) Also, graphing an equation helps us to visualize how the solution is related to other values of the variable. x 17 The Discovery Project on page 333 describes a numerical method for solving equations. E X AM P L E 4 | Solving a Quadratic Equation Algebraically and Graphically Solve the quadratic equations algebraically and graphically. (a) x 2 4x 2 0 (b) x 2 4x 4 0 (c) x 2 4x 6 0 ▼ SO LUTI O N 1: Algebraic We use the Quadratic Formula to solve each equation. Alan Turing (1912–1954) was at the center of two pivotal events of the 20th century: World War II and |
the invention of computers. At the age of 23 Turing made his mark on mathematics by solving an important problem in the foundations of mathematics that had been posed by David Hilbert at the 1928 International Congress of Mathematicians (see page 531). In this research he invented a theoretical machine, now called a Turing machine, which was the inspiration for modern digital computers. During World War II Turing was in charge of the British effort to decipher secret German codes. His complete success in this endeavor played a decisive role in the Allies’ victory. To carry out the numerous logical steps that are required to break a coded message, Turing developed decision procedures similar to modern computer programs. After the war he helped to develop the first electronic computers in Britain. He also did pioneering work on artificial intelligence and computer models of biological processes. At the age of 42 Turing died of poisoning after eating an apple that had mysteriously been laced with cyanide. SE CTI ON 2. 3 | Graphing Calculators: Solving Equations and Inequalities Graphically 161 The Quadratic Formula is discussed on page 89. (a 18 2 2 12 There are two solutions, and x 2 12. x 2 12 2 4 # 1 # 4 (b 10 2 2 There is just one solution, x 2. (c 18 2 There is no real solution. ▼ SO LUTI O N 2: Graphical We graph the equations y x 2 4x 2, y x 2 4x 4, and y x 2 4x 6 in Figure 6. By determining the x-intercepts of the graphs, we find the following solutions. (a) x 0.6 and x 3.4 (b) x 2 (c) There is no x-intercept, so the equation has no solution. 10 _5 _1 5 _1 10 _5 5 _1 10 _5 (a) y=≈-4x+2 (b) y=≈-4x+4 (c) y=≈-4x+6 FIGURE 6 ✎ Practice what you’ve learned: Do Exercise 35. 5 ▲ The graphs in Figure 6 show visually why a quadratic equation may have two solutions, one solution, or no real solution. We proved this fact algebraically in Section 1.3 when we studied the discriminant. E X AM P L E 5 | Another Graphical Method Solve the equation algebraically |
and graphically: 5 3x 8x 20 ▼ SO LUTI O N 1: Algebraic 5 3x 8x 20 3x 8x 25 11x 25 25 11 x 2 Subtract 5 Subtract 8x 3 11 Divide by –11 and simplify 162 CHAPTER 2 | Coordinates and Graphs 10 y⁄=5-3x _1 3 ▼ SO LUTI O N 2: Graphical We could move all terms to one side of the equal sign, set the result equal to y, and graph the resulting equation. But to avoid all this algebra, we graph two equations instead: y1 5 3x and y2 8x 20 y¤=8x-20 Intersection X=2.2727723 Y=-1.818182 _25 FIGURE 7 feature or the intersect command on a graphing calculator, we see from The solution of the original equation will be the value of x that makes y1 equal to y2; that is, the solution is the x-coordinate of the intersection point of the two graphs. Using the TRACE Figure 7 that the solution is x 2.27. ✎ Practice what you’ve learned: Do Exercise 31. ▲ In the next example we use the graphical method to solve an equation that is extremely difficult to solve algebraically. E X AM P L E 6 | Solving an Equation in an Interval Solve the equation x 3 6x 2 9x 1x 4 3. 1, 6 in the interval ▼ SO LUTI O N We are asked to find all solutions x that satisfy 1 x 6, so we will graph the equation in a viewing rectangle for which the x-values are restricted to this interval. x 3 6x 2 9x 1x We can also use the zero command to find the solutions, as shown in Figures 8(a) and 8(b). Figure 8 shows the graph of the equation 5, 5 rectangle in, we see that the solutions are x 2.18 and x 3.72. in the viewing. There are two x-intercepts in this viewing rectangle; zooming 1, 6 by 4 3 3 4 x 3 6x 2 9x 1x 0 Subtract y x 3 6x 2 9x 1x 1x 5 1 Zero X=2.1767162 Y=0 _5 (a) FIGURE 8 5 |
6 1 6 Zero X=3.7200502 Y=0 _5 (b) ✎ Practice what you’ve learned: Do Exercise 43. ▲ The equation in Example 6 actually has four solutions. You are asked to find the other two in Exercise 71. E X AM P L E 7 | Intensity of Light Two light sources are 10 m apart. One is three times as intense as the other. The light intensity L (in lux) at a point x meters from the weaker source is given by L 10 x 2 30 10 x 2 1 2 SE CTI ON 2. 3 | Graphing Calculators: Solving Equations and Inequalities Graphically 163 (See Figure 9.) Find the points at which the light intensity is 4 lux. FIGURE 9 x 10 − x ▼ SO LUTI O N We need to solve the equation y2 = +10 x2 30 (10 – x)2 The graphs of 4 10 x 2 30 10 x 1 2 2 y1 4 and y2 10 x 2 30 10 x 1 2 2 10 are shown in Figure 10. Zooming in (or using the intersect command) we find two solutions, x 1.67431 and x 7.1927193. So the light intensity is 4 lux at the points that are 1.67 m and 7.19 m from the weaker source. ✎ Practice what you’ve learned: Do Exercise 73. ▲ ■ Solving Inequalities Graphically Inequalities can be solved graphically. To describe the method, we solve x 2 5x 6 0 This inequality was solved algebraically in Section 1.6, Example 3. To solve the inequality graphically, we draw the graph of 5 y x 2 5x 6 Our goal is to find those values of x for which y 0. These are simply the x-values for which the graph lies below the x-axis. From Figure 11 we see that the solution of the inequality is the interval 2, 3. 3 4 10 y1 = 4 0 FIGURE 10 10 _1 _2 FIGURE 11 x 2 5x 6 0 5 y⁄ E X AM P L E 8 | Solving an Inequality Graphically Solve the inequality 3.7x 2 1.3x 1.9 2.0 1.4x. ▼ SO LUTI O N We graph the equations _3 3 y¤ _3 FIG |
URE 12 y1 y2 3.7x 2 1.3x 1.9 2.0 1.4x y1 3.7x 2 1.3x 1.9 and y2 2.0 1.4x in the same viewing rectangle in Figure 12. We are interested in those values of x for which y2; these are points for which the graph of y2 lies on or above the graph of y1. To dey1 termine the appropriate interval, we look for the x-coordinates of points where the graphs intersect. We conclude that the solution is (approximately) the interval ✎ Practice what you’ve learned: Do Exercise 59. 1.45, 0.72 3 ▲. 4 164 CHAPTER 2 | Coordinates and Graphs E X AM P L E 9 | Solving an Inequality Graphically Solve the inequality x 3 5x 2 8. ▼ SO LUTI O N We write the inequality as x 3 5x 2 8 0 and then graph the equation y x 3 5x 2 8 15, 15 in the viewing rectangle, as shown in Figure 13. The solution of the inequality consists of those intervals on which the graph lies on or above the x-axis. By moving the cursor to the x-intercepts, we find that, correct to one decimal place, the solution is 1.1, 1.5 4.6, q 6, 6 by. 4 3 4 3 3 4 3 2 15 _6 6 FIGURE 13 x 3 5x 2 8 0 _15 ✎ Practice what you’ve learned: Do Exercise 61. ▲ 2. ▼ CONCE PTS 1. The solutions of the equation x 2 2x 3 0 are the -intercepts of the graph of y x 2 2x 3. 2. The solutions of the inequality x 2 2x 3 0 are the x-coordinates of the points on the graph of y x 2 2x 3 that lie the x-axis. 3. The figure shows a graph of y x 4 3x 3 x 2 3x. Use the graph to do the following. (a) Find the solutions of the equation x 4 3x 3 x 2 3x 0. (b) Find the solutions of the inequality x 4 3x 3 x 2 3x 0. (a) Find the solutions of the equation 5x x 2 4. (b) Find the solutions of |
the inequality 5x x 2 4=5x-x2 y=4 -1 -1 -2 1 2 3 4 5 6 x y=x4-3x3-x2+3x y 8 6 4 2 -1-2 -2 -4 -6 -8 1 2 3 x 4 ✎ 4. The figure shows the graphs of y 5x x 2 and y 4. Use the graphs to do the following. ▼ SKI LLS 5–10 ■ Use a graphing calculator or computer to decide which viewing rectangle (a)–(d) produces the most appropriate graph of the equation. 5. y x 4 2 2, 2 4 by 0, 4 4 3 8, 8 by 40, 40 2, 2 3 0, 4 (a) (b) (c) (d) by 4 4 4 3 3 3 3 4 80, 800 3 4 4, 40 3 by 6. y x 2 7x 6 5, 5 by 4 0, 10 by 15, 8 4 10, 3 4 4 3 20, 100 by 4 3 100, 20 by 4 3 5, 5 3 20, 100 (a) (b) (c) (d) 3 3 3 3 4 4 4 SE CTI ON 2. 3 | Graphing Calculators: Solving Equations and Inequalities Graphically 165 31–40 ■ Solve the equation both algebraically and graphically. ✎ 31. x 4 5x 12 32. 1 2 x 3 6 2x 33. 2 x 1 2x 7 ✎ 35. x 2 32 0 37. x 2 9 0 39. 16x 4 625 34. 4 x 2 6 2x 5 2x 4 36. x 3 16 0 38. x 2 3 2x 40. 2x 5 243 0 41. x 5 1 2 4 80 0 42. 6 1 x 2 2 5 64 43–50 ■ Solve the equation graphically in the given interval. State each answer correct to two decimals. 4 ✎ 43. x 2 7x 12 0; 0, 6 4 3 44. x 2 0.75x 0.125 0; 3 7. y 100 x 2 by (a) (b) (c) (d) 4 4, 4 10, 10 4 15, 15 4 4, 4 by 4, 4 4 3 10, 10 by 3 4 30, 110 by 3 30, 110 4 4 4 3 8 |
. y 2x 2 1000 by 3 by 3 by 3 by 3 9. y 10 25x x 3 10, 10 4 10, 10 4 10, 10 4 25, 25 4 (a) (b) (c) (d) 3 3 3 3 4 10, 10 100, 100 4 1000, 1000 1200, 200 4 4 (a) (b) (c) (d) 4, 4 4, 4] by 4 3 10, 10 10, 10 by 3 4 20, 20 100, 100 by 3 4 100, 100 by 4 200, 200 4 3 4 y 28x x 2 4, 4 4, 4 (a) by 4 3 4 5, 5 (b) 0, 100 by 4 3 4 10, 10 10, 40 (c) by 4 3 2, 6 2, 10 (d) by 10. 11–22 ■ Determine an appropriate viewing rectangle for the equation, and use it to draw the graph. 11. y 100x 2 12. y 100x 2 13. y 4 6x x 2 14. y 0.3x 2 1.7x 3 15. y 24 256 x 2 16. y 212x 17 17. y 0.01x 3 x 2 5 18 19. y x 4 4x 3 20. y x x 2 25 2, 2 4 1, 4 4 3 2, 2 45. x 3 6x 2 11x 6 0; 46. 16x 3 16x 2 x 1; x 1x 1 0 ; 3 1, 5 3 1 1x 21 x 2 ; 4 1, 5 47. 48. 3 4 4 49. x 1/3 x 0; 3, 3 3 50. x 1/2 x 1/3 x 0; 4 1, 5 3 4 51–54 ■ Use the graphical method to solve the equation in the indicated exercise from Section 1.5. 51. Exercise 11 53. Exercise 31 52. Exercise 12 54. Exercise 32 21. y 1 x 1 0 0 22. y 2x x 2 5 0 0 55–58 ■ Find all real solutions of the equation, correct to two decimals. 23–26 ■ Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? ✎ 23. y 3x 2 6x 1 2, y 27 7 12 x 2 ; 4, 4 by 1, 3 24. y 249 x 2, y 1 51 41 3x ; 2 25 |
. y 6 4x x 2, y 3x 18; 26. y x 3 4x, y x 5; 4, 4 3 4 by 4 3 1, 8 3 8, 8 3 6, 2 3 by 4 by 3 5, 20 3 15, 15 3 4 4 4 4 ✎ 27. Graph the circle x 2 y 2 9 by solving for y and graphing two equations as in Example 3. 28. Graph the circle y 1 graphing two equations as in Example 3. 2 x 2 1 1 2 by solving for y and 55. x 3 2x 2 x 1 0 56. x 4 8x 2 2 0 57 58. x 4 16 x 3 4 59–66 ■ Find the solutions of the inequality by drawing appropriate graphs. State each answer correct to two decimals. ✎ ✎ 59. x 2 3x 10 60. 0.5x 2 0.875x 0.25 61. x 3 11x 6x 2 6 62. 16x 3 24x 2 9x 1 63. x 1/3 x 64. 20.5x 2 1 2 x 0 0 65 66. x 1 1 2 2 x 3 67–70 ■ Use the graphical method to solve the inequality in the indicated exercise from Section 1.6. 29. Graph the equation 4x 2 2y 2 1 by solving for y and graphing two equations corresponding to the negative and positive square roots. (This graph is called an ellipse.) 67. Exercise 43 69. Exercise 53 68. Exercise 44 70. Exercise 54 30. Graph the equation y 2 9x 2 1 by solving for y and graphing the two equations corresponding to the positive and negative square roots. (This graph is called a hyperbola.) 71. In Example 6 we found two solutions of the equation x 3 6x 2 9x 1x Find two more solutions, correct to two decimals., the solutions that lie between 1 and 6. 166 CHAPTER 2 | Coordinates and Graphs ▼ APPLICATIONS 72. Estimating Profit An appliance manufacturer estimates that the profit y (in dollars) generated by producing x cooktops per month is given by the equation ▼ DISCOVE RY • DISCUSSION • WRITI NG 74. Misleading Graphs Write a short essay describing different ways in which a graphing calculator might give a misleading graph of an equation. 75. Algebraic and Graphical Solution Methods |
Write a short essay comparing the algebraic and graphical methods for solving equations. Make up your own examples to illustrate the advantages and disadvantages of each method. 76. Equation Notation on Graphing Calculators When you enter the following equations into your calculator, how does what you see on the screen differ from the usual way of writing the equations? (Check your user’s manual if you’re not sure.) (a) y x 0 0 y x (c) x 1 (b) y 15 x (d) y x 3 13 x 2 77. Enter Equations Carefully A student wishes to graph the equations y x 1/3 and y x x 4 on the same screen, so he enters the following information into his calculator: Y1 X^1/3 Y2 X/X 4 The calculator graphs two lines instead of the equations he wanted. What went wrong? y 10x 0.5x 2 0.001x 3 5000 where 0 x 450. (a) Graph the equation. (b) How many cooktops must be produced to begin generating a profit? (c) For what range of values of x is the company’s profit greater than $15,000? ✎ 73. How Far Can You See? If you stand on a ship in a calm sea, then your height x (in ft) above sea level is related to the farthest distance y (in mi) that you can see by the equation y x 5280 b (a) Graph the equation for 0 x 100. (b) How high up do you have to be to be able to see 10 mi? 1.5x B a 2 x 2.4 Lines LEARNING OBJECTIVES After completing this section, you will be able to: ■ Find the slope of a line ■ Find the point-slope form of the equation of a line ■ Find the slope-intercept form of the equation of a line ■ Find equations of horizontal and vertical lines ■ Find the general equation of a line ■ Find equations for parallel and perpendicular lines ■ Model with linear equations: interpret slope as rate of change In this section we find equations for straight lines lying in a coordinate plane. The equations will depend on how the line is inclined, so we begin by discussing the concept of slope. SECTION 2.4 | Lines 167 ■ The Slope of a Line We first need a way to measure the “ste |
epness” of a line, or how quickly it rises (or falls) as we move from left to right. We define run to be the distance we move to the right and rise to be the corresponding distance that the line rises (or falls). The slope of a line is the ratio of rise to run: slope rise run Figure 1 shows situations in which slope is important. Carpenters use the term pitch for the slope of a roof or a staircase; the term grade is used for the slope of a road. 1 12 1 3 8 100 Slope of a ramp Slope= 1 12 FIGURE 1 Pitch of a roof Slope= 1 3 Grade of a road Slope= 8 100 If a line lies in a coordinate plane, then the run is the change in the x-coordinate and the rise is the corresponding change in the y-coordinate between any two points on the line (see Figure 2). This gives us the following definition of slope. y 2 1 0 Run FIGURE 2 y 2 1 0 Rise: change in y -coordinate (negative) Run x Rise: change in y-coordinate (positive) x SLOPE OF A LINE The slope m of a nonvertical line that passes through the points B is A x1, y1 2 1 and x2, y22 1 m rise run y2 x2 y1 x1 The slope of a vertical line is not defined. 168 CHAPTER 2 | Coordinates and Graphs The slope is independent of which two points are chosen on the line. We can see that this is true from the similar triangles in Figure 3: y2 x2 y1 x1 yœ 2 xœ 2 yœ 1 xœ 1 y B(x¤, y¤) y¤-y⁄ (rise) A(x⁄, y⁄) B'(x'¤, y'¤) x¤-x⁄ (run) A'(x'⁄, y'⁄) y'¤-y'⁄ x'¤-x'⁄ 0 FIGURE 3 x Figure 4 shows several lines labeled with their slopes. Notice that lines with positive slope slant upward to the right, whereas lines with negative slope slant downward to the right. The steepest lines are those for which the absolute value of the slope is the largest |
; a horizontal line has slope zero. y m=5 m=2 m=1 m= 1 2 m=0 m=_ 1 2 x 0 m=_5 m=_2 m=_1 FIGURE 4 Lines with various slopes E X AM P L E 1 | Finding the Slope of a Line Through Two Points y Q (8, 5) Find the slope of the line that passes through the points P 2, 1 1 2 and Q 8, 5. 2 1 ▼ SO LUTI O N through these two points. From the definition the slope is Since any two different points determine a line, only one line passes ) P 2, 1 ( FIGURE 5 x This says that for every 3 units we move to the right, the line rises 2 units. The line is drawn in Figure 5. ✎ Practice what you’ve learned: Do Exercise 5. ▲ m y2 x2 y1 x1 x, y) Run x – x1 Rise y – y1 x y P⁄(x⁄, y⁄) 0 FIGURE 6 SECTION 2.4 | Lines 169 ■ Point-Slope Form of the Equation of a Line Now let’s find the equation of the line that passes through a given point slope m. A point through P1 and P is equal to m (see Figure 6), that is, and has with x x1 lies on this line if and only if the slope of the line x1, y12 x, y P P 2 1 1 y y1 x x1 m This equation can be rewritten in the form also satisfied when x x1 and y y1. Therefore, it is an equation of the given line. ; note that the equation is 1 y y1 m x x12 POINT-SLOPE FORM OF THE EQUATION OF A LINE An equation of the line that passes through the point 1 x x12 y y1 m 1 x1, y12 and has slope m is E X AM P L E 2 | Finding the Equation of a Line with Given Point and Slope (a) Find an equation of the line through (b) Sketch the line. 1 1, 3 2 with slope 1 2. ▼ SO LUTI O N (a) Using the point-slope form with tion of the line as m 1 2, x1 1, and y1 3, we obtain |
an equa- y 3 1 21 2y 6 x 1 x 1 2 x 2y 5 0 1 2 From point-slope equation Multiply by 2 Rearrange (b) The fact that the slope is tells us that when we move to the right 2 units, the line y 1 0 3 x Run = 2 (1, _3) Rise = –1 FIGURE 7 drops 1 unit. This enables us to sketch the line in Figure 7. ✎ Practice what you’ve learned: Do Exercise 19. E X AM P L E 3 | Finding the Equation of a Line Through Two Given Points 1, 2 or, in the point-slope equation. We can use either point, 3, 4 1 We will end up with the same final answer. 2 2 1 Find an equation of the line through the points 1, 2 1 and 1 2 3, 4. 2 ▼ SO LUTI O N The slope of the line is 4 2 1 3 m 2 1 and y1 Using the point-slope form with x1 x 1 y 2 3 21 2y 4 3x 3 1 2 From point-slope equation Multiply by 2 3 6 2 4 2, we obtain y 0 (0, b) y=mx+b 3x 2y 1 0 Rearrange ✎ Practice what you’ve learned: Do Exercise 23. ■ Slope-Intercept Form of the Equation of a Line x FIGURE 8 Suppose a nonvertical line has slope m and y-intercept b (see Figure 8). This means that the line intersects the y-axis at the point, so the point-slope form of the equation of 0, b 1 2 ▲ ▲ 170 CHAPTER 2 | Coordinates and Graphs the line, with x 0 and y b, becomes 1 This simplifies to y mx b, which is called the slope-intercept form of the equation of a line. 2 y b m x 0 SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE An equation of the line that has slope m and y-intercept b is y mx b E X AM P L E 4 | Lines in Slope-Intercept Form (a) Find the equation of the line with slope 3 and y-intercept 2. (b) Find the slope and y-intercept of the line 3y 2x 1. ▼ SO |
LUTI O N (a) Since m 3 and b 2, from the slope-intercept form of the equation of a line we get y 3x 2 (b) We first write the equation in the form y mx b: 3y 2x 1 3y 2x 1 y 2 3 x 1 3 Add 2x Divide by 3 From the slope-intercept form of the equation of a line, we see that the slope is and the y-intercept is ✎ Practice what you’ve learned: Do Exercises 25 and 47. b 1 3. m 2 3 ▲ ■ Vertical and Horizontal Lines If a line is horizontal, its slope is m 0, so its equation is y b, where b is the y-intercept (see Figure 9). A vertical line does not have a slope, but we can write its equation as x a, where a is the x-intercept, because the x-coordinate of every point on the line is a. VERTICAL AND HORIZONTAL LINES An equation of the vertical line through a, b An equation of the horizontal line through is x a. 1 2 a, b 1 is y b. 2 E X AM P L E 5 | Vertical and Horizontal Lines (a) An equation for the vertical line through (3, 5) is x 3. (b) The graph of the equation x 3 is a vertical line with x-intercept 3. (c) An equation for the horizontal line through (8, 2) is y 2. (d) The graph of the equation y 2 is a horizontal line with y-intercept 2. The lines are graphed in Figure 10. ✎ Practice what you’ve learned: Do Exercises 29 and 33. ▲ Slope y-intercept y 2 3 x 1 3 y b 0 FIGURE 9 y 2 y=b (a, b) x=a a x x=3 _2 0 2 4 x y=_2 FIGURE 10 SECTION 2.4 | Lines 171 ■ General Equation of a Line A linear equation is an equation of the form Ax By C 0 where A, B, and C are constants and A and B are not both 0. The equation of a line is a linear equation: ■ A nonvertical line has the equation y mx b or mx y b 0, which is a linear equation with A m, B 1, |
and C b. ■ A vertical line has the equation x a or x a 0, which is a linear equation with A 1, B 0, and C a. Conversely, the graph of a linear equation is a line: ■ If B 0, the equation becomes y A B x C B Divide by B and this is the slope-intercept form of the equation of a line (with m A/B and b C/B). ■ If B 0, the equation becomes Ax C 0 Set B = 0 or x C/A, which represents a vertical line. We have proved the following. GENERAL EQUATION OF A LINE The graph of every linear equation Ax By C 0 (A, B not both zero) is a line. Conversely, every line is the graph of a linear equation. E X AM P L E 6 | Graphing a Linear Equation Sketch the graph of the equation 2x 3y 12 0. y 1 0 2x-3y-12=0 1 (6, 0) x (0, _4) FIGURE 11 ▼ SO LUTI O N 1 enough to find any two points on the line. The intercepts are the easiest points to find. Since the equation is linear, its graph is a line. To draw the graph, it is x-intercept: Substitute y 0, to get 2x 12 0, so x 6 y-intercept: Substitute x 0, to get 3y 12 0, so y 4 With these points we can sketch the graph in Figure 11. ▼ SO LUTI O N 2 We write the equation in slope-intercept form: 2x 3y 12 0 2x 3y 12 3y 2x 12 y 2 3 x 4 Add 12 Subtract 2x Divide by –3 172 CHAPTER 2 | Coordinates and Graphs y 1 0 (0, _4) 2x-3y-12=0 x 1 3 2 FIGURE 12 y D E F A FIGURE 13 l¤ l⁄ B C x This equation is in the form y mx b, so the slope is b 4. To sketch the graph, we plot the y-intercept and then move 3 units to the right and 2 units up as shown in Figure 12. ✎ Practice what you’ve learned: Do Exercise 53. and the y-intercept is m 2 3 ▲ ■ Parallel and Perpendicular Lines Since |
slope measures the steepness of a line, it seems reasonable that parallel lines should have the same slope. In fact, we can prove this. PARALLEL LINES Two nonvertical lines are parallel if and only if they have the same slope. ▼ P RO O F parallel, then the right triangles ABC and DEF are similar, so Let the lines l1 and l2 in Figure 13 have slopes m1 and m2. If the lines are m1 d d B, C A, C 1 1 2 2 d d E, F 1 D, F 1 2 2 m2 Conversely, if the slopes are equal, then the triangles will be similar, so BAC EDF ▲ and the lines are parallel. E X AM P L E 7 | Finding the Equation of a Line Parallel to a Given Line Find an equation of the line through the point 4x 6y 5 0. 1 5, 2 2 that is parallel to the line ▼ SO LUTI O N First we write the equation of the given line in slope-intercept form. 4x 6y 5 0 6y 4x 5 y 2 3 x 5 6 Subtract 4x + 5 Divide by 6 m 2 3. Since the required line is parallel to the given line, it also. From the point-slope form of the equation of a line, we get So the line has slope has slope m 2 3 x 5 y 2 2 31 2 3y 6 2x 10 2x 3y 16 0 Slope m = 2 3, point 5, 2 1 2 Multiply by 3 Rearrange Thus, the equation of the required line is 2x 3y 16 0. ✎ Practice what you’ve learned: Do Exercise 31. ▲ The condition for perpendicular lines is not as obvious as that for parallel lines. PERPENDICULAR LINES Two lines with slopes m1 and m2 are perpendicular if and only if is, their slopes are negative reciprocals: m1m2 1, that m2 1 m1 Also, a horizontal line (slope 0) is perpendicular to a vertical line (no slope). l¤ y O FIGURE 14 l⁄ A(1, m⁄) x B(1, m¤) SECTION 2.4 | Lines 173 ▼ P RO O F In Figure 14 we show two lines intersecting at the origin. (If the lines intersect at some other point, we consider lines parallel to these |
that intersect at the origin. These lines have the same slopes as the original lines.) If the lines l1 and l2 have slopes m1 and m2, then their equations are y m1x and lies on l2. By the Pythagorean Theo- y m2x. Notice that rem and its converse (see page 284), OA OB if and only if lies on l1 and 1, m12 1, m22 A B 1 1 O, A d 1 3 2 4 2 O, B d 1 3 2 4 2 A, B d 1 3 2 2 4 By the Distance Formula this becomes 12 m2 12 1 12 m2 22 m2 2 1 2 m2 1 1 1 2 1 m2 2 1 2 2m1m2 m2 2 m12 m2 1 2 2m1m2 m1m2 1 ▲ E X AM P L E 8 | Perpendicular Lines y 17 Q Show that the points P 3, 3, Q 1 2 1 8, 17 2, and R 11, 5 1 2 are the vertices of a right triangle. ▼ SO LUTI O N The slopes of the lines containing PR and QR are, respectively, m1 5 3 11 3 1 4 and m2 5 17 11 8 4 P 5 3 0 R 3 8 11 x FIGURE 15 1, these lines are perpendicular, so PQR is a right triangle. It is sketched Since m1m2 in Figure 15. ✎ Practice what you’ve learned: Do Exercise 57. ▲ E X AM P L E 9 | Finding an Equation of a Line Perpendicular to a Given Line Find an equation of the line that is perpendicular to the line through the origin. 4x 6y 5 0 and passes ▼ SO LUTI O N In Example 7 we found that the slope of the line 4x 6y 5 0 is. Since the required line passes through. Thus, the slope of a perpendicular line is the negative reciprocal, that is,, the point-slope form gives 0 21 y 3 2 x 2 ✎ Practice what you’ve learned: Do Exercise 35. ▲ E X AM P L E 10 | Graphing a Family of Lines Use a graphing calculator to graph the family of lines for b 2, 1, 0, 1, 2. What property do the lines share? y 0.5x b 174 CHAPTER 2 | Coordinates and Graphs |
▼ SO LUTI O N The lines are graphed in Figure 16 in the viewing rectangle by. The lines all have the same slope, so they are parallel. 6, 6 3 4 6, 6 4 3 FIGURE 16 y 0.5x b _6 6 _6 6 ✎ Practice what you’ve learned: Do Exercise 41. ▲ ■ Modeling with Linear Equations: Slope as Rate of Change When a line is used to model the relationship between two quantities, the slope of the line is the rate of change of one quantity with respect to the other. For example, the graph in Figure 17(a) gives the amount of gas in a tank that is being filled. The slope between the indicated points is m 6 gallons 3 minutes 2 gal/min The slope is the rate at which the tank is being filled, 2 gallons per minute. In Figure 17(b) the tank is being drained at the rate of 0.03 gallon per minute, and the slope is 0.03 18 15 12 9 6 3 0 1 2 6 gal 3 min 4 6 5 3 Time (min 18 15 12 9 6 3 0 −3 gal 100 min 20 100 Time (min) 200 x (a) Tank filled at 2 gal/min Slope of line is 2 (b) Tank drained at 0.03 gal/min Slope of line is −0.03 FIGURE 17 The next two examples give other situations in which the slope of a line is a rate of change. E X AM P L E 11 | Slope as Rate of Change A dam is built on a river to create a reservoir. The water level „ in the reservoir is given by the equation „ 4.5t 28 where t is the number of years since the dam was constructed and „ is measured in feet. (a) Sketch a graph of this equation. (b) What do the slope and „-intercept of this graph represent? SECTION 2.4 | Lines 175 „ „=4.5t+28 ▼ SO LUTI O N (a) This equation is linear, so its graph is a line. Since two points determine a line, we plot two points that lie on the graph and draw a line through them. 10 0 1 FIGURE 18 When t 0, then „ 4.5 When t 2, then „ 4.5 28 28, so 28 37, so 0 2 1 1 |
2 2 0, 28 2, 37 1 1 2 2 is on the line. is on the line. The line that is determined by these points is shown in Figure 18. t (b) The slope is m 4.5; it represents the rate of change of water level with respect to time. This means that the water level increases 4.5 ft per year. The „-intercept is 28 and occurs when t 0, so it represents the water level when the dam was constructed. ✎ Practice what you’ve learned: Do Exercise 69. ▲ E X AM P L E 12 | Linear Relationship Between Temperature and Elevation (a) As dry air moves upward, it expands and cools. If the ground temperature is 20 C and the temperature at a height of 1 km is 10 C, express the temperature T (in C) in terms of the height h (in kilometers). (Assume that the relationship between T and h is linear.) (b) Draw the graph of the linear equation. What does its slope represent? (c) What is the temperature at a height of 2.5 km? ▼ SO LUTI O N (a) Because we are assuming a linear relationship between T and h, the equation must be of the form T mh b where m and b are constants. When h 0, we are given that T 20, so 20 m 1 b 20 Thus, we have T mh 20 MATHEMATICS IN THE MODERN WORLD Changing Words, Sound, and Pictures into Numbers Pictures, sound, and text are routinely transmitted from one place to another via the Internet, fax machines, or modems. How can such things be transmitted through telephone wires? The key to doing this is to change them into numbers or bits (the digits 0 or 1). It’s easy to see how to change text to numbers. For example, we could use the correspondence A 00000001, B 00000010, C 00000011, D 00000100, E 00000101, and so on. The word “BED” then becomes 000000100000010100000100. By reading the digits in groups of eight, it is possible to translate this number back to the word “BED.” Changing sound to bits is more complicated. A sound wave can be graphed on an oscilloscope or a computer. The graph is then broken down mathematically into simpler components corresponding to the different frequencies of the original sound. (A branch |
of mathematics called Fourier analysis is used here.) The intensity of each component is a number, and the original sound can be reconstructed from these numbers. For example, music is stored on a CD as a sequence of bits; it may look like 101010001010010100101010 1000001011110101000101011.... (One second of music requires 1.5 million bits!) The CD player reconstructs the music from the numbers on the CD. Changing pictures into numbers involves expressing the color and brightness of each dot (or pixel) into a number. This is done very efficiently using a branch of mathematics called wavelet theory. The FBI uses wavelets as a compact way to store the millions of fingerprints they need on file. 176 CHAPTER 2 | Coordinates and Graphs T 20 10 When h 1, we have T 10, so T=_10h+20 The required expression is 1 20 10 m 2 m 10 20 10 1 0 1 3 h T 10h 20 (b) The graph is sketched in Figure 19. The slope is m 10C/km, and this represents the rate of change of temperature with respect to distance above the ground. So the temperature decreases 10C per kilometer of height. FIGURE 19 (c) At a height of h 2.5 km the temperature is T 10 ✎ Practice what you’ve learned: Do Exercise 73. 2.5 20 25 20 5 °C 2 1 ▲ 2. ▼ CONCE PTS 1. We find the “steepness,” or slope, of a line passing through two 11. P 1, 3 1, Q 1 2 1, 6 2 12. P 1 1, 4 6, 0, Q 1 2 2 13. Find the slopes of the lines l1, l2, l3, and l4 in the figure points by dividing the difference in the -coordinates of below. these points by the difference in the the line passing through the points 0, 1 2 1 -coordinates. So and has slope 2, 5 1 2. 2. A line has the equation y 3x 2. (a) This line has slope. (b) Any line parallel to this line has slope. (c) Any line perpendicular to this line has slope. 3. The point-slope form of the equation of the line with slope 3 |
passing through the point 1 1, 2 2 is. 4. (a) The slope of a horizontal line is. The equation of the horizontal line passing through 2, 3 2 1 is. (b) The slope of a vertical line is. The equation of y l⁄ l¤ 1 l‹ _2 0 2 x l› _2 14. (a) Sketch lines through and 1. 0, 0 1 2 with slopes 1, 0, 1 2, 2, (b) Sketch lines through 0, 0 1 2 with slopes 1 3, 1 2, 1 3, and 3. the vertical line passing through 2, 3 2 1 is. 15–18 ■ Find an equation for the line whose graph is sketched. ▼ SKI LLS 5–12 ■ Find the slope of the line through P and Q. ✎ 5. P 7. P 9. P 0, 0 2, 2 2, 2 2 10, 0 2 4, 3 2 6. P 8. P 10. P 1 1 1 1 1 0, 0, Q 2 2, 6 2 1, 2 2 2, 5, Q 3, 3 2 4, 3, Q 1 2 15. y 3 1 0 _2 1 3 5 x 2 16. 17. 18. y 3 _3 0 2 x y 1 0 _3 _3 SECTION 2.4 | Lines 177 1, 7 1 2 ; parallel to the line passing through 2, 5 1 2 and 37. Through 2, 1 1 2 38. Through through 2, 11 1 and 1, 1 1 2 39. (a) Sketch the line with slope. 2, 1 3 2 (b) Find an equation for this line. 2 ; perpendicular to the line passing 2 5, 1 1 2 that passes through the point 40. (a) Sketch the line with slope 2 that passes through the point 1 1 4, 1. 2 (b) Find an equation for this line. 41–44 ■ Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? 41. y 2x b for b 0, 1, 3, 6 42. y mx 3 for m 0, 0.25, 0.75, 1.5 ✎ 43. 44 for m 0, 0.25, 0.75, 1.5 1 for m 0, 0.5, 1, 2, 6 2 45–56 ■ Find |
the slope and y-intercept of the line and draw its graph. 45. x y 3 ✎ 47. x 3y 0 2 x 1 51. y 4 49. 1 3 y 1 0 ✎ 53. 3x 4y 12 46. 3x 2y 12 48. 2x 5y 0 50. 3x 5y 30 0 52. x 5 54. 4y 8 0 55. 3x 4y 1 0 56. 4x 5y 10 19–38 ■ Find an equation of the line that satisfies the given conditions. ✎ slope 7 2 0, 6 D 1 2 ✎ 19. Through 20. Through 21. Through 22. Through ✎ 23. Through 24. Through ✎ 25. Slope 3; 1 1 1 1 1 1 slope 5 slope 1 2 3 slope 2, 3 ; 2 2, 4 ; 2 1, 7 ; 2 3, 5 ; 2 and 1, 6 2, 1 2 1, 2 2 1 y-intercept 2 1 and 2 4, 3 2 2 26. Slope ; 5 y-intercept 4 27. x-intercept 1; 28. x-intercept 8; y-intercept 3 y-intercept 6 29. Through 30. Through 1 1 ; parallel to the x-axis 4, 5 2 ; parallel to the y-axis 4, 5 2 1, 6 31. Through 32. y-intercept 6; parallel to the line 2x 3y 4 0 ; parallel to the line x 2y 6 1 2 1, 2 33. Through 1 34. Through 35. Through 1 2, 6 2 1, 2 1 2x 5y 8 0 ; parallel to the line x 5 2 ; perpendicular to the line y 1 ; perpendicular to the line 2 ✎ ✎ ✎ ✎ 36. Through 1 2, 2 3B A ; perpendicular to the line 4x 8y 1 57. Use slopes to show that A 1 are vertices of a parallelogram. 1, 1 7, 7 5, 10 1 2, and 1 2 58. Use slopes to show that are vertices of a right triangle. A 3, 1 1 3, 3, B 1 2 2, and C 9, 8 1 2 59. Use slopes to show that, B are vertices of a rectangle. 1, 1 A 1 2 11, 3 10, 8, C 1 2 2, and 1 60. Use slopes to determine whether the |
given points are collinear (lie on a line). (a), 1, 1 2 1, 3 (b) 3, 9,, 2 1, 7 1 2 1 1 6, 21 1, 2 2 4, 15 2 61. Find an equation of the perpendicular bisector of the line 1 1 segment joining the points A 1, 4 and B 1 62. Find the area of the triangle formed by the coordinate axes and 1 2 2 7, 2. the line 2y 3x 6 0 63. (a) Show that if the x- and y-intercepts of a line are nonzero numbers a and b, then the equation of the line can be written in the form x a y b 1 This is called the two-intercept form of the equation of a line. (b) Use part (a) to find an equation of the line whose x-intercept is 6 and whose y-intercept is 8. x 70. Temperature Scales The relationship between the 178 CHAPTER 2 | Coordinates and Graphs 64. (a) Find an equation for the line tangent to the circle (b) What do the slope, the y-intercept, and the x-intercept of x 2 y 2 25 at the point 3, 4. (See the figure.) 1 (b) At what other point on the circle will a tangent line be 2 parallel to the tangent line in part (a)? y 0 (3, _4) ▼ APPLICATIONS 65. Grade of a Road West of Albuquerque, New Mexico, Route 40 eastbound is straight and makes a steep descent toward the city. The highway has a 6% grade, which means that its slope is. Driving on this road, you notice from elevation signs that you have descended a distance of 1000 ft. What is the change in your horizontal distance? 6 100 6% grade 1000 ft 66. Global Warming Some scientists believe that the average surface temperature of the world has been rising steadily. The average surface temperature can be modeled by T 0.02t 15.0 where T is temperature in C and t is years since 1950. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2050. 67. Drug Dosages If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, |
pharmacists use the equation c 0.0417D a 1 1 2 Suppose the dosage for an adult is 200 mg. (a) Find the slope. What does it represent? (b) What is the dosage for a newborn? 68. Flea Market The manager of a weekend flea market knows from past experience that if she charges x dollars for a rental space at the flea market, then the number y of spaces she can rent is given by the equation y 200 4x. (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be nonnegative quantities.) the graph represent? ✎ 69. Production Cost A small-appliance manufacturer finds that if he produces x toaster ovens in a month his production cost is given by the equation y 6x 3000 (where y is measured in dollars). (a) Sketch a graph of this linear equation. (b) What do the slope and y-intercept of the graph represent? Fahrenheit (F) and Celsius (C ) temperature scales is given 5 C 32. by the equation (a) Complete the table to compare the two scales at the given F 9 values. (b) Find the temperature at which the scales agree. [Hint: Suppose that a is the temperature at which the scales agree. Set F a and C a. Then solve for a.] C 30 20 10 0 F 50 68 86 71. Crickets and Temperature Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at 70 F and 168 chirps per minute at 80 F. (a) Find the linear equation that relates the temperature t and the number of chirps per minute n. (b) If the crickets are chirping at 150 chirps per minute, estimate the temperature. 72. Depreciation A small business buys a computer for $4000. After 4 years the value of the computer is expected to be $200. For accounting purposes the business uses linear depreciation to assess the value of the computer at a given time. This means that if V is the value of the computer at time t, then a linear equation is used to relate V and t. (a) Find a linear equation that relates V and t. (b) Sketch a graph of this linear equation |
. (c) What do the slope and V-intercept of the graph represent? (d) Find the depreciated value of the computer 3 years from the date of purchase. ✎ 73. Pressure and Depth At the surface of the ocean the water pressure is the same as the air pressure above the water, 15 lb/in2. Below the surface the water pressure increases by 4.34 lb/in2 for every 10 ft of descent. (a) Find an equation for the relationship between pressure and depth below the ocean surface. (b) Sketch a graph of this linear equation. (c) What do the slope and y-intercept of the graph represent? (d) At what depth is the pressure 100 lb/in2 SE CTI O N 2. 5 | Making Models Using Variation 179 $460 for 800 mi. Assume that there is a linear relationship between the monthly cost C of driving a car and the distance driven d. (a) Find a linear equation that relates C and d. (b) Use part (a) to predict the cost of driving 1500 mi per month. (c) Draw the graph of the linear equation. What does the slope of the line represent? (d) What does the y-intercept of the graph represent? (e) Why is a linear relationship a suitable model for this situation? 76. Manufacturing Cost The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Assuming that the relationship between cost and the number of chairs produced is linear, find an equation that expresses this relationship. Then graph the equation. (b) What is the slope of the line in part (a), and what does it represent? (c) What is the y-intercept of this line, and what does it represent? 74. Distance, Speed, and Time Jason and Debbie leave Detroit at 2:00 P.M. and drive at a constant speed, traveling west on I-90. They pass Ann Arbor, 40 mi from Detroit, at 2:50 P.M. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent? 75. Cost of Driving The monthly cost of driving a car depends on the number of miles driven. Lynn |
found that in May her driving cost was $380 for 480 mi and in June her cost was ▼ DISCOVE RY • DISCUSSION • WRITI NG 77. What Does the Slope Mean? Suppose that the graph of the outdoor temperature over a certain period of time is a line. How is the weather changing if the slope of the line is positive? If it is negative? If it is zero? 78. Collinear Points Suppose you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method? 2.5 Making Models Using Variation LEARNING OBJECTIVES After completing this section, you will be able to: ■ Find equations for direct variation ■ Find equations for inverse variation ■ Find equations for joint variation Mathematical models are discussed in more detail in Focus on Modeling, which begins on page 192. When scientists talk about a mathematical model for a real-world phenomenon, they often mean an equation that describes the relationship between two quantities. For instance, the model might describe how the population of an animal species varies with time or how the pressure of a gas varies as its temperature changes. In this section we study a kind of modeling called variation. ■ Direct Variation Two types of mathematical models occur so often that they are given special names. The first is called direct variation and occurs when one quantity is a constant multiple of the other, so we use an equation of the form y kx to model this dependence. 180 CHAPTER 2 | Coordinates and Graphs y k 0 FIGURE 1 y=kx (k>0) 1 x DIRECT VARIATION If the quantities x and y are related by an equation y kx for some constant k 0, we say that y varies directly as x, or y is directly proportional to x, or simply y is proportional to x. The constant k is called the constant of proportionality. Recall that the graph of an equation of the form y mx b is a line with slope m and y-intercept b. So the graph of an equation y kx that describes direct variation is a line with slope k and y-intercept 0 (see Figure 1). E X AM P L E 1 | Direct Variation During a thunderstorm you see the lightning before you hear the thunder because light travels much faster than sound. The distance between you and |
the storm varies directly as the time interval between the lightning and the thunder. (a) Suppose that the thunder from a storm 5400 ft away takes 5 s to reach you. Determine the constant of proportionality, and write the equation for the variation. (b) Sketch the graph of this equation. What does the constant of proportionality represent? (c) If the time interval between the lightning and thunder is now 8 s, how far away is the storm? ▼ SO LUTI O N (a) Let d be the distance from you to the storm, and let t be the length of the time inter- val. We are given that d varies directly as t, so where k is a constant. To find k, we use the fact that t 5 when d 5400. Substituting these values in the equation, we get d kt d 6000 4000 2000 5400 k 5 2 1 k 5400 5 Substitute 1080 Solve for k d=1080t Substituting this value of k in the equation for d, we obtain as the equation for d as a function of t. d 1080t 0 2 4 6 8 t FIGURE 2 (b) The graph of the equation d 1080t is a line through the origin with slope 1080 and is shown in Figure 2. The constant k 1080 is the approximate speed of sound (in ft/s). (c) When t 8, we have d 1080 # 8 8640 So the storm is 8640 ft 1.6 mi away. ✎ Practice what you’ve learned: Do Exercises 17 and 29. ▲ ■ Inverse Variation Another equation that is frequently used in mathematical modeling is y k/x, where k is a constant. SE CTI O N 2. 5 | Making Models Using Variation 181 y 0 y= k x (k>0) INVERSE VARIATION If the quantities x and y are related by the equation y k x x for some constant k 0, we say that y is inversely proportional to x or y varies inversely as x. FIGURE 3 Inverse variation The graph of y k/x for x 0 is shown in Figure 3 for the case k 0. It gives a picture of what happens when y is inversely proportional to x. E X AM P L E 2 | Inverse Variation Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure of the gas is |
inversely proportional to the volume of the gas. (a) Suppose the pressure of a sample of air that occupies 0.106 m3 at 25 C is 50 kPa. Find the constant of proportionality, and write the equation that expresses the inverse proportionality. (b) If the sample expands to a volume of 0.3 m3, find the new pressure. ▼ SO LUTI O N (a) Let P be the pressure of the sample of gas and let V be its volume. Then, by the definition of inverse proportionality, we have P k V where k is a constant. To find k, we use the fact that P 50 when V 0.106. Substituting these values in the equation, we get 50 k 0.106 k 50 1 2 1 0.106 2 5.3 Substitute Solve for k Putting this value of k in the equation for P, we have (b) When V 0.3, we have P 5.3 V P 5.3 0.3 17.7 So the new pressure is about 17.7 kPa. ✎ Practice what you’ve learned: Do Exercises 19 and 35. ▲ ■ Joint Variation A physical quantity often depends on more than one other quantity. If one quantity is proportional to two or more other quantities, we call this relationship joint variation. 182 CHAPTER 2 | Coordinates and Graphs JOINT VARIATION If the quantities x, y, and z are related by the equation z kxy where k is a nonzero constant, we say that z varies jointly as x and y or z is jointly proportional to x and y. In the sciences, relationships between three or more variables are common, and any combination of the different types of proportionality that we have discussed is possible. For example, if we say that z is proportional to x and inversely proportional to y. z k x y E X AM P L E 3 | Newton’s Law of Gravitation Newton’s Law of Gravitation says that two objects with masses m1 and m2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express Newton’s Law of Gravitation as an equation. ▼ SO LUTI O N Using the definitions of joint and inverse variation and the traditional notation G for the |
gravitational constant of proportionality, we have F G m1m2 2 r 5 ✎ Practice what you’ve learned: Do Exercises 21 and 41. ▲ If m1 and m2 are fixed masses, then the gravitational force between them is F C/r 2 (where C Gm1m2 is a constant). Figure 4 shows the graph of this equation for r 0 with C 1. Observe how the gravitational attraction decreases with increasing distance. 1.5 0 FIGURE 4 Graph of F 1 2 r 2. ▼ CONCE PTS 1. If the quantities x and y are related by the equation y 3x, ▼ SKI LLS 5–16 ■ Write an equation that expresses the statement. then we say that y is to x and the 5. T varies directly as x. constant of is 3. 6. P is directly proportional to „. 2. If the quantities x and y are related by the equation y 3 x, then 7. √ is inversely proportional to z. we say that y is to x and the constant of 8. „ is jointly proportional to m and n. is 3. 9. y is proportional to s and inversely proportional to t. 3. If the quantities x, y, and z are related by the equation z 3 x y, 10. P varies inversely as T. then we say that z is to x and 11. z is proportional to the square root of y. to y. 12. A is proportional to the square of t and inversely proportional 4. If z is jointly proportional to x and y and if z is 10 when x is 4 and y is 5, then x, y, and z are related by the equation z =. to the cube of x. 13. V is jointly proportional to l, „, and h. 14. S is jointly proportional to the squares of r and u. SE CTI O N 2. 5 | Making Models Using Variation 183 (a) Express this relationship by writing an equation. (b) To double the period, how would we have to change the length l? l 31. Printing Costs The cost C of printing a magazine is jointly proportional to the number of pages p in the magazine and the number of magazines printed m. (a) Write an equation that expresses this joint variation. (b) Find the constant of proportionality if the printing cost is $60,000 for 4000 copies of |
a 120-page magazine. (c) How much would the printing cost be for 5000 copies of a 92-page magazine? 32. Boyle’s Law The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. (a) Write an equation that expresses this variation. (b) Find the constant of proportionality if 100 L of gas exerts a pressure of 33.2 kPa at a temperature of 400 K (absolute temperature measured on the Kelvin scale). (c) If the temperature is increased to 500 K and the volume is decreased to 80 L, what is the pressure of the gas? 33. Power from a Windmill The power P that can be obtained from a windmill is directly proportional to the cube of the wind speed s. (a) Write an equation that expresses this variation. (b) Find the constant of proportionality for a windmill that produces 96 watts of power when the wind is blowing at 20 mi/h. (c) How much power will this windmill produce if the wind speed increases to 30 mi/h? 34. Power Needed to Propel a Boat The power P (measured in horse power, hp) needed to propel a boat is directly proportional to the cube of the speed s. An 80-hp engine is needed to propel a certain boat at 10 knots. Find the power needed to drive the boat at 15 knots. 15. R is proportional to i and inversely proportional to P and t. 16. A is jointly proportional to the square roots of x and y. 17–28 ■ Express the statement as an equation. Use the given information to find the constant of proportionality. 17. y is directly proportional to x. If x 6, then y 42. 18. z varies inversely as t. If t 3, then z 5. 19. R is inversely proportional to s. If s 4, then R 3. 20. P is directly proportional to T. If T 300, then P 20. 21. M varies directly as x and inversely as y. If x 2 and y 6, ✎ ✎ ✎ then M 5. 22. S varies jointly as p and q. If p 4 and q 5, then S 180. 23. W is inversely proportional to the square of r. If r 6, then W 10. 24. t is jointly proportional to x and y, and inversely proportional to r. If |
x 2, y 3, and r 12, then t 25. 25. C is jointly proportional to l, „, and h. If l „ h 2, then C 128. 26. H is jointly proportional to the squares of l and „. If l 2 and „ 1 3, then H 36. 27. s is inversely proportional to the square root of t. If s 100, then t 25. 28. M is jointly proportional to a, b, and c and inversely proportional to d. If a and d have the same value and if b and c are both 2, then M 128. ▼ APPLICATIONS 29. Hooke’s Law Hooke’s Law states that the force needed to ✎ keep a spring stretched x units beyond its natural length is directly proportional to x. Here the constant of proportionality is called the spring constant. (a) Write Hooke’s Law as an equation. (b) If a spring has a natural length of 10 cm and a force of 40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant. (c) What force is needed to keep the spring stretched to a length of 14 cm? 5 cm 30. Law of the Pendulum The period of a pendulum (the time elapsed during one complete swing of the pendulum) varies directly with the square root of the length of the pendulum. (a) Write an equation that expresses this variation. (b) A car weighing 1600 lb travels around a curve at 60 mi/h. The next car to round this curve weighs 2500 lb and requires the same force as the first car to keep from skidding. How fast is the second car traveling? 184 CHAPTER 2 | Coordinates and Graphs ✎ 35. Loudness of Sound The loudness L of a sound (measured in decibels, dB) is inversely proportional to the square of the distance d from the source of the sound. A person who is 10 ft from a lawn mower experiences a sound level of 70 dB. How loud is the lawn mower when the person is 100 ft away? 36. Stopping Distance The stopping distance D of a car after the brakes have been applied varies directly as the square of the speed s. A certain car traveling at 50 mi/h can stop in 240 ft. What is the maximum speed it can be traveling if it needs to |
stop in 160 ft? 37. A Jet of Water The power P of a jet of water is jointly proportional to the cross-sectional area A of the jet and to the cube of the velocity √. If the velocity is doubled and the cross-sectional area is halved, by what factor will the power increase? ✎ 41. Electrical Resistance The resistance R of a wire varies directly as its length L and inversely as the square of its diameter d. (a) Write an equation that expresses this joint variation. (b) Find the constant of proportionality if a wire 1.2 m long and 0.005 m in diameter has a resistance of 140 ohms. (c) Find the resistance of a wire made of the same material that is 3 m long and has a diameter of 0.008 m. 42. Kepler’s Third Law Kepler’s Third Law of planetary motion states that the square of the period T of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance d from the sun. (a) Express Kepler’s Third Law as an equation. (b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (c) The planet Neptune is about 2.79 10 9 mi from the sun. Find the period of Neptune. 43. Radiation Energy The total radiation energy E emitted by a heated surface per unit area varies as the fourth power of its absolute temperature T. The temperature is 6000 K at the surface of the sun and 300 K at the surface of the earth. (a) How many times more radiation energy per unit area is produced by the sun than by the earth? (b) The radius of the earth is 3960 mi and the radius of the sun is 435,000 mi. How many times more total radiation does the sun emit than the earth? 44. Value of a Lot The value of a building lot on Galiano Island is jointly proportional to its area and the quantity of water produced by a well on the property. A 200 ft by 300 ft lot has a well producing 10 gallons of water per minute, and is valued at $48,000. What is the value of a 400 ft by 400 ft lot if the well on the lot produces 4 gallons of water per minute? 38. Aerodynamic Lift The lift L on an airplane wing at takeoff varies jointly |
as the square of the speed s of the plane and the area A of its wings. A plane with a wing area of 500 ft2 traveling at 50 mi/h experiences a lift of 1700 lb. How much lift would a plane with a wing area of 600 ft2 traveling at 40 mi/h experience? Lift 39. Drag Force on a Boat The drag force F on a boat is jointly proportional to the wetted surface area A on the hull and the square of the speed s of the boat. A boat experiences a drag force of 220 lb when traveling at 5 mi/h with a wetted surface area of 40 ft2. How fast must a boat be traveling if it has 28 ft2 of wetted surface area and is experiencing a drag force of 175 lb? 40. Skidding in a Curve A car is traveling on a curve that forms a circular arc. The force F needed to keep the car from skidding is jointly proportional to the weight „ of the car and the square of its speed s, and is inversely proportional to the radius r of the curve. CHAPTER 2 | Review 185 47. Frequency of Vibration The frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string. (a) Write an equation that represents this variation. (b) What effect does doubling the length of the string have on the frequency of its vibration? 48. Spread of a Disease The rate r at which a disease spreads in a population of size P is jointly proportional to the number x of infected people and the number P x who are not infected. An infection erupts in a small town that has population P 5000. (a) Write an equation that expresses r as a function of x. (b) Compare the rate of spread of this infection when 10 people are infected to the rate of spread when 1000 people are infected. Which rate is larger? By what factor? (c) Calculate the rate of spread when the entire population is infected. Why does this answer make intuitive sense? ▼ DISCOVE RY • DISCUSSION • WRITI NG 49. Is Proportionality Everything? A great many laws of physics and chemistry are expressible as proportionalities. Give at least one example of a function that occurs in the sciences that is not a proportionality. 45. Growing Cabbages In the short growing season of the Canadian arctic territory of Nunavut |
, some gardeners find it possible to grow gigantic cabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives and inversely proportional to the number of other cabbages surrounding it. A cabbage that received 20 oz of nutrients and had 12 other cabbages around it grew to 30 lb. What size would it grow to if it received 10 oz of nutrients and had only 5 cabbage “neighbors”? 46. Heat of a Campfire The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire and inversely proportional to the cube of his distance from the fire. If he is 20 ft from the fire and someone doubles the amount of wood burning, how far from the fire would he have to be so that he feels the same heat as before? x CHAPTER 2 | REVIEW ▼ P R O P E RTI LAS The Distance Formula (p. 139) The distance between the points and B A 1 x22 The Midpoint Formula (p. 141) The midpoint of the line segment from 2 A, B x1 d 1 1 2 x1, y12 2 x2, y22 1 y22 2 y1 1 x1, y12 to B x2, y22 1 is x1 x2 2 y1, a A 1 y2 2 b Intercepts (p. 148) To find the x-intercepts of the graph of an equation, set solve for x. y 0 and To find the y-intercepts of the graph of an equation, set solve for y. x 0 and Circles (p. 149) The circle with center (0, 0) and radius r has equation x2 y2 r2 The circle with center (h, k) and radius r has equation is r2 Symmetry (p. 152) The graph of an equation is symmetric with respect to the x-axis if the equation remains unchanged when you replace y by y. The graph of an equation is symmetric with respect to the y-axis if the equation remains unchanged when you replace x by x. The graph of an equation is symmetric with respect to the origin if the equation remains unchanged when you replace x by x and y by y. Slope of |
a Line (p. 167) The slope of the nonvertical line that contains the points and is B A x1, y1 2 1 x2, y22 1 m rise run y2 x2 y1 x1 186 CHAPTER 2 | Coordinates and Graphs Equations of Lines (pp. 169–171) If a line has slope m, has y-intercept b, and contains the point x1, y12, then: 1 the point-slope form of its equation is x x12 the slope-intercept form of its equation is y y1 m 1 y mx b The equation of any line can be expressed in the general form Ax By C 0 (where A and B can’t both be 0). Vertical and Horizontal Lines (p. 170) The vertical line containing the point equation x a. a, b 2 1 has the are m2 Parallel and Perpendicular Lines (p. 172) Two lines with slopes m1 and m2 parallel if and only if m1 perpendicular if and only if m1 m2 1 Variation (p. 179) If y is directly proportional to x, then y kx If y is inversely proportional to x, then y k x If z is jointly proportional to x and y, then z kxy In each case, k is the constant of proportionality. The horizontal line containing the point equation y b. 1 a, b 2 has the ▼ CO N C E P T S U M MARY Section 2.1 ■ Graph points and regions in the coordinate plane ■ Use the Distance Formula ■ Use the Midpoint Formula Section 2.2 ■ Graph equations by plotting points ■ Find intercepts of the graph of an equation ■ Identify the equation of a circle ■ Graph circles in a coordinate plane ■ Determine symmetry properties of an equation Section 2.3 ■ Use a graphing calculator to graph equations ■ Solve equations graphically ■ Solve inequalities graphically Section 2.4 ■ Find the slope of a line ■ Find the point-slope form of the equation of a line Review Exercises 1(a)–4(a), 5, 6 1(b)–4(b), 7 1(c)–4(c) Review Exercises 15–24 25–30 1(e)–4(e), 8–14, 63, 64 1(e)–4(e), 11–14 25–30 Review Exercises 31–34 |
35–38 39–42 Review Exercises 1(d)–4(d) 1(d)–4(d) ■ Find the slope-intercept form of the equation of a line 1(d)–4(d), 43–52, 63, 64 ■ Find equations for horizontal and vertical lines ■ Find the general form for the equation of a line ■ Find equations for parallel or perpendicular lines ■ Model with linear equations: interpret slope as rate of change Section 2.5 ■ Find equations for direct variation ■ Find equations for inverse variation ■ Find equations for joint variation 47–48 43–52 49–52 53–54 Review Exercises 55, 59, 60 56–58 61–62 ▼ E X E RC I S E S 1–4 ■ Two points P and Q are given. (a) Plot P and Q on a coordinate plane. (b) Find the distance from P to Q. (c) Find the midpoint of the segment PQ. (d) Find the slope of the line determined by P and Q, and find equations for the line in point-slope form and in slope-intercept form. Then sketch a graph of the line. (e) Sketch the circle that passes through Q and has center P, and find the equation of this circle. 1. P 3. P 1 1 0, 3 2 6, 2 2, Q 3, 7 1, Q 2 4, 14 1 2. P 4. P 1 1 2 2, 0 2 5, 2 2, Q 4, 8 1, Q 2 3, 6 1 2 5–6 ■ Sketch the region given by the set. 5. x, y 2 0 51 4 x 4 and 2 y 2 6 x 4 or y 2 6. x, y 2 51 0 7. Which of the points 6 or 5, 3 B 1 2 is closer to the point 4, 4 A 1 2 1, 3 C 1? 2 8. Find an equation of the circle that has center radius 12. 9. Find an equation of the circle that has center passes through the origin. 2, 5 and 2 5, 1 and 2 1 1 10. Find an equation of the circle that contains the points 2, 3 2 and has the midpoint of the segment PQ as its P 1 Q and 1 center. 1, 8 2 11–14 ■ (a) Complete the square to determine whether the equation represents a circle or a point |
or has no graph. (b) If the equation is that of a circle, find its center and radius, and sketch its graph. 11. x 2 y 2 2x 6y 9 0 12. 2x 2 2y 2 2x 8y 13. x 2 y 2 72 12x 14. x 2 y 2 6x 10y 34 0 1 2 15–24 ■ Sketch the graph of the equation by making a table and plotting points. 15. y 2 3x 17. x 3y 21 16. 2x y 1 0 18. x 2y 12 19. x 2 y 7 1 21. y 16 x 2 23. x 1y 20. x 4 y 5 0 22. 8x y 2 0 24. y 21 x 2 25–30 ■ (a) Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin. (b) Find the x- and y-intercepts of the graph of the equation. 25. 27. y 9 x2 x2 y 1 1 2 1 2 29. 9x2 16y2 144 26. 28. 30. 6x y2 36 x4 16 y y 4 x CHAPTER 2 | Review 187 31–34 ■ Use a graphing device to graph the equation in an appropriate viewing rectangle. 31. y x 2 6x 32. 33. y x 3 4x 2 5x 34. y 25 x x 2 4 y 2 1 35–38 ■ Solve the equation graphically. 35. x 2 4x 2x 7 37. x 4 9x 2 x 9 38. 36. 2x 4 x x 3 5 0 2 5 2 @ 39–42 ■ Solve the inequality graphically. 39. 4x 3 x 2 2 1 4 4x 2 x 1 42. 41. x x 40. x 3 4x 2 5x 2 2 16 10 0 0 0 @ 0 43–52 ■ A description of a line is given. Find an equation for the line in (a) slope-intercept form and (b) general form. 43. The line that has slope 2 and y-intercept 6 44. The line that has slope (6, 3) 1 2 and passes through the point 45. The line that passes through the points (1, 6) and (2, 4) 46. The line that has x-intercept 4 and y-intercept 12 47. The vertical |
line that passes through the point (3, 2) 48. The horizontal line with y-intercept 5 49. The line that passes through the point (1, 1) and is parallel to the line 2x 5y 10 50. The line that passes through the origin and is parallel to the line containing (2, 4) and (4,4) 51. The line that passes through the origin and is perpendicular to the line y 1 2 x 10 52. The line that passes through the point (1, 7) and is perpendicu- lar to the line x 3y 16 0 53. Hooke’s Law states that if a weight „ is attached to a hanging spring, then the stretched length s of the spring is linearly related to „. For a particular spring we have s 0.3„ 2.5 where s is measured in inches and „ in pounds. (a) What do the slope and s-intercept in this equation represent? (b) How long is the spring when a 5-lb weight is attached? 54. Margarita is hired by an accounting firm at a salary of $60,000 per year. Three years later her annual salary has increased to $70,500. Assume that her salary increases linearly. (a) Find an equation that relates her annual salary S and the number of years t that she has worked for the firm. (b) What do the slope and S-intercept of her salary equation represent? (c) What will her salary be after 12 years with the firm? 55. Suppose that M varies directly as z and that M 120 when z 15. Write an equation that expresses this variation. 188 CHAP TER 2 | Coordinates and Graphs 56. Suppose that z is inversely proportional to y and that z 12 when y 16. Write an equation that expresses z in terms of y. 57. The intensity of illumination I from a light varies inversely as the square of the distance d from the light. (a) Write this statement as an equation. (b) Determine the constant of proportionality if it is known that a lamp has an intensity of 1000 candles at a distance of 8 m. (c) What is the intensity of this lamp at a distance of 20 m? 58. The frequency of a vibrating string under constant tension is inversely proportional to its length. If a violin string 12 inches long vibrates |
440 times per second, to what length must it be shortened to vibrate 660 times per second? 59. The terminal velocity of a parachutist is directly proportional to the square root of his weight. A 160-lb parachutist attains a terminal velocity of 9 mi/h. What is the terminal velocity for a parachutist who weighs 240 lb? 60. The maximum range of a projectile is directly proportional to the square of its velocity. A baseball pitcher throws a ball at 60 mi/h, with a maximum range of 242 ft. What is his maximum range if he throws the ball at 70 mi/h? 61. Suppose that F is jointly proportional to q1 and q2 and that 12. Find an equation that F 0.006 when q1 expresses F in terms of q1 and q2. 4 and q2 62. The kinetic energy E of a moving object is jointly proportional to the object’s mass m and the square of its speed √. A rock with mass 10 kg that is moving at 6 m/s has a kinetic energy of 180 J (joules). What is the kinetic energy of a car with mass 1700 kg that is moving at 30 m/s? 63–64 ■ Find equations for the circle and the line in the figure. 63. (_5, 12) y 0 x 64. y 5 0 (8, 1 ) x 5 ■ CHAPTER 2 | TEST 1. Let P and Q 7, 5 be two points in the coordinate plane. 1, 3 2 1 1 (a) Plot P and Q in the coordinate plane. 2 (b) Find the distance between P and Q. (c) Find the midpoint of the segment PQ. (d) Find the slope of the line that contains P and Q. (e) Find the perpendicular bisector of the line that contains P and Q. (f) Find an equation for the circle for which the segment PQ is a diameter. 2. Find the center and radius of each circle, and sketch its graph. (a) x 2 y 2 25 4 (bc) x 2 6x y 2 2y 6 0 3. Test each equation for symmetry. Find the x- and y-intercepts, and sketch a graph of the equation. (a) x 4 y 2 (b) y 0 x 2 0 4. A line has the general linear equation 3x 5y 15. (a) Find the x- and |
y-intercepts of the graph of this line. (b) Graph the line. Use the intercepts that you found in part (a) to help you. (c) Write the equation of the line in slope-intercept form. (d) What is the slope of the line? (e) What is the slope of any line perpendicular to the given line? 5. Find an equation for the line with the given property. (a) It passes through the point 3, 6 and is parallel to the line 3x y 10 0. (b) It has x-intercept 6 and y-intercept 4. 1 2 6. A geologist uses a probe to measure the temperature T (in C) of the soil at various depths below the surface, and finds that at a depth of x cm, the temperature is given by the linear equation T 0.08x 4. (a) What is the temperature at a depth of one meter (100 cm)? (b) Sketch a graph of the linear equation. (c) What do the slope, the x-intercept, and the T-intercept of the graph of this equation represent? 7. Solve the equation or inequality graphically, correct to two decimals. (a) x 3 9x 1 0 2 x 2 2x (b) 1 2 1 8. The maximum weight M that can be supported by a beam is jointly proportional to its width „ and the square of its height h and inversely proportional to its length L. (a) Write an equation that expresses this proportionality. (b) Determine the constant of proportionality if a beam 4 in. wide, 6 in. high, and 12 ft long can support a weight of 4800 lb. (c) If a 10-ft beam made of the same material is 3 in. wide and 10 in. high, what is the maximum weight it can support? L h „ 189 ● CUMUL ATIVE REVIE W TEST | CHAPTERS 1 and 2 1. Johanna earns 4.75% simple interest on her bank account annually. If she earns $380 interest in a given year, what amount of money did she have in her account at the beginning of the year? 2. Calculate each complex number and write the result in the form a bi. (b) 212 28 23 22 (a) 3 5 2 i 10 c) 15i 4 3i ( |
d) The complex conjugate of (a) x 7 2 3 2 x 3. Find all solutions, real and complex, of each equation. 4x 1 2x 3 x 2 2x 6 3x2 4x 2 0 x 0.5 2.25 x2 5x 6 x4 16 (d) (b) (c) (e) (f) 2 4 23 2 4. Solve each inequality. Graph the solution on a real number line, and express the solution in interval notation. (a) x 5 4 2x (c) 2x2 x 1 0 (b) 7 2 x 1 1 (d) x x 3 1 5. (a) Use a graphing calculator to graph the equation y x4 2x3 7x2 8x 12. (b) Use your graph to solve the equation (c) Use your graph to solve the inequality x4 2x3 7x2 8x 12 0. x 4 2x 3 7x 2 8x 12 0. 6. Let P 3, 6 and Q 10, 1 be two points in the coordinate plane. 1 1 (a) Find the distance between P and Q. 2 2 (b) Find the slope-intercept form of the equation of the line that contains P and Q. (c) Find an equation of the circle that contains P and Q and whose center is the midpoint of the segment PQ. (d) Find an equation for the line that contains P and that is perpendicular to the segment PQ. (e) Find an equation for the line that passes through the origin and that is parallel to the segment PQ. 7. Sketch a graph of each equation. Label all x- and y-intercepts. If the graph is a circle, find its center and radius. 2x 5y 20 (a) (b) x2 6x y2 0 8. Find an equation for each graph. (a) (b) y 6 0 3 x (_3, 4) y 1 1 x 190 CUMUL ATIVE REVIE W TEST | Chapters 1 and 2 191 9. A sailboat departed from an island and traveled north for 2 hours, then east for hours, all at the same speed. At the end of the trip it was 20 miles from where it started. At what speed did the boat travel? 11 2 10. A survey finds that the average starting salary for young people |
in their first full-time job is proportional to the square of the number of years of education they have completed. College graduates with 16 years of education have an average starting salary of $48,000. (a) Write an equation that expresses the relationship between years of education x and average starting salary S. (b) What is the average starting salary of a person who drops out of high school after completing the tenth grade? (c) A person with a master’s degree has an average starting salary of $60,750. How many years of education does this represent FITTING LINES TO DATA In Section 2.4 we used linear equations to model relationships between varying quantities. In practice, such relationships are discovered by collecting data about the quantities being studied. But data seldom fall into a precise line. Because of measurement errors or other random factors, a scatter plot of real-world data may appear to lie more or less on a line but not exactly. For example, the scatter plot in Figure 1(a) shows the results of a study on childhood obesity; the graph plots the body mass index (BMI) versus the number of hours of television watched per day for 25 adolescent subjects. Of course, we would not expect an exact relationship between these variables as in Figure 1(b), but clearly, the scatter plot in Figure 1(a) indicates a linear trend: The more hours a subject spends watching TV, the higher the BMI. So although we cannot fit a line exactly through the data points, a line like the one in Figure 1(a) shows the general trend of the data. BMI 30 20 10 BMI 30 20 10 a) Line of best fit (b) Line fits data exactly FIGURE 1 Fitting lines to data is one of the most important tools available to researchers who need to analyze numerical data. In this section we learn how to find and use lines that best fit data that exhibits a linear trend. ■ The Line That Best Fits the Data Until recently, infant mortality in the United States was declining steadily. Table 1 gives the nationwide infant mortality rate for the period from 1950 to 2000; the rate is the number of infants who died before reaching their first birthday, out of every 1000 live births. Over this half century the mortality rate was reduced by over 75%, a remarkable achievement in neonatal care. TABLE 1 U.S. Infant Mortality Year 1950 |
1960 1970 1980 1990 2000 Rate 29.2 26.0 20.0 12.6 9.2 6.9 y 30 20 10 0 10 20 30 40 50 x FIGURE 2 U.S. infant mortality rate 192 Fitting Lines to Data 193 The scatter plot in Figure 2 shows that the data lie roughly on a straight line. We can try to fit a line visually to approximate the data points, but since the data aren’t exactly linear, there are many lines that might seem to work. Figure 3 shows two attempts at “eyeballing” a line to fit the data. y 30 20 10 0 10 20 30 40 50 x FIGURE 3 Visual attempts to fit line to data y 0 FIGURE 4 Distance from the data points to the line Of all the lines that run through these data points, there is one that “best” fits the data, in the sense that it provides the most accurate linear model for the data. We now describe how to find this line. It seems reasonable that the line of best fit is the line that is as close as possible to all the data points. This is the line for which the sum of the vertical distances from the data points to the line is as small as possible (see Figure 4). For technical reasons it is better to use the line where the sum of the squares of these distances is smallest. This is called the regression line. The formula for the regression line is found by using calculus, but fortunately, the formula is programmed into most graphing calculators. In Example 1 we see how to use a TI-83 calculator to find the regression line for the infant mortality data described above. (The process for other calculators is similar.) x E X AM P L E 1 | Regression Line for U.S. Infant Mortality Rates (a) Find the regression line for the infant mortality data in Table 1. (b) Graph the regression line on a scatter plot of the data. (c) Use the regression line to estimate the infant mortality rates in 1995 and 2006. ▼ SO LUTI O N (a) To find the regression line using a TI-83 calculator, we must first enter the data into the lists L1 and L2, which are accessed by pressing the. Figure 5 shows the calculator screen after the data have been entered. (Note that we |
are letting x 0 correspond to the year 1950, so that x 50 corresponds to 2000. This makes the equations easier to work with.) key and selecting STAT EDIT L3 ------- 1 L2 29.2 26 20 12.6 9.2 6.9 L1 0 10 20 30 40 50 ------L2(7)= FIGURE 5 Entering the data 194 Focus on Modeling STAT We then press the provides the output shown in Figure 6(a). This tells us that the regression line is y 0.48x 29.4 key again and select, then 4:LinReg(ax+b), which CALC Here x represents the number of years since 1950, and y represents the corresponding infant mortality rate. LinReg y=ax+b a= -.4837142857 b=29.40952381 30 0 0 55 (a) Output of the LinReg command FIGURE 6 (b) Scatter plot and regression line (b) The scatter plot and the regression line have been plotted on a graphing calculator screen in Figure 6(b). (c) The year 1995 is 45 years after 1950, so substituting 45 for x, we find that 45 29.4 7.8 y 0.48. So the infant mortality rate in 1995 was about 7.8. Similarly, substituting 56 for x, we find that the infant mortality rate predicted for 2006 was about 29.4 2.5. 0.48 56 1 2 ▲ 1 2 An Internet search shows that the actual infant mortality rate was 7.6 in 1995 and 6.4 in 2006. So the regression line is fairly accurate for 1995 (the actual rate was slightly lower than the predicted rate), but it is considerably off for 2006 (the actual rate was more than twice the predicted rate). The reason is that infant mortality in the United States stopped declining and actually started rising in 2002, for the first time in more than a century. This shows that we have to be very careful about extrapolating linear models outside the domain over which the data are spread. Examples of Regression Analysis Since the modern Olympic Games began in 1896, achievements in track and field events have been improving steadily. One example in which the winning records have shown an upward linear trend is the pole vault. Pole vaulting began in the northern Netherlands as a practical activity: When traveling from village to village, people would vault across the many canals that crisscrossed |
the area to avoid having to go out of their way to find a bridge. Households maintained a supply of wooden poles of lengths appropriate for each member of the family. Pole vaulting for height rather than distance became a collegiate track and field event in the mid-1800s and was one of the events in the first modern Olympics. In the next example we find a linear model for the gold-medal-winning records in the men’s Olympic pole vault Tim Mack, 2004 Olympic Gold Medal Winner Fitting Lines to Data 195 E X AM P L E 2 | Regression Line for Olympic Pole Vault Records Table 2 gives the men’s Olympic pole vault records up to 2004. (a) Find the regression line for the data. (b) Make a scatter plot of the data, and graph the regression line. Does the regression line appear to be a suitable model for the data? (c) What does the slope of the regression line represent? (d) Use the model to predict the winning pole vault height for the 2008 Olympics. TABLE 2 Men’s Olympic Pole Vault Records Year 1896 1900 1904 1906 1908 1912 1920 1924 1928 1932 1936 1948 1952 x 4 0 4 6 8 12 20 24 28 32 36 48 52 Gold medalist Height (m) William Hoyt, USA Irving Baxter, USA Charles Dvorak, USA Fernand Gonder, France A. Gilbert, E. Cook, USA Harry Babcock, USA Frank Foss, USA Lee Barnes, USA Sabin Can, USA William Miller, USA Earle Meadows, USA Guinn Smith, USA Robert Richards, USA 3.30 3.30 3.50 3.50 3.71 3.95 4.09 3.95 4.20 4.31 4.35 4.30 4.55 Year 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 x 56 60 64 68 72 76 80 84 88 92 96 100 104 Gold medalist Height (m) Robert Richards, USA Don Bragg, USA Fred Hansen, USA Bob Seagren, USA W. Nordwig, E. Germany Tadeusz Slusarski, Poland W. Kozakiewicz, Poland Pierre Quinon, France Sergei Bubka, USSR M. Tarassob, Unified Team Jean Jaffione, France Nick Hysong, USA Timothy Mack, USA 4.56 4.70 5.10 5.40 5.64 |
5.64 5.78 5.75 5.90 5.87 5.92 5.90 5.95 LinReg y=ax+b a=.0265652857 b=3.400989881 ▼ SO LUTI O N (a) Let x year 1900, so 1896 corresponds to x 4, 1900 to x 0, and so on. Using a calculator, we find the following regression line: y 0.0266x 3.40 Output of the LinReg function on the TI-83 (b) The scatter plot and the regression line are shown in Figure 7. The regression line appears to be a good model for the data. (c) The slope is the average rate of increase in the pole vault record per year. So on average, the pole vault record increased by 0.0266 m/yr. y 6 4 2 Height (m) FIGURE 7 Scatter plot and regression line for pole vault data 0 20 60 40 Years since 1900 80 100 x (d) The year 2008 corresponds to x 108 in our model. The model gives y 0.0266 6.27 3.40 108 2 1 So the model predicts that in 2008 the winning pole vault will be 6.27 m. ▲ 196 Focus on Modeling If you are reading this after the 2008 Olympics, look up the actual record for 2008 and compare it with this prediction. Such predictions are reasonable for points close to our data, but we can’t predict too far away from the data. Is it reasonable to use this model to predict the record 100 years from now? (In Exercise 10 we find a regression line for the pole vault data from 1972 to 2004. Do the exercise to see whether more recent data provide a better predictor of future records.) In the next example we see how linear regression is used in medical research to investigate potential causes of diseases such as cancer. E X AM P L E 3 | Regression Line for Links Between Asbestos and Cancer When laboratory rats are exposed to asbestos fibers, some of the rats develop lung tumors. Table 3 lists the results of several experiments by different scientists. (a) Find the regression line for the data. (b) Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data? (c) What does the y-intercept of the regression line represent? TABLE 3 Asbestos–Tumor Data Asbestos exposure ( |
fibers/mL) Percent that develop lung tumors 50 400 500 900 1100 1600 1800 2000 3000 2 6 5 10 26 42 37 28 50 ▼ SO LUTI O N (a) Using a calculator, we find the following regression line (see Figure 8(a)): y 0.0177x 0.5405 (b) The scatter plot and regression line are graphed in Figure 8(b). The regression line appears to be a reasonable model for the data. LinReg y=ax+b a=.0177212141 b=.5404689256 55 0 0 3100 (a) Output of the LinReg command (b) Scatter plot and regression line FIGURE 8 Linear Regression for the asbestos–tumor data (c) The y-intercept is the percentage of rats that develop tumors when no asbestos fibers are present. In other words, this is the percentage that normally develop lung tumors ▲ (for reasons other than asbestos). Fitting Lines to Data 197 How Good Is the Fit? The Correlation Coefficient For any given set of two-variable data it is always possible to find a regression line, even if the data points do not tend to lie on a line and even if the variables don’t seem to be related at all. Look at the three scatter plots in Figure 9. In the first scatter plot the data points lie close to a line. In the second plot, there is still a linear trend but the points are more scattered. In the third plot there doesn’t seem to be any trend at all, linear or otherwise. y r=0.98 y r=0.84 y r=0.09 x x x FIGURE 9 A graphing calculator can give us a regression line for each of these scatter plots. But how well do these lines represent or “fit” the data? To answer this question, statisticians have invented the correlation coefficient, usually denoted r. The correlation coefficient is a number between 1 and 1 that measures how closely the data follow the regression line—or, in other words, how strongly the variables are correlated. Many graphing calculators give the value of r when they compute a regression line. If r is close to 1 or 1, then the variables are strongly correlated—that is, the scatter plot follows |
the regression line closely. If r is close to 0, then the variables are weakly correlated or not correlated at all. (The sign of r depends on the slope of the regression line.) The correlation coefficients of the scatter plots in Figure 9 are indicated on the graphs. For the first plot r is close to 1 because the data are very close to linear. The second plot also has a relatively large r, but not as large as the first, because the data, while fairly linear, are more diffuse. The third plot has an r close to 0, since there is virtually no linear trend in the data. There are no hard and fast rules for deciding what values of r are sufficient for deciding that a linear correlation is “significant.” The correlation coefficient is only a rough guide in helping us decide how much faith to put into a given regression line. In Example 1 the correlation coefficient is 0.99, indicating a very high level of correlation, so we can safely say that the drop in infant mortality rates from 1950 to 2000 was strongly linear. (The value of r is negative, since infant mortality trended down over this period.) In Example 3 the correlation coefficient is 0.92, which also indicates a strong correlation between the variables. So exposure to asbestos is clearly associated with the growth of lung tumors in rats. Does this mean that asbestos causes lung cancer? If two variables are correlated, it does not necessarily mean that a change in one variable causes a change in the other. For example, the mathematician John Allen Paulos points out that shoe size is strongly correlated to mathematics scores among schoolchildren. Does this mean that big feet cause high math scores? Certainly not—both shoe size and math skills increase independently as children get older. So it is important not to jump to conclusions: Correlation and causation are not the same thing. Correlation is a useful tool in bringing important cause-and-effect relationships to light; but to prove causation, we must explain the mechanism by which one variable affects the other. For example, the link between smoking and lung cancer was observed as a correlation long before science found the mechanism through which smoking causes lung cancer. 198 Focus on Modeling Femur Problems 1. Femur Length and Height Anthropologists use a linear model that relates femur length to height. The model allows an anthropologist |
to determine the height of an individual when only a partial skeleton (including the femur) is found. In this problem we find the model by analyzing the data on femur length and height for the eight males given in the table. (a) Make a scatter plot of the data. (b) Find and graph a linear function that models the data. (c) An anthropologist finds a femur of length 58 cm. How tall was the person? Femur length Height (cm) (cm) 50.1 48.3 45.2 44.7 44.5 42.7 39.5 38.0 178.5 173.6 164.8 163.7 168.3 165.0 155.4 155.8 2. Demand for Soft Drinks A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the table. (a) Make a scatter plot of the data. (b) Find and graph a linear function that models the data. (c) Use the model to predict soft-drink sales if the temperature is 95F. High temperature (F) Number of cans sold 55 58 64 68 70 75 80 84 340 335 410 460 450 610 735 780 3. Tree Diameter and Age To estimate ages of trees, forest rangers use a linear model that relates tree diameter to age. The model is useful because tree diameter is much easier to measure than tree age (which requires special tools for extracting a representative cross section of the tree and counting the rings). To find the model, use the data in the table, which were collected for a certain variety of oaks. (a) Make a scatter plot of the data. (b) Find and graph a linear function that models the data. (c) Use the model to estimate the age of an oak whose diameter is 18 in. Diameter (in.) Age (years) 2.5 4.0 6.0 8.0 9.0 9.5 12.5 15.5 15 24 32 56 49 76 90 89 Fitting Lines to Data 199 4. Carbon Dioxide Levels The Mauna Loa Observatory, located on the island of Hawaii, has been monitoring carbon dioxide (CO2) levels in the atmosphere since 1958. The table lists the average annual CO2 levels measured in parts per million (ppm) from 1984 to 2006. (a) Make a scatter plot of the data. ( |
b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the CO2 level in the atmosphere in 2005. Compare your answer with the actual CO2 level of 379.7 that was measured in 2005. Year CO2 level (ppm) 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 344.3 347.0 351.3 354.0 356.3 358.9 362.7 366.5 369.4 372.0 377.5 380.9 Temperature Chirping rate (F) (chirps/min) 5. Temperature and Chirping Crickets Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. 50 55 60 65 70 75 80 85 90 20 46 79 91 113 140 173 198 211 (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 100F. 6. Extent of Arctic Sea Ice The National Snow and Ice Data Center monitors the amount of ice in the Arctic year round. The table gives approximate values for the sea ice extent in millions of square kilometers from 1980 to 2006, in two-year intervals. (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the ice extent in the year 2010. Ice extent Year (million km2) Year Ice extent (million km2) 1980 1982 1984 1986 1988 1990 1992 7.9 7.4 7.2 7.6 7.5 6.2 7.6 1994 1996 1998 2000 2002 2004 2006 7.1 7.9 6.6 6.3 6.0 6.1 5.7 Flow rate (%) Mosquito positive rate (%) 0 10 40 60 90 100 22 16 12 11 6 2 7. Mosquito Prevalence The table lists the relative abundance of mosquitoes (as measured by the mosquito positive rate) versus the flow rate (measured as a percentage of maximum flow) of canal networks in Saga City, Japan. (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the mosquito positive |
rate if the canal flow is 70% of maximum. 200 Focus on Modeling 8. Noise and Intelligibility Audiologists study the intelligibility of spoken sentences under different noise levels. Intelligibility, the MRT score, is measured as the percent of a spoken sentence that the listener can decipher at a certain noise level in decibels (dB). The table shows the results of one such test. (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Find the correlation coefficient. Is a linear model appropriate? (d) Use the linear model in part (b) to estimate the intelligibility of a sentence at a 94-dB noise level. Noise level (dB) MRT score (%) 80 84 88 92 96 100 104 99 91 84 70 47 23 11 9. Life Expectancy The average life expectancy in the United States has been rising steadily over the past few decades, as shown in the table. (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model you found in part (b) to predict the life expectancy in the year 2006. (d) Search the Internet or your campus library to find the actual 2006 average life expectancy. Compare to your answer in part (c). Year Life expectancy 1920 1930 1940 1950 1960 1970 1980 1990 2000 54.1 59.7 62.9 68.2 69.7 70.8 73.7 75.4 76.9 10. Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event, there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 to complete the table on the next page of winning pole vault heights. (Note that we are using x 0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part (a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? Fitting Lines to |
Data 201 (d) What does the regression line predict as the winning pole vault height for the 2008 Olympics? Research the Internet or other sources to find the actual winning height for 2008, and compare it to this predicted value. Has this new regression line provided a better prediction than the line in Example 2? Year x Height (m) 5.64 0 4 8 1972 1976 1980 1984 1988 1992 1996 2000 2004 11. Olympic Swimming Records The tables give the gold medal times in the men’s and women’s 100-m freestyle Olympic swimming event. (a) Find the regression lines for the men’s data and the women’s data. (b) Sketch both regression lines on the same graph. When do these lines predict that the women will overtake the men in the event? Does this conclusion seem reasonable? MEN WOMEN Year Gold medalist Time (s) C. Daniels, USA D. Kahanamoku, USA D. Kahanamoku, USA J. Weissmuller, USA J. Weissmuller, USA Y. Miyazaki, Japan F. Csik, Hungary 1908 1912 1920 1924 1928 1932 1936 1948 W. Ris, USA C. Scholes, USA 1952 J. Henricks, Australia 1956 J. Devitt, Australia 1960 D. Schollander, USA 1964 1968 M. Wenden, Australia 1972 M. Spitz, USA 1976 J. Montgomery, USA 1980 J. Woithe, E. Germany R. Gaines, USA 1984 1988 M. Biondi, USA A. Popov, Russia 1992 A. Popov, Russia 1996 P. van den Hoogenband, Netherlands 2000 P. van den Hoogenband, Netherlands 2004 65.6 63.4 61.4 59.0 58.6 58.2 57.6 57.3 57.4 55.4 55.2 53.4 52.2 51.22 49.99 50.40 49.80 48.63 49.02 48.74 48.30 48.17 Year 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 Gold medalist Time (s) F. Durack, Australia E. Bleibtrey, USA E. Lackie, USA A. Osipowich, USA H. Madison, USA H. Mastenbroek, Holland G. Andersen, Denmark K. Szoke, Hungary D. Fraser, Australia D. |
Fraser, Australia D. Fraser, Australia J. Henne, USA S. Nielson, USA K. Ender, E. Germany B. Krause, E. Germany (Tie) C. Steinseifer, USA N. Hogshead, USA K. Otto, E. Germany Z. Yong, China L. Jingyi, China I. DeBruijn, Netherlands J. Henry, Australia 82.2 73.6 72.4 71.0 66.8 65.9 66.3 66.8 62.0 61.2 59.5 60.0 58.59 55.65 54.79 55.92 55.92 54.93 54.64 54.50 53.83 53.84 202 Focus on Modeling 12. Shoe Size and Height Do you think that shoe size and height are correlated? Find out by surveying the shoe sizes and heights of people in your class. (Of course, the data for men and women should be separate.) Find the correlation coefficient. 13. Demand for Candy Bars In this problem you will determine a linear demand equation that describes the demand for candy bars in your class. Survey your classmates to determine what price they would be willing to pay for a candy bar. Your survey form might look like the sample to the left. (a) Make a table of the number of respondents who answered “yes” at each price level. (b) Make a scatter plot of your data. (c) Find and graph the regression line y mp b, which gives the number of responents y who would buy a candy bar if the price were p cents. This is the demand equation. Why is the slope m negative? (d) What is the p-intercept of the demand equation? What does this intercept tell you about pricing candy bars? 3.1 What Is a Function? 3.2 Graphs of Functions 3.3 Getting Information from the Graph of a Function 3.4 Average Rate of Change of a Function 3.5 Transformations of Functions 3.6 Combining Functions 3.7 One-to-One Functions and Their Inverses CHAPTER 3 Functions Do you know the rule? We need to know a lot of rules for our everyday living—such as the rule that relates the amount of gas left in the gas tank to the distance we have driven or the rule that relates the grade we get in our algebra course to our exam scores. If we’re more adventuresome, |
like these sky divers, we may also need to know the rule that relates the distance fallen to the time we’ve been falling. Rules like these are modeled in algebra by using functions. In fact, a function is simply a rule that relates two quantities. One of the most famous such rules says that the distance d that an object falls in t seconds is 16t 2 ft (see Example 2 in Section 3.4). This rule works in a vacuum, where there is no air resistance. But for these sky divers it’s air resistance that makes the experience so much fun! (We’ll study the rule that takes air resistance into account in Chapter 5.) 203203 203 204 CHAPTER 3 | Functions 3.1 What Is a Function? LEARNING OBJECTIVES After completing this section, you will be able to: ■ Recognize functions in the real world ■ Work with function notation ■ Find domains of functions ■ Represent functions verbally, algebraically, graphically, and numerically Perhaps the most useful mathematical idea for modeling the real world is the concept of function, which we study in this chapter. In this section we explore the idea of a function and then give the mathematical definition of function. ■ Functions All Around Us In nearly every physical phenomenon we observe that one quantity depends on another. For example, your height depends on your age, the temperature depends on the date, the cost of mailing a package depends on its weight (see Figure 1). We use the term function to describe this dependence of one quantity on another. That is, we say the following: ■ Height is a function of age. ■ Temperature is a function of date. ■ Cost of mailing a package is a function of weight. The U.S. Post Office uses a simple rule to determine the cost of mailing a first-class parcel on the basis of its weight. But it’s not so easy to describe the rule that relates height to age or the rule that relates temperature to date. Height (in ft 15 10 Age (in years) 20 25 * F 80 60 40 20 0 Daily high temperature Columbia, MO, April 1995 5 10 15 20 25 30 Date „ (ounces) Postage (dollars) 0 < „≤1 1 < „≤2 2 < „≤3 3 < „≤4 4 < „≤5 5 < „≤6 1.13 |
1.30 1.47 1.64 1.81 1.98 FIGURE 1 Height is a function of age. T emperature is a function of date. Postage is a function of weight. Can you think of other functions? Here are some more examples: ■ The area of a circle is a function of its radius. ■ The number of bacteria in a culture is a function of time. ■ The weight of an astronaut is a function of her elevation. ■ The price of a commodity is a function of the demand for that commodity. The rule that describes how the area A of a circle depends on its radius r is given by the formula A pr 2. Even when a precise rule or formula describing a function is not available, we can still describe the function by a graph. For example, when you turn on a hot water faucet, the temperature of the water depends on how long the water has been running. So we can say ■ Temperature of water from the faucet is a function of time. SE CTI ON 3.1 | What Is a Function? 205 Figure 2 shows a rough graph of the temperature T of the water as a function of the time t that has elapsed since the faucet was turned on. The graph shows that the initial temperature of the water is close to room temperature. When the water from the hot water tank reaches the faucet, the water’s temperature T increases quickly. In the next phase, T is constant at the temperature of the water in the tank. When the tank is drained, T decreases to the temperature of the cold water supply. T (°F) 110 100 90 80 70 60 50 0 t ■ Definition of Function A function is a rule. To talk about a function, we need to give it a name. We will use letters such as f, g, h,... to represent functions. For example, we can use the letter f to represent a rule as follows: “f ” is the rule “square the number” FIGURE 2 Graph of water temperature T as a function of time t We have previously used letters to stand for numbers. Here we do something quite different. We use letters to represent rules. When we write f f 2 2 4. Similarly, we mean “apply the rule f to the number 2.” Applying the rule gives 3 2 9, f 4 2 16, and in general f x 2. x 1 2 4 |
1 2 DEFINITION OF A FUNCTION A function f is a rule that assigns to each element x in a set A exactly one element, called f x, in a set B. 2 1 1 2 We usually consider functions for which the sets A and B are sets of real numbers. The is read “f of x” or “f at x” and is called the value of ff at x, or the image of x symbol f under ff. The set A is called the domain of the function. The range of f is the set of all possible values of f as x varies throughout the domain, that is, x x 2 1 range of f f x ƒ x A 1 5 The symbol that represents an arbitrary number in the domain of a function f is called an independent variable. The symbol that represents a number in the range of f is called a dependent variable. So if we write y f, then x is the independent variable and y is x the dependent variable. 6 2 2 1 It is helpful to think of a function as a machine (see Figure 3). If x is in the domain of the function f, then when x enters the machine, it is accepted as an input and the machine produces an output f according to the rule of the function. Thus, we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. x 2 1 FIGURE 3 Machine diagram of f x input f Ï output Another way to picture a function is by an arrow diagram as in Figure 4 on the next page. Each arrow connects an element of A to an element of B. The arrow indicates that is associated with a, and so on. f is associated with x, f a x 1 2 1 2 10 10 key on your calculator is a The good example of a function as a machine. First you input x into the display. Then you press the key labeled. (On most graphing calculators the order of these operations is reversed.) If x 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x 0, then an approximation to correct to a certain number of decimal 10 places. (Thus, the culator is not quite the same as the exact mathematical function f defined by f appears in the display, key on your cal- 1x 1x.) x 1 2 206 CHAPTER 3 |
| Functions A x a B Ï f(a) FIGURE 4 Arrow diagram of f f E X AM P L E 1 | Analyzing a Function A function f is defined by the formula x 2 4 f x 1 2 (a) Express in words how f acts on the input x to produce the output f 15 (b) Evaluate f, f 2 (c) Find the domain and range of f. (d) Draw a machine diagram for f., and ▼ SO LUTI O N (a) The formula tells us that f first squares the input x and then adds 4 to the result. So f is the function “square, then add 4” (b) The values of f are found by substituting for x in the formula f 32 4 13 2 2 4 8 Replace x by 3 3 2 2 Replace x by –2 f f 1 1 2 15 1 2 15 2 1 f 1 2 4 9 2 Replace x by 5 x 2 4. x 1 2 (c) The domain of f consists of all possible inputs for f. Since we can evaluate the for x 2 4 for every real number x, the domain of f is the set of all real x mula f 2 numbers. 1 The range of f consists of all possible outputs of f. Because x 2 0 for all real 4. Thus, numbers x, we have x 2 4 4, so for every output of f we have f 4, q the range of f is d) A machine diagram for f is shown in Figure 5. ✎ Practice what you’ve learned: Do Exercises 9, 13, 17, and 43. ▲ x input 3 _2 square and add 4 square and add 4 square and add 4 x2+4 output 13 8 FIGURE 5 Machine diagram ■ Evaluating a Function In the definition of a function the independent variable x plays the role of a “placeholder.” For example, the function f 3x 2 x 5 can be thought of as x 2 To evaluate f at a number, we substitute the number for the placeholder AM P L E 2 | Evaluating a Function Let f 1 (a) f x 2 2 3x 2 x 5. Evaluate each function value. (d) f (b) f (c 2B A SE CTI ON 3.1 | What Is a Function? 207 To evaluate f at a number, we |
substitute the number for x in the defini- 1 2 ▼ SO LUTI O N tion of f. 3 # 2 2 (a) f 2 2 2 3 # 02 0 5 5 3 # 2 4 5 47 2 3 # 2 1 5 15 2 4 2 (d) (b) (c ✎ Practice what you’ve learned: Do Exercise 19. ▲ E X AM P L E 3 | A Piecewise Defined Function A cell phone plan costs $39 a month. The plan includes 400 free minutes and charges 20¢ for each additional minute of usage. The monthly charges are a function of the number of minutes used, given by C x 1 2 39 39 0.20 x 400 1 2 if 0 x 400 if x 400 Find C 100 1 2, C 400 1 2, and C. 2 480 1 b ▼ SO LUTI O N Remember that a function is a rule. Here is how we apply the rule for this function. First we look at the value of the input x. If 0 x 400, then the value of C. is 39. On the other hand, if x 400, then the value of C is 39 0.20 x 400 x 1 2 1 2 1 x 2 Since 100 400, we have C Since 400 400, we have C Since 480 400, we have C 39. 39. 39 0.20 100 1 400 1 480 1 2 2 2 480 400 1 2 55. A piecewise-defined function is defined by different formulas on different parts of its domain. The function C of Example 3 is piecewise defined. Thus, the plan charges $39 for 100 minutes, $39 for 400 minutes, and $55 for 480 minutes. ✎ Practice what you’ve learned: Do Exercise 27. ▲ E X AM P L E 4 | Evaluating a Function Expressions like the one in part (d) of Example 4 occur frequently in calculus; they are called difference quotients, and they represent the average change in the value of f between x a and x a h. 2x 2 3x 1 a f If (a) 1 x f (c) f 2 1 1 2 a h 2, evaluate the following. f f (b) (d ▼ SO LUTI O N (a) a 2 a 2 a h (b) (c) f f 1 1 1 2 2 |
a2 3a a2 2ah h2 2 1 2a2 4ah 2h2 3a 3h 1 1 2a2 3a d) Using the results from parts (c) and (a), we have 2a2 4ah 2h2 3a 3h 1 h 1 2 2a2 3a 1 2 4ah 2h2 3h h 4a 2h 3 ✎ Practice what you’ve learned: Do Exercise 35. ▲ 208 CHAPTER 3 | Functions E X AM P L E 5 | The Weight of an Astronaut If an astronaut weighs 130 pounds on the surface of the earth, then her weight when she is h miles above the earth is given by the function 130 „ h 1 2 2 3960 3960 h b a (a) What is her weight when she is 100 mi above the earth? (b) Construct a table of values for the function „ that gives her weight at heights from 0 to 500 mi. What do you conclude from the table? ▼ SO LUTI O N (a) We want the value of the function „ when h 100; that is, we must calculate 100 „ 1. 2 100 „ 1 2 130 3960 3960 100 b a 2 123.67 So at a height of 100 mi she weighs about 124 lb. (b) The table gives the astronaut’s weight, rounded to the nearest pound, at 100-mile increments. The values in the table are calculated as in part (a). The weight of an object on or near the earth is the gravitational force that the earth exerts on it. When in orbit around the earth, an astronaut experiences the sensation of “weightlessness” because the centripetal force that keeps her in orbit is exactly the same as the gravitational pull of the earth. h 0 100 200 300 400 500 „„ h 11 22 130 124 118 112 107 102 The table indicates that the higher the astronaut travels, the less she weighs. ✎ Practice what you’ve learned: Do Exercise 71. ▲ ■ The Domain of a Function Recall that the domain of a function is the set of all inputs for the function. The domain of a function may be stated explicitly. For example, if we write Domains of algebraic expressions are discussed on page 45 then the domain is the set of all real numbers x for which 0 x 5. If the function is given by an algebraic expression and the domain |
is not stated explicitly, then by convention the domain of the function is the domain of the algebraic expression—that is, the set of all real numbers for which the expression is defined as a real number. For example, consider the functions 1x The function f is not defined at x 4, so its domain is 5 fined for negative x, so its domain is. The function g is not de- E X AM P L E 6 | Finding Domains of Functions Find the domain of each function. (ab) g x 1 2 29 x 2 (c) h t 1 2 t 1t 1 SE CTI ON 3.1 | What Is a Function? 209 ▼ SO LUTI O N (a) The function is not defined when the denominator is 0. Since is not defined when x 0 or x 1. Thus, the domain of f is 2 we see that f x 1 2 The domain may also be written in interval notation as x 5 0 x 0, x 1 6 q, 0 0, 1 1, q 1 (b) We can’t take the square root of a negative number, so we must have 9 x 2 0. Using the methods of Section 1.6, we can solve this inequality to find that 3 x 3. Thus, the domain of g is, 3 4 (c) We can’t take the square root of a negative number, and we can’t divide by 0, so we must have t 1 0, that is, t 1. So the domain of h is 0 5 ✎ Practice what you’ve learned: Do Exercises 47 and 51. 6 1 2 t t 1 1, q ▲ ■ Four Ways to Represent a Function To help us understand what a function is, we have used machine and arrow diagrams. We can describe a specific function in the following four ways: ■ verbally (by a description in words) ■ algebraically (by an explicit formula) ■ visually (by a graph) ■ numerically (by a table of values) A single function may be represented in all four ways, and it is often useful to go from one representation to another to gain insight into the function. However, certain functions are described more naturally by one method than by the others. An example of a verbal description is the following rule for converting between temperature scales: � |
�To find the Fahrenheit equivalent of a Celsius temperature, multiply the Celsius temperature by, then add 32.” 9 5 In Example 7 we see how to describe this verbal rule or function algebraically, graphically, and numerically. A useful representation of the area of a circle as a function of its radius is the algebraic formula pr 2 A r 1 2 The graph produced by a seismograph (see the box on the next page) is a visual representation of the vertical acceleration function a t of the ground during an earthquake. As a 1 final example, consider the function C, which is described verbally as “the cost of mail2 ing a first-class letter with weight „.” The most convenient way of describing this function is numerically—that is, using a table of values. „ 1 2 We will be using all four representations of functions throughout this book. We sum- marize them in the following box. 210 CHAPTER 3 | Functions FOUR WAYS TO REPRESENT A FUNCTION Verbal Using words: Algebraic Using a formula: “To convert from Celsius to Fahrenheit, multiply the Celsius temperature by, then add 32.” 9 5 pr 2 A r 1 2 Relation between Celsius and Fahrenheit temperature scales Area of a circle Visual Using a graph: a (cm/s2) 100 50 −50 5 10 15 20 25 30 t (s) Source: Calif. Dept. of Mines and Geology Numerical Using a table of values: „„ (ounces) C(„„) (dollars.13 1.30 1.47 1.64 1.81 o Vertical acceleration during an earthquake Cost of mailing a first-class parcel E X AM P L E 7 | Representing a Function Verbally, Algebraically, Numerically, and Graphically 2 C 1 Let F be the Fahrenheit temperature corresponding to the Celsius temperature C. (Thus, F is the function that converts Celsius inputs to Fahrenheit outputs.) The box above gives a verbal description of this function. Find ways to represent this function (a) Algebraically (using a formula) (b) Numerically (using a table of values) (c) Visually (using a graph) ▼ SO LUTI O N (a) The verbal description tells us that we should first multiply the input C by and then 9 |
5 add 32 to the result. So we get (b) We use the algebraic formula for F that we found in part (a) to construct a table of values: C F 1 2 9 5 C 32 C (Celsius) F (Fahrenheit) 10 0 10 20 30 40 14 32 50 68 86 104 SE CTI ON 3.1 | What Is a Function? 211 (c) We use the points tabulated in part (b) to help us draw the graph of this function in Figure 6. F 100 90 80 70 60 50 40 30 20 10 _10 0 30 ✎ Practice what you’ve learned: Do Exercise 65. 20 10 40 C ▲ FIGURE 6 Celsius and Fahrenheit 3. ▼ CONCE PTS 1. If a function f is given by the formula y f of f at x a. 7. Subtract 5, then square x, then f 2 1 a 2 1 is the 8. Take the square root, add 8, then multiply by 1 3 9–12 ■ Express the function (or rule) in words. 2. For a function f, the set of all possible inputs is called the of f, and the set of all possible outputs is called the of f. ✎ 9. h 11 10. k x 1 2 12. g x 1 2 1x 2 x 3 4 3. (a) Which of the following functions have 5 in their domain? x 5 x 2x 10 x 2 3x b) For the functions from part (a) that do have 5 in their do- main, find the value of the function at 5. 4. A function is given algebraically by the formula f 2 3. Complete these other ways to represent f: x 4 x 1 2 2 1 (a) Verbal: “Subtract 4, then (b) Numerical: and. 22 ff x 11 19 x 0 2 4 6 ▼ SKI LLS 5–8 ■ Express the rule in function notation. (For example, the rule x 2 5.) “square, then subtract 5” is expressed as the function f x 1 2 5. Add 3, then multiply by 2 6. Divide by 7, then subtract 4 13–14 ■ Draw a machine diagram for the function. 3 1x 1 14. 13 15–16 ■ Complete the table. 15 16. g x 1 2 0 2x 3 0 x ff 11 22 x 1 0 1 |
2 3 gg x 11 22 x 3 2 0 1 3 17–26 ■ Evaluate the function at the indicated values 10 1, f 2 0. x2 6; x3 2x; 2x 1 ; 2, f A 2 x 2 2x 1, f 1 2 B ; ✎ 17. f 18. f ✎ 19. 20 ab a ; 39. f x 1 2 x x 1 40. f x 1 2x x 1 x 3 2 2 41. f x 3 5x 4x 2 2 1 1 43–64 ■ Find the domain of the function. 42. f x 2x 2x, 1 x 5 x 2 1, 0 x 5 44. f x 1 2 x 2 1 212 CHAPTER 3 | Functions 2x 2 3x 4x 2 ; 1 1, f 1 12, 21. g g 22. h h f f f f f f 23. 24. 25. 26 27–30 ■ Evaluate the piecewise defined function at the indicated values. ✎ 27. f f 28. f f 29 2x 2x x 1 3 2 B, f 1 30. f x 1 2 • 3x x 1 x 2 2 2 1 0 if x 0 if if x 2 if if x 1 if 1 x 1 if x 1 1 2, f 0, f 25 2 1 2 1 if x 0 if 0 x 2 if 31–34 ■ Use the function to evaluate the indicated expressions and simplify. 1 1 1 2 2 31. 32. 33. f f f 34; f 3x 1; f x 4; f 2x 1 x 2 2, 2 1 6x 18; f a 2, 2f 35–42 ■ Find, where h 0., and the difference quotient ✎ 35. f 37. f x x 1 1 2 2 3x 2 5 36. f 38 ✎ 43. 45. 46. f f f ✎ 47. f ✎ 49. f 51. 53. f f 55. h 57. g 59 2x 5 23 t 1 22x 5 22 x 3 x 24 x 2 6x 61. f x 1 2 63. f x 1 2 1 3 2x 4 x 1 2 22x 1 2 48. f 50. f 52 54. 56. g 1 G x 2 x 1 2 58. g 60. g 62 64. f x 1 2 1 3x 6 x 4 x 2 x 6 24 x 9 |
27 3x 2x 2 9 1x 2x 2 x 1 2x 2 2x 8 x 2 26 x x 24 9 x 2 65–68 ■ A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. ✎ 65. To evaluate f 66. To evaluate g result by. 3 4 x x 1 1, divide the input by 3 and add 2 2 3 to the result., subtract 4 from the input and multiply the 2 67. Let T x 1 2 be the amount of sales tax charged in Lemon County on a purchase of x dollars. To find the tax, take 8% of the purchase price. 68. Let V 1 2 d be the volume of a sphere of diameter d. To find the volume, take the cube of the diameter, then multiply by p and divide by 6. ▼ APPLICATIONS 69. Production Cost The cost C in dollars of producing x yards of a certain fabric is given by the function C x 1500 3x 0.02x 2 0.0001x 3 2 10 1 1 (a) Find C 100 1 (b) What do your answers in part (a) represent? (c) Find C and C. 2 2 0 1. (This number represents the fixed costs.) 2 70. Area of a Sphere The surface area S of a sphere is a func- tion of its radius r given by 4pr2 S r 1 2 (a) Find S (b) What do your answers in part (a) represent? and S 2 1 3 1. 2 2 ✎ 71. Torricelli’s Law A tank holds 50 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli’s Law gives the volume of water remaining in the tank after t minutes as (a) Find, R 2 1 (b) Make a table of values of, and 10 R R 1 1 2 1.. 100 2 x R 2 1 SE CTI ON 3.1 | What Is a Function? 213 V t 1 2 50 a 2 1 t 20 b 0 t 20 (a) Find V(0) and V(20). (b) What do your answers to part (a) represent? (c) Make a |
table of values of V t for t 0, 5, 10, 15, 20. 1 2 72. How Far Can You See? Because of the curvature of the earth, the maximum distance D that you can see from the top of a tall building or from an airplane at height h is given by the function 22rh h2 D h 1 2 where r 3960 mi is the radius of the earth and D and h are measured in miles. (a) Find D(0.1) and D(0.2). (b) How far can you see from the observation deck of Toronto’s CN Tower, 1135 ft above the ground? (c) Commercial aircraft fly at an altitude of about 7 mi. How far can the pilot see? 73. Blood Flow As blood moves through a vein or an artery, its velocity √ is greatest along the central axis and decreases as the distance r from the central axis increases (see the figure). The formula that gives √ as a function of r is called the law of laminar flow. For an artery with radius 0.5 cm, we have 0 r 0.5 1 (a) Find √ (b) What do your answers to part (a) tell you about the flow 0.25 r2 18,500 2 0.1 1 1 0.4 1 and √. 2 √ r 2 2 of blood in this artery? (c) Make a table of values of √(r) for r 0, 0.1, 0.2, 0.3, 0.4, 0.5. 0.5 cm r 74. Pupil Size When the brightness x of a light source is in- creased, the eye reacts by decreasing the radius R of the pupil. The dependence of R on x is given by the function R x 1 2 B 13 7x 0.4 1 4x 0.4 R 75. Relativity According to the Theory of Relativity, the length L of an object is a function of its velocity √ with respect to an observer. For an object whose length at rest is 10 m, the function is given by L √ 1 2 10 B 1 √ 2 c 2 where c is the speed of light. (a) Find 0.5c, and (b) How does the length of an object change as its velocity 0.75c 0.9c |
, increases? 76. Income Tax In a certain country, income tax T is assessed according to the following function of income x: T x 1 2 • 0 0.08x 1600 0.15x if 0 x 10,000 if 10,000 x 20,000 if 20,000 x (a) Find (b) What do your answers in part (a) represent? 12,000 25,000 5,000, and, 77. Internet Purchases An Internet bookstore charges $15 shipping for orders under $100 but provides free shipping for orders of $100 or more. The cost C of an order is a function of the total price x of the books purchased, given by C x 1 2 x 15 x e if x 100 if x 100 (a) Find, C (b) What do your answers in part (a) represent?, and 100 105, C 90 75 78. Cost of a Hotel Stay A hotel chain charges $75 each night for the first two nights and $50 for each additional night’s stay. The total cost T is a function of the number of nights x that a guest stays. (a) Complete the expressions in the following piecewise defined function. T x 1 2 if 0 x 2 if x 2 e (b) Find T(2), T(3), and T(5). (c) What do your answers in part (b) represent? 79. Speeding Tickets In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum is 40. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum. 214 CHAPTER 3 | Functions (a) Complete the expressions in the following piecewise defined function, where x is the speed at which you are driving. F x 1 2 • if 0 x 40 if 40 x 65 if x 65 (b) Find F(30), F(50), and F(75). (c) What do your answers in part (b) represent? 80. Height of Grass A home owner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period beginning on a Sunday. Georgia, on March 18, 1996. The time t was measured in hours from midnight. Sketch a rough graph of T as a function of t. t 0 2 4 6 8 |
10 12 T 58 57 53 50 51 57 61 83. Population Growth The population P (in thousands) of San Jose, California, from 1988 to 2000 is shown in the table. (Midyear estimates are given.) Draw a rough graph of P as a function of time t. t 1988 1990 1992 1994 1996 1998 2000 P 733 782 800 817 838 861 895 81. Temperature Change You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Sketch a rough graph of the temperature of the pie as a function of time. 82. Daily Temperature Change Temperature readings T (in °F) were recorded every 2 hours from midnight to noon in Atlanta, ▼ DISCOVE RY • DISCUSSION • WRITI NG 84. Examples of Functions At the beginning of this section we discussed three examples of everyday, ordinary functions: Height is a function of age, temperature is a function of date, and postage cost is a function of weight. Give three other examples of functions from everyday life. 85. Four Ways to Represent a Function In the box on page 210 we represented four different functions verbally, algebraically, visually, and numerically. Think of a function that can be represented in all four ways, and write the four representations. 3.2 Graphs of Functions LEARNING OBJECTIVES After completing this section, you will be able to: ■ Graph a function by plotting points ■ Graph a function using a graphing calculator ■ Graph piecewise defined functions ■ Use the Vertical Line Test ■ Determine whether an equation defines a function The most important way to visualize a function is through its graph. In this section we investigate in more detail the concept of graphing functions. SE CTI ON 3.2 | Graphs of Functions 215 ■ Graphing Functions by Plotting Points To graph a function f, we plot the points plot the points 1 sponding output of the function. x, y x, f 22 x 1 2 1 in a coordinate plane. In other words, we whose x-coordinate is an input and whose y-coordinate is the corre- y Óx, ÏÔ THE GRAPH OF A FUNCTION If f is a function with domain A, then the graph of f is the set of ordered pairs f(2) Ï f(1) 0 1 2 x x FIGURE 1 The height of the |
graph x above the point x is the value of f 1. 2 x, f x 1 22 0 51 x A 6 x, y In other words, the graph of f is the set of all points 1 y f x is, the graph of f is the graph of the equation. 1 2 such that y f ; that x 1 2 2 2 x tion. We can read the value of f the point x (see Figure 1). The graph of a function f gives a picture of the behavior or “life history” of the funcfrom the graph as being the height of the graph above A function f of the form f 1 mx b is called a linear function because its graph is the graph of the equation y mx b, which represents a line with slope m and y-intercept b. A special case of a linear function occurs when the slope is m 0. The function f b, where b is a given number, is called a constant function because all its values are the same number, namely, b. Its graph is the horizontal line y b. Figure 2 shows the graphs of the constant function f 3 and the linear function f 2x 1=3 _2 0 2 4 6 x y 1 0 y=2x+1 1 x FIGURE 2 The constant function Ï=3 The linear function Ï=2x+1 E X AM P L E 1 | Graphing Functions by Plotting Points Sketch the graphs of the following functions. (a) f x3 x2 (b) g x x (c) 1 2 1 2 1x h x 1 2 ▼ SO LUTI O N We first make a table of values. Then we plot the points given by the table and join them by a smooth curve to obtain the graph. The graphs are sketched in Figure 3 on the next page. x 0 1 2 1 2 3 ff(x) x2 gg(x) x3 (x) 1x 0 1 12 13 2 15 216 CHAPTER 3 | Functions (_2, 4) y 3 (2, 4) y=≈ (_1, 1) (1, 1) y 2 (2, 8) y=x£ ) (1, 1 0 1!_1, _1) 1 x (_2, _8) y 1 0 (2, ) œ∑2 (1, 1) 1 y=œ∑x (4, 2) |
FIGURE 3 (a) Ï=≈ (b) ˝=x£ (c) h(x)=œ∑x ✎ Practice what you’ve learned: Do Exercises 11, 15, and 19. x ▲ ■ Graphing Functions with a Graphing Calculator A convenient way to graph a function is to use a graphing calculator. Because the graph of a function f is the graph of the equation y f, we can use the methods of Section 2.3 2 to graph functions on a graphing calculator. x 1 E X AM P L E 2 | Graphing a Function with a Graphing Calculator Use a graphing calculator to graph the function f rectangle. 1 x 2 x3 8x 2 in an appropriate viewing To graph the function f x3 8x 2, we graph the equation ▼ SO LUTI O N 1 y x3 8x 2. On the TI-83 graphing calculator the default viewing rectangle gives the graph in Figure 4(a). But this graph appears to spill over the top and bottom of the screen. We need to expand the vertical axis to get a better representation of the graph. The viewing rectangle gives a more complete picture of the graph, as shown in 3 Figure 4(b). 100, 100 4, 10 by 3 x 2 4 4 10 100 _10 10 _4 10 _10 (a) _100 (b) ✎ Practice what you’ve learned: Do Exercise 29. ▲ E X AM P L E 3 | A Family of Power Functions FIGURE 4 Graphing the function f x3 8x 2 x 1 2 (a) Graph the functions f by 1, 3. 4 (b) Graph the functions for n 2, 4, and 6 in the viewing rectangle x n for n 1, 3, and 5 in the viewing rectangle 2, 2 2, 2 4 4 3 3 by 2, 2 3. 4 (c) What conclusions can you draw from these graphs? SE CTI ON 3.2 | Graphs of Functions 217 ▼ SO LUTI O N graphs for parts (a) and (b) are shown in Figure 5. To graph the function f x 1 2 x n, we graph the equation y x n. The 3 x§ x¢ x™ 2 x ∞ x £ x _2 _2 2 2 _1 _2 (a) Even powers of x (b) Odd powers |
of x (c) We see that the general shape of the graph of f even or odd. x n x 1 2 depends on whether n is FIGURE 5 A family of power functions f x n x 1 2 If n is even, the graph of If n is odd, the graph of is similar to the parabola y x 2. is similar to that of y x3. ✎ Practice what you’ve learned: Do Exercise 69. ▲ Notice from Figure 5 that as n increases, the graph of y x n becomes flatter near 0 and steeper when x 1. When 0 x 1, the lower powers of x are the “bigger” functions. But when x 1, the higher powers of x are the dominant functions. ■ Graphing Piecewise Defined Functions A piecewise defined function is defined by different formulas on different parts of its domain. As you might expect, the graph of such a function consists of separate pieces. E X AM P L E 4 | Graph of a Piecewise Defined Function Sketch the graph of the function. f x 1 2 x 2 2x 1 e if x 1 if x 1 On many graphing calculators the graph in Figure 6 can be produced by using the logical functions in the calculator. For example, on the TI-83 the following equation gives the required graph: Y1 1 X 1 X2 2 1 5 X 1 2 1 2X 1 2 1, 3 2 1 If x 1, then f x 2, so the part of the graph to the left of x 1 ▼ SO LUTI O N coincides with the graph of y x 2, which we sketched in Figure 3. If x 1, then 2x 1, so the part of the graph to the right of x 1 coincides with the line f 1 y 2x 1, which we graphed in Figure 2. This enables us to sketch the graph in Figure 6. indicates that this point is included in the graph; the open dot at The solid dot at x x 2 2 1 1, 1 2 1 indicates that this point is excluded from the graph. y 1 0 f (x) = 2x + 1 if x > 1 1 x f(x) = x2 if x ≤ 1 _2 2 _1 (To avoid the extraneous vertical line between the two parts of the graph, put the calculator in Dot mode.) ✎ |
Practice what you’ve learned: Do Exercise 35. FIGURE 6 x 2 2x 1 e x f 1 2 if x 1 if x 1 ▲ 218 CHAPTER 3 | Functions E X AM P L E 5 | Graph of the Absolute Value Function Sketch the graph of the absolute value function f ▼ SO LUTI O N Recall that if x 0 if x 0 Using the same method as in Example 4, we note that the graph of f coincides with the line y x to the right of the y-axis and coincides with the line y x to the left of the y-axis (see Figure 7). y 1 0 1 x ✎ Practice what you’ve learned: Do Exercise 23. ▲ The greatest integer function is defined by FIGURE 7 Graph of f x 1 2 x 0 0 “ x‘ greatest integer less than or equal to x “2.3‘ 2, “0.002‘ 0, “1.999‘ 1, “3.5‘ 4, and For example, “0.5‘ 1 “2‘ 2,. E X AM P L E 6 | Graph of the Greatest Integer Function Sketch the graph of f x “ x‘. 1 2 ▼ SO LUTI O N is constant between consecutive integers, so the graph between integers is a horizontal line segment, as shown in Figure 8. The table shows the values of f for some values of x. Note that FIGURE 8 The greatest integer function, y “ x‘ ▲ The greatest integer function is an example of a step function. The next example gives a real-world example of a step function. E X AM P L E 7 | The Cost Function for Long-Distance Phone Calls The cost of a long-distance daytime phone call from Toronto to Mumbai, India, is 69 cents for the first minute and 58 cents for each additional minute (or part of a minute). Draw the graph of the cost C (in dollars) of the phone call as a function of time t (in minutes). C 1 0 1 t SE CTI ON 3.2 | Graphs of Functions 219 be the cost for t minutes. Since t 0, the domain of the func- ▼ SO LUTI O N tion is 0, q Let C t 1 2 1. From the given information we have.69 0.69 0.58 1 |
.27 0.69 2 0.69 3 if 0 t 1 if 1 t 2 1.85 if 2 t 3 2.43 if 3 t 4 0.58 0.58 2 1 1 2 FIGURE 9 Cost of a long-distance call and so on. The graph is shown in Figure 9. ✎ Practice what you’ve learned: Do Exercise 81. ▲ ■ The Vertical Line Test The graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test. THE VERTICAL LINE TEST A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once. We can see from Figure 10 why the Vertical Line Test is true. If each vertical line, then exactly one functional value is defined by a, c, then the curve and at 1 x a intersects a curve only once at f cannot represent a function because a function cannot assign two different values to a. b. But if a line x a intersects the curve twice, at a, b 1 a=a (a, b) a x y 0 x=a (a, c) (a, b) a x FIGURE 10 Vertical Line Test Graph of a function Not a graph of a function E X AM P L E 8 | Using the Vertical Line Test Using the Vertical Line Test, we see that the curves in parts (b) and (c) of Figure 11 represent functions, whereas those in parts (a) and (d) do not. y 0 x (a) FIGURE 11 y 0 (b) y x 0 x (c) ✎ Practice what you’ve learned: Do Exercise 51. y 0 (d) x ▲ 220 CHAPTER 3 | Functions ■ Equations That Define Functions Any equation in the variables x and y defines a relationship between these variables. For example, the equation y x 2 0 defines a relationship between y and x. Does this equation define y as a function of x? To find out, we solve for y and get y x 2 We see that the equation defines a rule, or function, that gives one value of y for each value of x. We can express this rule in function notation as 1 But not every |
equation defines y as a function of x, as the following example shows. 2 f x x 2 E X AM P L E 9 | Equations That Define Functions Does the equation define y as a function of x? (a) y x 2 2 (b) x 2 y 2 4 ▼ SO LUTI O N (a) Solving for y in terms of x gives Image not available due to copyright restrictions y x 2 2 y x 2 2 Add x 2 Add x 2 Donald Knuth was born in Milwaukee in 1938 and is Professor Emeritus of Computer Science at Stanford University. While still a graduate student at Caltech, he started writing a monumental series of books entitled The Art of Computer Programming. President Carter awarded him the National Medal of Science in 1979. When Knuth was a high school student, he became fascinated with graphs of functions and laboriously drew many hundreds of them because he wanted to see the behavior of a great variety of functions. (Today, of course, it is far easier to use computers and graphing calculators to do this.) Knuth is famous for his invention of TEX, a system of computer-assisted typesetting. This system was used in the preparation of the manuscript for this textbook. Dr. Knuth has received numerous honors, among them election as an associate of the French Academy of Sciences, and as a Fellow of the Royal Society. The last equation is a rule that gives one value of y for each value of x, so it defines y as a function of x. We can write the function as f x 2 2. x 1 2 (b) We try to solve for y in terms of x Subtract x 2 y 24 x 2 Take square roots The last equation gives two values of y for a given value of x. Thus, the equation does not define y as a function of x. ✎ Practice what you’ve learned: Do Exercises 57 and 61. ▲ The graphs of the equations in Example 9 are shown in Figure 12. The Vertical Line Test shows graphically that the equation in Example 9(a) defines a function but the equation in Example 9(b) does not. y 1 0 y-≈=2 x 1 (a) FIGURE 12 ≈+¥=4 1 x y 1 0 (b) SE CTI ON 3.2 | Graphs of Functions |
221 The following table shows the graphs of some functions that you will see frequently in this book. SOME FUNCTIONS AND THEIR GRAPHS Linear functions Ï=mx+b Power functions x n Ï= Root functions Ï= nœ∑x Reciprocal functions Ï=1/x n Absolute value function Ï=|x | y b Ï=b y Ï=≈ y Ï=œ∑x y Ï=|x | 3. ▼ CONCE PTS 1. To graph the function f, we plot the points (x, coordinate plane. To graph f x 2 ). So the point (2, 1 (x, ) in a x3 2, we plot the points ) is on the graph of f. y b Ï=mx+b y Ï=x£ y Ï= £œ∑x y Ï=x¢ y x x Ï=x∞ y x x Ï= ¢œ∑x Ï= ∞œ∑x Greatest integer function Ï=“x‘ y 1 1 x Ï=“x‘ The height of the graph of f above the x-axis when x 2 is. 2. If f 2 1 2 3, then the point (2, ) is on the graph of f. 222 CHAPTER 3 | Functions 3. If the point (2, 3) is on the graph of f, then f 2 1 2. 30. 4. Match the function with its graph. (a) f (c 1x (b) f (d III II IV ▼ SKI LLS 5–28 ■ Sketch the graph of the function by first making a table of values. 5. f 7. f 9. f 10. f 11. f 13. h 15. g 17 2x 4 x 3, 3 x 3 x 3 2 x 2 16 x 2 x 3 8 x 2 2x 1 1x 1x 19. f x 1 21. 23 25. G x 1 2 2x 0 x 0 0 0 x ✎ ✎ ✎ 27. f x 1 2 0 2x 2 0 6. f 8 3x 12. f 14. g 16. g 18. h 20 22. 24. g 1 H x 2 x 1 2 26. G x 1 2 28 4x 2 x4 1x 4 1x 29 |
–32 ■ Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. ✎ 2 29. f x 1 (a) (b) (c) (d) 8x x 2 5, 5 by 4 10, 10 2, 10 4 10, 10 3 3 3 3 5, 5 3 by 4 by 4 10, 10 3 5, 20 3 4 100, 100 by a) (b) (c) (d) x h 1 (a) (b) (c) (d) x k 1 (a) (b) (c) (d) 4 10, 10 3 4 4 100, 100 3 4 4 4 x 2 x 20 2, 2 5, 5 by 3 10, 10 by 7, 7 25, 20 3 10, 10 by 4 by 4 x 3 5x 4 2, 2 2, 2 by 4 3 4 10, 10 3, 3 by 3 4 10, 5 3, 3 by 3 4 10, 10 by 4 4 10, 10 4 4 3 32 x 4 x 2 2 1, 1 by 3 2, 2 by 3 5, 5 by 3 by 1 1, 1 3 2, 2 3 5, 5 3 10, 10 3 4 4 4 10, 10 3 4 4 4 4 4 31. 32. 33–46 ■ Sketch the graph of the piecewise defined function. 33. f x 1 2 34. f x 1 2 ✎ 35. f x 1 2 36. f x 1 2 37. f x 1 2 38. f x 1 2 0 1 e if x 2 if if x 1 if x 1 if x 2 if x 2 if x 2 if x 2 if x 0 if x 0 2x 3 3 x e if x 1 if x 1 39. f x 1 2 • 40 if x 1 if 1 x 1 if x 1 if x 1 if 1 x 1 if x 1 41. f x 1 2 42. f x 1 2 43. f x 1 2 44. f x 1 2 45. f x 1 2 46. f x 1 2 2 x 2 e if x 1 if x 1 1 x 2 x e if x 2 if x 2 0 3 e x 2 1 e if if 0 0 if if x2 x 6 x 9 x 2 x 3 • • if x 2 if 2 x 2 if x 2 if x 0 if 0 x 3 if x 3 49. 50 |
. y 2 0 y 2 0 47–48 ■ Use a graphing device to draw the graph of the piecewise defined function. (See the margin note on page 217.) 47 if x 1 if x 1 48. f x 1 2 e 2x x 2 x 1 3 1 2 if x 1 if x 1 49–50 ■ The graph of a piecewise defined function is given. Find a formula for the function in the indicated form. f x 1 2 • if x 2 if 2 x 2 if x 2 55. 2 x SE CTI ON 3.2 | Graphs of Functions 223 53–56 ■ Use the Vertical Line Test to determine whether the curve is the graph of a function x. If it is, state the domain and range of the function. 53. y 2 0 y 1 0 54. 56 • if x 1 if 1 x 2 if x 2 1 x 51–52 ■ Use the Vertical Line Test to determine whether the curve is the graph of a function of x. ✎ 51. (a) (c) 52. (a) (cb) (d) (b) (d ✎ ✎ ✎ 57–68 ■ Determine whether the equation defines y as a function of x. (See Example 9.) 57. x 2 2y 4 59. x y2 61. x y2 9 63. x 2y y 1 58. 3x 7y 21 60. x 2 (y 1)2 4 62. x 2 y 9 1x y 12 64. y 0 x 65. 2 0 67. x y3 0 2x 66. 0 68. x y4 y 0 0 69–74 ■ A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. 69. f 2 x x 2 c 1 (a) c 0, 2, 4, 6; (b) c 0, 2, 4, 6; 10, 10 (c) How does the value of c affect the graph? 10, 10 by 4 3 5, 5 by 3 5, 5 3 4 3 4 4 70. 71a) c 0, 1, 2, 3; (b) c 0, 1, 2, 3; (c) How does the value of c affect the graph? 10, 10 |
by 4 3 5, 5] by 3 3 5, 5 3 10, 10a) c 0, 2, 4, 6; (b) c 0, 2, 4, 6; (c) How does the value of c affect the graph? 10, 10] by 3 3 10, 10] by 3 10, 10] 3 10, 10] 72. f x cx 2 1 2 (a) c 1, 1 2 (b) c 1, 1,, 2; (c) How does the value of c affect the graph? [5, 5] by [10, 10] [5, 5] by [10, 10], 2, 4; 1 2 73. f 2 x c c 1 2, 1 c 1, 1 x 1 (a) (b) (c) How does the value of c affect the graph? [1, 4] by [1, 3] [3, 3] by [2, 2] 4, 1 ; 6 3, 1 ; 5 224 CHAPTER 3 | Functions 74. 2 x 1/x n f 1 (a) n 1, 3; 4 (b) n 2, 4; 4 (c) How does the value of n affect the graph? 3, 3 3 3, 3 3 3, 3 3 3, 3 3 by by 4 4 75–78 ■ Find a function whose graph is the given curve. 75. The line segment joining the points (2, 1) and (4, 6) 76. The line segment joining the points (3, 2) and (6, 3) 77. The top half of the circle x 2 y2 9 78. The bottom half of the circle x 2 y2 9 ▼ APPLICATIONS 79. Weather Balloon As a weather balloon is inflated, the thickness T of its rubber skin is related to the radius of the balloon by T r 1 2 0.5 r 2 where T and r are measured in centimeters. Graph the function T for values of r between 10 and 100. 80. Power from a Wind Turbine The power produced by a wind turbine depends on the speed of the wind. If a windmill has blades 3 meters long, then the power P produced by the turbine is modeled by 14.1√ 3 P √ 1 2 where P is measured in watts (W) and √ in meters per second. Graph the function P for wind speeds between 1 m/s and 10 m |
/s. ✎ 81. Utility Rates Westside Energy charges its electric customers a base rate of $6.00 per month, plus 10¢ per kilowatt-hour (kWh) for the first 300 kWh used and 6¢ per kWh for all usage over 300 kWh. Suppose a customer uses x kWh of electricity in one month. (a) Express the monthly cost E as a function of x. (b) Graph the function E for 0 x 600. 82. Taxicab Function A taxi company charges $2.00 for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a function of the distance x traveled (in miles) for 0 x 2, and sketch the graph of this function. 83. Postage Rates The domestic postage rate for first-class let- ters weighing 3.5 oz or less is 41 cents for the first ounce (or less), plus 17 cents for each additional ounce (or part of an ounce). Express the postage P as a function of the weight x of a letter, with 0 x 3.5, and sketch the graph of this function. ▼ DISCOVE RY • DISCUSSION • WRITI NG 84. When Does a Graph Represent a Function? For every integer n, the graph of the equation y x n is the graph of a function, namely f(x) x n. Explain why the graph of x y2 is not the graph of a function of x. Is the graph of x y3 the graph of a function of x? If so, of what function of x is it the graph? Determine for what integers n the graph of x y n is the graph of a function of x. 85. Step Functions In Example 7 and Exercises 82 and 83 we are given functions whose graphs consist of horizontal line segments. Such functions are often called step functions, because their graphs look like stairs. Give some other examples of step functions that arise in everyday life. x “ x‘, g 86. Stretched Step Functions Sketch graphs of the functions on separate graphs., and x h f 1 How are the graphs related? If n is a positive integer, what does k the graph of look like? “ nx‘ “2x‘ “3x 87. |
Graph of the Absolute Value of a Function (a) Draw the graphs of the functions f x x 2 x 6. How are the graphs of f and g 1 2 x g and 2 relatedb) Draw the graphs of the functions x 4 6x 2 g 2 (c) In general, if. How are the graphs of f and g related? 0, how are the graphs of f and g f 1 related? Draw graphs to illustrate your answer. x 4 6x 2 and DISCOVERY PR OJECT RELATIONS AND FUNCTIONS A function f can be represented as a set of ordered pairs (x, y), where x is the input and y f is the output. For example, the function that squares each natural number can be represented by the ordered pairs {(1, 1), (2, 4), (3, 9),...}. x 1 2 A relation is any collection of ordered pairs. If we denote the ordered pairs in a relation by (x, y) then the set of x-values (or inputs) is the domain, and the set of y-values (or outputs) is the range. With this terminology a function is a relation where for each x-value there is exactly one y-value (or for each input there is exactly one output). The correspondences in the figure below are relations; the first is a function but the second is not because the input 7 in A corresponds to two different outputs, 15 and 17, in B. A 1 2 3 4 B 10 20 30 A 7 8 9 B 15 17 18 19 Function Not a function We can describe a relation by listing all the ordered pairs in the relation or giving the rule of correspondence. Also, since a relation consists of ordered pairs, we can sketch its graph. Let’s consider the following relations and try to decide which are functions. (a) The relation that consists of the ordered pairs {(1, 1), (2, 3), (3, 3), (4, 2)}. (b) The relation that consists of the ordered pairs {(1, 2), (1, 3), (2, 4), (3, 2)}. (c) The relation whose graph is shown to the left. (d) The relation whose input values are days in January 2005 and whose output values are the maximum temperature in Los Angeles on that day. (e) The relation whose input values are days in January 2005 and whose |
output values are the persons born in Los Angeles on that day. The relation in part (a) is a function because each input corresponds to exactly one output. But the relation in part (b) is not, because the input 1 corresponds to two different outputs (2 and 3). The relation in part (c) is not a function because the input 1 corresponds to two different outputs (1 and 2). The relation in (d) is a function because each day corresponds to exactly one maximum temperature. The relation in (e) is not a function because many people (not just one) were born in Los Angeles on most days in January 2005. 1. Let A {1, 2, 3, 4} and B {1, 0, 1}. Is the given relation a function from y 3 2 1 0 _1 1 2 3 x A to B? (a) {(1, 0), (2, 1), (3, 0), (4, 1)} (b) {(1, 0), (2, 1), (3, 0), (3, 1), (4, 0)} 2. Determine if the correspondence is a function. A A B (b) (aCONTINUES) 225 RELATIONS AND FUNCTIONS (CONTINUED) 3. The following data were collected from members of a college precalculus class. Is the set of ordered pairs (x, y) a function? 6'6" 6'0" 5'6" 5'0" (a) x Height y Weight (b) (c) 72 in. 60 in. 60 in. 63 in. 70 in. x Age 19 21 40 21 21 180 lb 204 lb 120 lb 145 lb 184 lb y ID Number 82-4090 80-4133 66-8295 64-9110 20-6666 x Year of graduation y Number of graduates 2005 2006 2007 2008 2009 2 12 18 7 1 4. An equation in x and y defines a relation, which might or might not be a function (see page 220). Decide whether the relation consisting of all ordered pairs of real numbers (x, y) satisfying the given condition is a function. (a) y x 2 (c) x y (b) x y 2 (d) 2x 7y 11 5. In everyday life we encounter many relations that might or might not define functions. For example, we match up people with their telephone number(s), |
baseball players with their batting averages, or married men with their wives. Does this last correspondence define a function? In a society in which each married man has exactly one wife, the rule is a function. But in a polygamous society the rule is not a function. Which of the following everyday relations are functions? (a) x is the daughter of y (x and y are women in the United States) (b) x is taller than y (x and y are people in California) (c) x has received dental treatment from y (x and y are millionaires in the United States) (d) x is a digit (0 to 9) on a telephone dial and y is a corresponding letter 226 SE CTI O N 3. 3 | Getting Information from the Graph of a Function 227 Getting Information from the Graph of a Function 3.3 LEARNING OBJECTIVES After completing this section, you will be able to: ■ Find function values from a graph ■ Find the domain and range of a function from a graph ■ Find where a function is increasing or decreasing from a graph ■ Find local maxima and minima of functions from a graph Many properties of a function are more easily obtained from a graph than from the rule that describes the function. We will see in this section how a graph tells us whether the values of a function are increasing or decreasing and also where the maximum and minimum values of a function are. ■ Values of a Function; Domain and Range A complete graph of a function contains all the information about a function, because the graph tells us which input values correspond to which output values. To analyze the graph of a function, we must keep in mind that the height of the graph is the value of the function. So we can read off the values of a function from its graph. E X AM P L E 1 | Finding the Values of a Function from a Graph T (*F) 40 30 20 10 0 1 2 3 4 5 6 x FIGURE 1 Temperature function The function T graphed in Figure 1 gives the temperature between noon and 6:00 P.M. at a certain weather station. (a) Find T, and T 3 2 (c) Find the value(s) of x for which T (d) Find the value(s) of x for which T (b) Which is larger, T 25. 25., T 2 or ▼ SO LUTI O N (a) T(1) is the temperature at 1:00 |
P.M. It is represented by the height of the graph above the x-axis at x 1. Thus, 1 5 2 1 2 (b) Since the graph is higher at x 2 than at x 4, it follows that T is larger than T 4 1 (c) The height of the graph is 25 when x is 1 and when x is 4. In other words, the tem- 30 and T 2. Similarly, T 20. 25 T 3 2 2 1 1 1. 2 perature is 25 at 1:00 P.M. and 4:00 P.M. (d) The graph is higher than 25 for x between 1 and 4. In other words, the temperature was 25 or greater between 1:00 P.M. and 4:00 P.M. ✎ Practice what you’ve learned: Do Exercise 5. ▲ The graph of a function helps us to picture the domain and range of the function on the x-axis and y-axis, as shown in Figure 2. y Range y=Ï FIGURE 2 Domain and range of f 0 Domain x 228 CHAPTER 3 | Functions E X AM P L E 2 | Finding the Domain and Range from a Graph 24 x 2 (a) Use a graphing calculator to draw the graph of (b) Find the domain and range of f. x f. 1 2 ▼ SO LUTI O N (a) The graph is shown in Figure 3. FIGURE 3 Graph of f 24 x 2 x 1 2 Range=[0, 2] _2 0 2 Domain=[_2, 2] (b) From the graph in Figure 3 we see that the domain is 3 ✎ Practice what you’ve learned: Do Exercise 15. 2, 2 4 and the range is 0, 2. 4 3 ▲ ■ Increasing and Decreasing Functions It is very useful to know where the graph of a function rises and where it falls. The graph shown in Figure 4 rises, falls, then rises again as we move from left to right: It rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing when its graph rises and decreasing when its graph falls. y f is increasing. B f is decreasing. D A 0 a y=Ï b C c f is increasing. d x a, b 3 b, c 4 4. We have the following definition. DE |
FINITION OF INCREASING AND DECREASING FUNCTIONS FIGURE 4 f is increasing on. f is decreasing on and c, d 3 4 3 f is increasing on an interval I if f f is decreasing on an interval I if 1 f x12 x12 1 f 1 f x22 x22 1 whenever x1 whenever x1 x2 x2 in I. in I. y y f f(x⁄) f(x¤) f f( x⁄) f(x¤) 0 x⁄ x¤ x 0 x⁄ x¤ x f is increasing f is decreasing SE CTI O N 3. 3 | Getting Information from the Graph of a Function 229 E X AM P L E 3 | Intervals on Which a Function Increases and Decreases The graph in Figure 5 gives the weight W of a person at age x. Determine the intervals on which the function W is increasing and on which it is decreasing. W (lb) 200 150 100 50 0 10 20 30 40 50 60 70 80 x (yr) FIGURE 5 Weight as a function of age The function W is increasing on. It is decreasing on. The function W is constant (neither increasing nor decreasing) on and 25, 30 4. This means that the person gained weight until age 25, then gained weight again 4 ▼ SO LUTI O N 40, 50 3 50, 80 3 between ages 35 and 40. He lost weight between ages 40 and 50. ✎ Practice what you’ve learned: Do Exercise 45. 35, 40 0, 25 and ▲ 4 4 4 3 3 3 E X AM P L E 4 | Finding Intervals Where a Function Increases and Decreases (a) Sketch a graph of the function f (b) Find the domain and range of f. (c) Find the intervals on which f increases and decreases. x 1 2 12x 2 4x 3 3x 4. ▼ SO LUTI O N (a) We use a graphing calculator to sketch the graph in Figure 6. (b) The domain of f is because f is defined for all real numbers. Using the feature on the calculator, we find that the highest value is f of f is q, 32. 2 1 2 (c) From the graph we see that f is increasing on the intervals and is decreasing on 1, 0 2, q. q, 1 1 and 0, |
2 3 4 4 and 4 3 2 1 4 3 TRACE 32. So the range FIGURE 6 Graph of f 12x 2 4x 3 3x 4 x 1 2 40 _2.5 3.5 _40 ✎ Practice what you’ve learned: Do Exercise 23. ▲ E X AM P L E 5 | Finding Intervals Where a Function Increases and Decreases (a) Sketch the graph of the function (b) Find the domain and range of the function. (c) Find the intervals on which f increases and decreases. x f 1 2 x2/3. 230 CHAPTER 3 | Functions ▼ SO LUTI O N (a) We use a graphing calculator to sketch the graph in Figure 7. (b) From the graph we observe that the domain of f is (c) From the graph we see that f is decreasing on q, 0 and the range is 0, q and increasing on 3. 2 0, q 3. 2 1 4 FIGURE 7 Graph of f x2/3 x 1 2 _20 10 _1 20 ✎ Practice what you’ve learned: Do Exercise 29. ▲ ■ Local Maximum and Minimum Values of a Function Finding the largest or smallest values of a function is important in many applications. For example, if a function represents revenue or profit, then we are interested in its maximum value. For a function that represents cost, we would want to find its minimum value. (See Focus on Modeling: Modeling with Functions on pages 280–289 for many such examples.) We can easily find these values from the graph of a function. We first define what we mean by a local maximum or minimum. LOCAL MAXIMUMS AND MINIMUMS OF A FUNCTION 1. The function value f 1 a is a local maximum value of f if f x when x is near a 2 x (This means that f 2 In this case we say that f has a local maximum at x a. a 1 2 1 2 1 for all x in some open interval containing a.) 2. The function value f 1 a is a local minimum of f if f x when x is near a 2 x (This means that f 2 In this case we say that f has a local minimum at x a. a 1 1 2 1 2 for all x in some open interval containing a.) Local maximum Local maximum f Local minimum |
x Local minimum We can find the local maximum and minimum values of a function using a graphing calculator. If there is a viewing rectangle such that the point the graph of f within the viewing rectangle (not on the edge), then the number f 1 f cal maximum value of f (see Figure 8 on the next page). Notice that f 1 numbers x that are close to a. 1 2 a 2 x 2 a, f a 1 22 1 is the highest point on is a lofor all a SE CTI O N 3. 3 | Getting Information from the Graph of a Function 231 y 0 FIGURE 8 Local maximum value f(a) Local minimum value f(b) a b x b, f Similarly, if there is a viewing rectangle such that the point 1 on the graph of f within the viewing rectangle, then the number f x value of f. In this case, f for all numbers x that are close to b. f b 1 b b 1 2 is the lowest point 22 is a local minimum 1 2 1 2 E X AM P L E 6 | Finding Local Maxima and Minima from a Graph 20 _5 5 _20 FIGURE 9 Graph of x 3 8x 1 f x 1 2 Find the local maximum and minimum values of the function f to three decimals. 1 x 2 x3 8x 1, correct ▼ SO LUTI O N The graph of f is shown in Figure 9. There appears to be one local maximum between x 2 and x 1, and one local minimum between x 1 and x 2. Let’s find the coordinates of the local maximum point first. We zoom in to enlarge the area near this point, as shown in Figure 10. Using the feature on the graphing device, we move the cursor along the curve and observe how the y-coordinates change. The local maximum value of y is 9.709, and this value occurs when x is 1.633, correct to three decimals. TRACE We locate the minimum value in a similar fashion. By zooming in to the viewing rectangle shown in Figure 11, we find that the local minimum value is about 7.709, and this value occurs when x 1.633. 9.71 _7.7 1.6 _1.7 _1.6 9.7 _7.71 FIGURE 10 ✎ Practice what you’ve learned: Do Exercise 35. FIG |
URE 11 1.7 ▲ The maximum and minimum commands on a TI-82 or TI-83 calculator provide another method for finding extreme values of functions. We use this method in the next example. E X AM P L E 7 | A Model for the Food Price Index A model for the food price index (the price of a representative “basket” of foods) between 1990 and 2000 is given by the function 0.0113t 3 0.0681t 2 0.198t 99.1 I t 1 2 where t is measured in years since midyear 1990, so 0 t 10, and I that I 3 1 1990–2000. is scaled so 100. Estimate the time when food was most expensive during the period t 2 1 2 232 CHAPTER 3 | Functions ▼ SO LUTI O N The graph of I as a function of t is shown in Figure 12(a). There appears to be a maximum between t 4 and t 7. Using the maximum command, as shown in Figure 12(b), we see that the maximum value of I is about 100.38, and it occurs when t 5.15, which corresponds to August 1995. 102 0 96 102 10 0 96 Maximum X=5.1514939 Y=100.38241 10 FIGURE 12 (a) (b) ✎ Practice what you’ve learned: Do Exercise 53. ▲ 3. ▼ CONCE PTS 1–4 ■ These exercises refer to the graph of the function f shown below. y 3 0 f 3 x 4. (a) A function value f a 1 is a local maximum value of f if f(a) is the ing a. From the graph of f we see that one local maximum 2 value of f on some interval contain- value of f is is. and that this value occurs when x (b) The function value f a 1 2 is a local minimum value of f if a is the f value of f on some interval containing a. From the graph of f we see that one local minimum 1 2 value of f is is. and that this value occurs when x 1. To find a function value f a from the graph of f, we find the 1 2 height of the graph above the x-axis at x. From the graph of f we see that f 3 1 2. 2. The domain of the function f is all the -values of the points on the |
graph, and the range is all the corresponding -values. From the graph of f we see that the domain of f is the interval and the range of f is the interval. 3. (a) If f is increasing on an interval, then the y-values of the as the x-values increase. points on the graph From the graph of f we see that f is increasing on the intervals and. (b) If f is decreasing on an interval, then y-values of the points on the graph graph of f we see that f is decreasing on the intervals as the x-values increase. From the and. ✎ ▼ SKI LLS 5. The graph of a function h is given. (a) Find b) Find the domain and range of h. (c) Find the values of x for which h (d) Find the values of x for which h, and. 3 _3 3 x 6. The graph of a function g is given at the top of the next page. 7). 1 2 1, g 2 (a) Find g 0), and g 1 (b) Find the domain and range of g. (c) Find the values of x for which g (d) Find the values of x for which g x) 4. 1 x) 4. 1 g y 4 0 4 x 7. The graph of a function g is given., g 2 2 (a) Find g 1 (b) Find the domain and range of g, and _3 3 x 8. Graphs of the functions f and g are given. (a) Which is larger, f (b) Which is larger, f (c) For which values of x is f 0 2 3 1 1 2 or g(0? 2 3 or _2 f _2 g 2 x 9–18 ■ A function f is given. (a) Use a graphing calculator to draw the graph of f. (b) Find the domain and range of f from the graph. 9. f 11. f 13. f 15. 17 216 x 2 1x 1 ✎ 10. f 12. f 14. f 16. 18(x 1) x 2 x 2 4 225 x 2 2x 2 19–22 ■ The graph of a function is given. Determine the intervals on which the function is (a) increasing and (b) decreasing. SE CTI O N 3. 3 | Getting Information from the Graph of a Function 233 21. y |
22. 1 0 1 x y 1 1 x 23–30 ■ A function f is given. (a) Use a graphing device to draw the graph of f. (b) State approximately the intervals on which f is increasing and on which f is decreasing. ✎ 23. f 25. f 27. f 28. f ✎ 29 5x 2x 3 3x 2 12x x 3 2x 2 x 2 24. f 26 4x x4 16x 2 x4 4x 3 2x 2 4x 3 x 2/5 30. f x 1 2 4 x 2/3 31–34 ■ The graph of a function is given. (a) Find all the local maximum and minimum values of the function and the value of x at which each occurs. (b) Find the intervals on which the function is increasing and on which the function is decreasing. 31. 32. y 1 0 1 x 33. y 34 35–42 ■ A function is given. (a) Find all the local maximum and minimum values of the function and the value of x at which each occurs. State each answer correct to two decimals. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimals. ✎ 35. f x 3 x x 1 2 36. f x 1 2 19. y 20 39. 37. g x 1 U 2 x 1 2 41. V x 1 2 x4 2x 3 11x 2 x16 x 1 x2 x3 3 x x 2 x 3 x5 8x 3 20x x2x x 2 38. g x 40. 1 U 2 x 1 2 42 234 CHAPTER 3 | Functions ▼ APPLICATIONS 43. Power Consumption The figure shows the power consump- tion in San Francisco for September 19, 1996 (P is measured in megawatts; t is measured in hours starting at midnight). (a) What was the power consumption at 6 A.M.? At 6 P.M.? (b) When was the power consumption the lowest? (c) When was the power consumption the highest? P (MW) 800 600 400 200 0 3 6 9 12 15 18 21 t (h) Source: Pacific Gas & Electric 44. Earthquake The graph shows the vertical acceleration of the ground from the 1994 Northridge earthquake in Los Angeles, as measured by a seismograph. (Here t represents |
the time in seconds.) (a) At what time t did the earthquake first make noticeable movements of the earth? (b) At what time t did the earthquake seem to end? (c) At what time t was the maximum intensity of the earth- quake reached? a (cm/s2) 100 50 −50 5 10 15 20 25 30 t (s) Source: Calif. Dept. of Mines and Geology ✎ 45. Weight Function The graph gives the weight W of a person at age x. (a) Determine the intervals on which the function W is in- creasing and on which it is decreasing. (b) What do you think happened when this person was 30 years old? Weight (pounds) W 200 150 100 50 (b) Describe in words what the graph indicates about his travels on this day. Distance from home (miles) 8 A.M. 10 NOON 2 4 6 P.M. Time (hours) 47. Changing Water Levels The graph shows the depth of water W in a reservoir over a one-year period as a function of the number of days x since the beginning of the year. (a) Determine the intervals on which the function W is in- creasing and on which it is decreasing. (b) At what value of x does W achieve a local maximum? A local minimum? W (ft) 100 75 50 25 0 100 200 300 x (days) 48. Population Growth and Decline The graph shows the population P in a small industrial city from 1950 to 2000. The variable x represents the number of years since 1950. (a) Determine the intervals on which the function P is increas- ing and on which it is decreasing. (b) What was the maximum population, and in what year was it attained? P (thousands) 50 40 30 20 10 0 10 20 30 40 50 x (years) 49. Hurdle Race Three runners compete in a 100-meter hurdle race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race? What do you think happened to runner B? y (m) 100 A B C 0 10 20 30 40 50 Age (years) 60 70 x 46. Distance Function The graph gives a sales representative’s distance from his home as a function of time on a |
certain day. (a) Determine the time intervals on which his distance from home was increasing and on which it was decreasing. 0 20 t (s) SE CTI O N 3. 3 | Getting Information from the Graph of a Function 235 50. Gravity Near the Moon We can use Newton’s Law of Gravitation to measure the gravitational attraction between the moon and an algebra student in a space ship located a distance x above the moon’s surface: F x 1 2 350 x 2 Here F is measured in newtons (N), and x is measured in millions of meters. (a) Graph the function F for values of x between 0 and 10. (b) Use the graph to describe the behavior of the gravitational attraction F as the distance x increases. 51. Radii of Stars Astronomers infer the radii of stars using the Stefan Boltzmann Law: E T 1 2 1 5.67 108 T 4 2 where E is the energy radiated per unit of surface area measured in watts (W) and T is the absolute temperature measured in kelvins (K). (a) Graph the function E for temperatures T between 100 K and 300 K. (b) Use the graph to describe the change in energy E as the temperature T increases. 52. Migrating Fish A fish swims at a speed √ relative to the water, against a current of 5 mi/h. Using a mathematical model of energy expenditure, it can be shown that the total energy E required to swim a distance of 10 mi is given by 2.73√ 3 E √ 1 2 10 √ 5 Biologists believe that migrating fish try to minimize the total energy required to swim a fixed distance. Find the value of that minimizes energy required. √ NOTE This result has been verified; migrating fish swim against a current at a speed 50% greater than the speed of the current. ✎ 53. Highway Engineering A highway engineer wants to esti- mate the maximum number of cars that can safely travel a particular highway at a given speed. She assumes that each car is 17 ft long, travels at a speed s, and follows the car in front of it at the “safe following distance” for that speed. She finds that the number N of cars that can pass a given point per minute is modeled by the function N s 1 2 88s 17 17 2 s |
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