Chapter 1 Section 3 Linear Functions Show that the vertex of the parabola y Ax 2 Bx C (A 0) occurs at the

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1 Chapter 1 Section 3 Linear Functions Show that the verte of the parabola A 2 B C (A 0) occurs at the B point where. [Hint: First verif that 2A Then note that the largest or smallest value of B must occur where. 2A A 2 B C A 2A B 2 C B2 A 4A 2 f() A 2 B C Verte (high point) B 2A Verte (low point) B 2A 3 Linear Functions 42. Graph f() Determine the values of for which the function is defined. 43. Graph f() Determine the values of for which the function is 2 1 defined. In man practical situations, the rate at which one quantit changes with respect to another is constant. Here is a simple eample from economics. EXAMPLE 3.1 A manufacturer s total cost consists of a fied overhead of $200 plus production costs of $50 per unit. Epress the total cost as a function of the number of units produced and draw the graph.

2 30 Chapter 1 Functions, Graphs, and Limits Eplore! Input the cost function, Y1 50 {200, 300, 400} into the equation editor of our graphing calculator. Use the WINDOW menu to set the viewing window to [1, 5]1 b [ 100, 700]100. Eplain the effect of varing the overhead value. Solution Let denote the number of units produced and C() the corresponding total cost. Then, where Hence, Total cost = (cost per unit)(number of units) + overhead Cost per unit 50 Number of units Overhead 200 C() The graph of this cost function is sketched in Figure C() (2,300) (3,350) (0, 200) FIGURE 1.15 The cost function C() The total cost in Eample 3.1 increases at a constant rate of $50 per unit. As a result, its graph in Figure 1.15 is a straight line that increases in height b 50 units for each 1-unit increase in. In general, a function whose value changes at a constant rate with respect to its independent variable is said to be a linear function. This is because the graph of such a function is a straight line. In algebraic terms, a linear function is a function of the form f() a 0 a 1

3 Chapter 1 Section 3 Linear Functions 31 where a 0 and a 1 are constants. For eample, the functions f() 3 2, f() 5, 2 and f() 12 are all linear. Linear functions are traditionall written in the form m b where m and b are constants. This standard notation will be used in the discussion that follows. Linear Functions A linear function is a function that changes at a constant rate with respect to its independent variable. The graph of a linear function is a straight line. The equation of a linear function can be written in the form m b where m and b are constants. THE SLOPE OF A LINE A surveor might sa that a hill with a rise of 2 feet for ever foot of run has a slope of m rise run The steepness of a line can be measured b slope in much the same wa. In particular, suppose ( 1, 1 ) and ( 2, 2 ) lie on a line as indicated in Figure Between these points, changes b the amount 2 1 and b the amount 2 1. The slope is the ratio change in Slope change in It is sometimes convenient to use the smbol instead of 2 1 to denote the change in. The smbol is read delta. Similarl, the smbol is used to denote the change 2 1. The Slope of a Line The slope of the nonvertical line passing through the points ( 1, 1 ) and ( 2, 2 ) is given b the formula Slope

4 32 Chapter 1 Functions, Graphs, and Limits ( 2, 2 ) ( 1, 1 ) 2 1 = (Run) 2 1 = (Rise) FIGURE 1.16 Slope The use of this formula is illustrated in the following eample. Eplore! Construct a famil of straight lines through the origin b storing Y1 L1*X in the equation editor and entering the values 2, 1,.5,.5, 1, 2 into L1 using the STAT menu and the Edit option. Graph using the Decimal Window and trace the values of the different lines for 1. Find the slope of the line joining the points ( 2, 5) and (3, 1). Solution EXAMPLE 3.2 The situation is illustrated in Figure Slope ( 2) 5 ( 2, 5) = 1 5 = 6 = 3 ( 2) = 5 (3, 1) FIGURE 1.17 The line joining ( 2, 5) and (3, 1).

5 Chapter 1 Section 3 Linear Functions 33 The sign and magnitude of the slope of a line indicate the line s direction and steepness, respectivel. The slope is positive if the height of the line increases as increases and is negative if the height decreases as increases. The absolute value of the slope is large if the slant of the line is severe and small if the slant of the line is gradual. The situation is illustrated in Figure m = 2 m = 1 m = 1 2 m = 1 2 m = 1 m = 2 FIGURE 1.18 The direction and steepness of a line. HORIZONTAL AND VERTICAL LINES Horizontal and vertical lines (Figures 1.19a and 1.19b) have particularl simple equations. The coordinates of all of the points on a horizontal line are the same. Hence, a horizontal line is the graph of a linear function of the form b, where b is a constant. The slope of a horizontal line is zero, since changes in produce no changes in. (0, b) = b = c (c, 0) (a) (b) FIGURE 1.19 Horizontal and vertical lines.

6 34 Chapter 1 Functions, Graphs, and Limits The coordinates of all the points on a vertical line are equal. Hence, vertical lines are characterized b equations of the form c, where c is a constant. The slope of a vertical line is undefined. This is because onl the coordinates of points change in on a vertical line can change, and so the denominator of the quotient is change in zero. THE SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE (0, b) FIGURE 1.20 The slope and intercept of the line m b. 1 m The constants m and b in the equation m b of a nonvertical line have geometric interpretations. The coefficient m is the slope of the line. To see this, suppose that ( 1, 1 ) and ( 2, 2 ) are two points on the line m b. Then, 1 m 1 b and 2 m 2 b, and so Slope 2 1 (m 2 b) (m 1 b) m 2 m 1 m( 2 1 ) m The constant b in the equation m b is the value of corresponding to 0. Hence, b is the height at which the line m b crosses the ais, and the corresponding point (0, b) is the intercept of the line. The situation is illustrated in Figure Because the constants m and b in the equation m b correspond to the slope and intercept, respectivel, this form of the equation of a line is known as the slope-intercept form. Eplore! The Slope-Intercept Form of the Equation of a Line The equation = m + b is the equation of the line whose slope is m and whose intercept is (0, b). Store Y1 L1*X 1 in the equation editor, and determine the slopes needed to create the screen shown. The slope-intercept form of the equation of a line is particularl useful when geometric information about a line (such as its slope or intercept) is to be determined from the line s algebraic representation. Here is a tpical eample. EXAMPLE 3.3 Find the slope and intercept of the line and draw the graph.

7 Chapter 1 Section 3 Linear Functions 35 Solution First put the equation in slope-intercept form m b. To do this, solve for to get (0, 2) or FIGURE 1.21 The line (3, 0) It follows that the slope is 2 and the intercept is (0, 2). 3 To graph a linear function, plot two of its points and draw a straight line through them. In this case, ou alread know one point, the intercept (0, 2). A convenient choice for the coordinate of the second point is 3, since the corresponding coordinate is 2 (3) 2 0. Draw a line through the points (0, 2) and (3, 0) 3 to obtain the graph shown in Figure THE POINT-SLOPE FORM OF THE EQUATION OF A LINE Geometric information about a line can be obtained readil from the slope-intercept formula m b. There is another form of the equation of a line, however, that is usuall more efficient for problems in which the geometric properties of a line are known and the goal is to find the equation of the line. Eplore! Write Y1.5X L1 in the equation editor and determine the values of the list L1 needed to create the screen shown. The Point-Slope Form of the Equation of a Line The equation 0 m( 0 ) is an equation of the line that passes through the point ( 0, 0 ) and that has slope equal to m. The point-slope form of the equation of a line is simpl the formula for slope in disguise. To see this, suppose the point (, ) lies on the line that passes through a given point ( 0, 0 ) and that has slope m. Using the points (, ) and ( 0, 0 ) to compute the slope, ou get 0 m 0 which ou can put in point-slope form 0 m( 0 ) b simpl multipling both sides b 0. The use of the point-slope form of the equation of a line is illustrated in the net two eamples.

8 36 Chapter 1 Functions, Graphs, and Limits (5, 1) (0, 23 /2) FIGURE 1.22 The line EXAMPLE 3.4 Find the equation of the line that passes through the point (5, 1) and whose slope is 1 equal to. 2 Solution 1 Use the formula 0 m( 0 ) with ( 0, 0 ) (5, 1) and m 2 which ou can rewrite as 1 1 ( 5) 2 to get The graph is shown in Figure Instead of the point-slope form, the slope-intercept form could have been used to solve the problem in Eample 3.4. For practice, solve the problem this wa. Notice that the solution based on the point-slope formula is more efficient. The net eample illustrates how ou can use the point-slope form to find an equation of a line that passes through two given points. 10 (1, 6) EXAMPLE 3.5 Find the equation of the line that passes through the points (3, 2) and (1, 6). (3, 2) FIGURE 1.23 The line Solution First compute the slope m 6 ( 2) 1 3 Then use the point-slope formula with (1, 6) as the given point ( 0, 0 ) to get 6 4( 1) Convince ourself that the resulting equation would have been the same if ou had chosen (3, 2) to be the given point ( 0, 0 ). The graph is shown in Figure or 4 10

9 Chapter 1 Section 3 Linear Functions 37 PRACTICAL APPLICATIONS If the rate of change of one quantit with respect to a second quantit is constant, the function relating the quantities must be linear. The constant rate of change is the slope of the corresponding line. The net two eamples illustrate techniques ou can use to find the appropriate linear functions in such situations. Since the beginning of the ear, the price of whole wheat bread at a local discount supermarket has been rising at a constant rate of 2 cents per month. B November first, the price had reached $1.56 per loaf. Epress the price of the bread as a function of time and determine the price at the beginning of the ear. Solution Let denote the number of months that have elapsed since the first of the ear and the price of a loaf of bread (in cents). Since changes at a constant rate with respect to, the function relating to must be linear, and its graph is a straight line. Since the price increases b 2 each time increases b 1, the slope of the line must be 2. The fact that the price was 156 cents ($1.56) on November first, 10 months after the first of the ear, implies that the line passes through the point (10, 156). To write an equation defining as a function of, use the point-slope formula with to get EXAMPLE ( 10) 0 m( 0 ) m 2, 0 10, The corresponding line is shown in Figure Notice that the intercept is (0, 136), which implies that the price of bread at the beginning of the ear was $1.36 per loaf. or (10, 156) (0, 136) 10 (Jan. 1) (Nov. 1) FIGURE 1.24 The rising price of bread:

10 38 Chapter 1 Functions, Graphs, and Limits Sometimes it is hard to tell how two quantities, and, in a data set are related b simpl eamining the data. In such cases, it ma be useful to graph the data to see if the points (, ) follow a clear pattern (sa, lie along a line). Here is an eample of this procedure. Eplore! Place the data in Table 1.2 into L1 and L2, where L1 is the ear beond 1900 and L2 is the number of federal civilian emploees (in millions). Set the graph tpe to Scatterplot b accessing the STAT PLOT menu, and verif that the line fits the data well, as claimed in the note on page 39. Table 1.2 lists federal emploment information for the United States in the period Plot a graph with federal emploment on the ais and time (measured from 1950) on the ais. Do the points follow a clear pattern? Based on this data, what do ou think the federal emploment will be in the ear 2000? TABLE 1.2 Emploment Information for Year EXAMPLE 3.7 Federal Civilian Emploees (Millions) Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 1990, p Solution The graph is shown in Figure Note that it is roughl linear. Does this mean federal emploment is linearl related to time? Not reall, but it does suggest that we ma be able to get useful information b using a line to approimate the data. Such a line is shown in Figure Note that b etending the line to the right, we might guess that federal emploment will reach approimatel 3.4 million in the ear 2000.

11 Chapter 1 Section 3 Linear Functions 39 Millions of emploees Years past FIGURE 1.25 Growth of federal civilian emploment in the United States ( ). Note It is often useful to find a line that best approimates data in some meaningful wa. A procedure for doing this, called least-squares approimation, is developed in Chapter 7. When least-squares approimation is applied to the data in this eample, it ields the line , so in the ear 2000 (when 50), we ma epect the federal civilian emploment to be 0.027(50) million. PARALLEL AND PERPENDICULAR LINES In applications, it is sometimes necessar or useful to know whether two given lines are parallel or perpendicular. A vertical line is parallel onl to other vertical lines and is perpendicular to an horizontal line. Cases involving nonvertical lines can be handled b the following criteria. Parallel and Perpendicular Lines Let m 1 and m 2 be the slopes of the nonvertical lines L 1 and L 2. Then L 1 and L 2 are parallel if and onl if m 1 m 2 L 1 and L 2 are perpendicular if and onl if m 2 1 m 1 These criteria are illustrated in Figure Geometric proofs are outlined in Eercises 54 and 55. We close this section with an eample illustrating one wa the criteria can be used.

12 40 Chapter 1 Functions, Graphs, and Limits L 1 L 2 L 1 L 2 (a) Parallel lines have m 1 m 2 (b) Perpendicular lines have m 2 1/m 1 FIGURE 1.26 Eplore! Store f() A* 2 and g() ( 1/A)* 5 into the equation editor of our graphing calculator. On the home screen, store different values into A, and then graph the functions using a square viewing window. What do ou notice for different A values? Can ou predict where the point of intersection will be? Let L be the line (a) Find the equation of a line L 1 parallel to L through P( 1, 4). (b) Find the equation of a line L 2 perpendicular to L through Q(2, 3). Solution EXAMPLE 3.8 B rewriting the equation in the slope-intercept form 4, we 3 2 see that L has slope m L 4. 3 (a) An line parallel to L must also have slope m 4. The required line L 1 contains 3 P( 1, 4), so 4 4 ( 1) (b) A line perpendicular to L must have slope m 1/m L line L 2 contains Q(2, 3), we have 3. Since the required ( 2)

13 Chapter 1 Section 3 Linear Functions 41 The given line L and the required lines L 1 and L 2 are shown in Figure L 1 L L 2 P( 1, 4) Q(2, 3) FIGURE 1.27 Lines parallel and perpendicular to a given line L. P. R. O. B. L. E. M. S 1.3 P. R. O. B. L. E. M. S 1.3 In Problems 1 through 6, find the slope (if possible) of the line that passes through the given pair of points. 1. (2, 3) and (0, 4) 2. ( 1, 2) and (2, 5) 3. (2, 0) and (0, 2) 4. (5, 1) and ( 2, 1) 5. (2, 6) and (2, 4) 6. and In Problems 7 through 18, find the slope and intercepts of the given line and draw a graph , , 1 8 1

14 42 Chapter 1 Functions, Graphs, and Limits In Problems 19 through 34, write an equation for the line with the given properties. 19. Through (2, 0) with slope Through ( 1, 2) with slope 21. Through (5, 2) with slope Through (0, 0) with slope MANUFACTURING COST CAR RENTAL COURSE REGISTRATION MEMBERSHIP FEES 23. Through (2, 5) and parallel to the 24. Through (2, 5) and parallel to the ais ais 25. Through (1, 0) and (0, 1) 26. Through (2, 5) and (1, 2) 1 5, 1 2 3, Through and 28. Through ( 2, 3) and (0, 5) 29. Through (1, 5) and (3, 5) 30. Through (1, 5) and (1, 4) 31. Through (4, 1) and parallel to the 32. Through ( 2, 3) and parallel to the line 2 3 line , Through (3, 5) and perpendicular 34. Through and perpendicular to the line 4 to the line = A manufacturer s total cost consists of a fied overhead of $5,000 plus production costs of $60 per unit. Epress the total cost as a function of the number of units produced and draw the graph. 36. A certain car rental agenc charges $35 per da plus 55 cents per mile. (a) Epress the cost of renting a car from this agenc for 1 da as a function of the number of miles driven and draw the graph. (b) How much does it cost to rent a car for a 1-da trip of 50 miles? (c) How man miles were driven if the dail rental cost was $72? 37. Students at a state college ma preregister for their fall classes b mail during the summer. Those who do not preregister must register in person in September. The registrar can process 35 students per hour during the September registration period. Suppose that after 4 hours in September, a total of 360 students (including those who preregistered) have been registered. (a) Epress the number of students registered as a function of time and draw the graph. (b) How man students were registered after 3 hours? (c) How man students preregistered during the summer? 38. Membership in a swimming club costs $250 for the 12-week summer season. If a member joins after the start of the season, the fee is prorated; that is, it is reduced linearl. (a) Epress the membership fee as a function of the number of weeks that have elapsed b the time the membership is purchased and draw the graph.

15 Chapter 1 Section 3 Linear Functions 43 (b) Compute the cost of a membership that is purchased 5 weeks after the start of the season. LINEAR DEPRECIATION LINEAR DEPRECIATION WATER CONSUMPTION CAR POOLING TEMPERATURE CONVERSION COLLEGE ADMISSIONS 39. A doctor owns $1,500 worth of medical books which, for ta purposes, are assumed to depreciate linearl to zero over a 10-ear period. That is, the value of the books decreases at a constant rate so that it is equal to zero at the end of 10 ears. Epress the value of the books as a function of time and draw the graph. 40. A manufacturer bus $20,000 worth of machiner that depreciates linearl so that its trade-in value after 10 ears will be $1,000. (a) Epress the value of the machiner as a function of its age and draw the graph. (b) Compute the value of the machiner after 4 ears. (c) When does the machiner become worthless? The manufacturer might not wait this long to dispose of the machiner. Discuss the issues the manufacturer ma consider in deciding when to sell. 41. Since the beginning of the month, a local reservoir has been losing water at a constant rate. On the 12th of the month the reservoir held 200 million gallons of water, and on the 21st it held onl 164 million gallons. (a) Epress the amount of water in the reservoir as a function of time and draw the graph. (b) How much water was in the reservoir on the 8th of the month? 42. To encourage motorists to form car pools, the transit authorit in a certain metropolitan area has been offering a special reduced rate at toll bridges for vehicles containing four or more persons. When the program began 30 das ago, 157 vehicles qualified for the reduced rate during the morning rush hour. Since then, the number of vehicles qualifing has been increasing at a constant rate, and toda 247 vehicles qualified. (a) Epress the number of vehicles qualifing each morning for the reduced rate as a function of time and draw the graph. (b) If the trend continues, how man vehicles will qualif during the morning rush hour 14 das from now? 43. (a) Temperature measured in degrees Fahrenheit is a linear function of temperature measured in degrees Celsius. Use the fact that 0 Celsius is equal to 32 Fahrenheit and 100 Celsius is equal to 212 Fahrenheit to write an equation for this linear function. (b) Use the function ou obtained in part (a) to convert 15 Celsius to Fahrenheit. (c) Convert 68 Fahrenheit to Celsius. 44. The average scores of incoming students at an eastern liberal arts college in the SAT mathematics eamination have been declining at a constant rate in recent ears. In 1990, the average SAT score was 575, while in 1995 it was 545. (a) Epress the average SAT score as a function of time. (b) If the trend continues, what will the average SAT score of incoming students be in 2000? (c) If the trend continues, when will the average SAT score be 527?

16 44 Chapter 1 Functions, Graphs, and Limits APPRECIATION OF ASSETS AIR POLLUTION NUTRITION ALCOHOL ABUSE CONTROL 45. The value of a certain rare book doubles ever 10 ears. In 1900, the book was worth $100. (a) How much was it worth in 1930? In 1990? What about the ear 2000? (b) Is the value of the book a linear function of its age? Answer this question b interpreting an appropriate graph. 46. In certain parts of the world, the number of deaths N per week have been observed to be linearl related to the average concentration of sulfur dioide in the air. Suppose there are 97 deaths when 100 mg/m 3 and 110 deaths when 500 mg/m 3. (a) What is the functional relationship between N and? (b) Use the function in part (a) to find the number of deaths per week when 300 mg/m 3. What concentration of sulfur dioide corresponds to 100 deaths per week? (c) Research data on how air pollution affects the death rate in a population.* Summarize our results in a one-paragraph essa. 47. Each ounce of Food I contains 3 gm of carbohdrate and 2 gm of protein, and each ounce of Food II contains 5 gm of carbohdrate and 3 gm of protein. Suppose ounces of Food I are mied with ounces of Food II. The foods are combined to produce a blend that contains eactl 73 gm of carbohdrate and 46 gm of protein. (a) Eplain wh there are 3 5 gm of carbohdrate in the blend and wh we must have Find a similar equation for protein. Sketch the graphs of both equations. (b) Where do the two graphs in part (a) intersect? Interpret the significance of this point of intersection. 48. Ethl alcohol is metabolized b the human bod at a constant rate (independent of concentration). Suppose the rate is 10 ml per hour. (a) How much time is required to eliminate the effects of a liter of beer containing 3% ethl alcohol? (b) Epress the time T required to metabolize the effects of drinking ethl alcohol as a function of the amount A of ethl alcohol consumed. (c) Discuss how the function in part (b) can be used to determine a reasonable cutoff value for the amount of ethl alcohol A that each individual ma be served at a part. 49. Graph and on the same set of coordinate aes using [ 10, 10]1 b [ 10, 10]1. Are the two lines parallel? * You ma find the following articles helpful: D. W. Docker, J. Schwartz, and J. D. Spengler, Air Pollution and Dail Mortalit: Associations with Particulates and Acid Aerosols, Environ. Res., Vol. 59, 1992, pp ; Y. S. Kim, Air Pollution, Climate, Socioeconomics Status and Total Mortalit in the United States, Sci. Total Environ., Vol. 42, 1985, pp

17 Chapter 1 Section 3 Linear Functions Graph and on the same set of coordinate aes using [ 10, 10]1 b [ 10, 10]1 for a starting range. Adjust the range settings until both lines are displaed. Are the two lines parallel? 51. A rental compan rents a piece of equipment for a $60.00 flat fee plus an hourl fee of $5.00 per hour. (a) Make a chart showing the number of hours the equipment is rented and the cost for renting the equipment for 2 hours, 5 hours, 10 hours, and t hours of time. (b) Write an algebraic epression representing the cost as a function of the number of hours t. Assume t can be measured to an decimal portion of an hour. (In other words, assume t is an nonnegative real number.) (c) Graph the epression from part (b). (d) Use the graph to approimate, to two decimal places, the number of hours the equipment was rented if the bill is $ (before taes). ASTRONOMY 52. The following table gives the length of ear L (in earth ears) of each planet in the solar sstem along with the mean (average) distance D of the planet from the sun, in astronomical units (1 astronomical unit is the mean distance of the earth from the sun). (a) Plot the point (D, L) for each planet on a coordinate plane. Do these quantities appear to be linearl related? (b) For each planet, compute the ratio L 2 D 3. Interpret what ou find b epressing L as a function of D. (c) What ou have discovered in part (b) is one of Kepler s laws, named for the German astronomer Johannes Kepler ( ). Read an article about Kepler and describe his place in the histor of science. Mean Distance Length of Planet from Sun, D Year, L Mercur Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Source: Kendrick Frazier, The Solar Sstem, Aleandria, VA: Time/Life Books, p. 37.

18 46 Chapter 1 Functions, Graphs, and Limits EMPLOYMENT RATES PARALLEL LINES PERPENDICULAR LINES 53. In the note after Eample 3.7, we observed that the line that best approimates the data in the eample has the equation Interpret the slope of this line in terms of the rate of growth of federal emploment. Would ou epect the rate of growth of private emploment to be larger or smaller than the federal growth rate? Write a paragraph on the relationship between federal and private emploment. 54. Show that two nonvertical lines are parallel if and onl if the have the same slope. 55. Show that if a nonvertical line L 1 with slope m 1 is perpendicular to a line L 2 with slope m 2, then m 2 1/m 1. [Hint: Find epressions for the slopes of the lines L 1 and L 2 in the accompaning figure. Then appl the Pthagorean theorem along with the distance formula from Problem 39, Section 2 of this chapter, to the right triangle OAB to obtain the required relationship between m 1 and m 2.] L 1 A (a, b) 0 B (a, c) 4 Functional Models ELIMINATION OF VARIABLES A mathematical representation of a practical situation is called a mathematical model. In preceding sections, ou saw models representing such quantities as manufacturing cost, air pollution levels, population size, suppl, and demand. In this section, ou will see eamples illustrating some of the techniques ou can use to build mathematical models of our own. In the first two eamples, the quantit ou are seeking is epressed most naturall in terms of two variables. You will have to eliminate one of these variables before ou can write the quantit as a function of a single variable. L 2 EXAMPLE 4.1 The highwa department is planning to build a picnic area for motorists along a major highwa. It is to be rectangular with an area of 5,000 square ards and is to be fenced off on the three sides not adjacent to the highwa. Epress the number of ards of fencing required as a function of the length of the unfenced side.