1.3 LINES IN THE PLANE AND SLOPE

Transcription

1 000_00.qd //0 : AM Page CHAPTER Functions, Graphs, and Limits. LINES IN THE PLANE AND SLOPE Use the slope-intercept form of a linear equation to sketch graphs. Find slopes of lines passing through two points. Use the point-slope form to write equations of lines. Find equations of parallel and perpendicular lines. Use linear equations to model and solve real-life problems. TECHNOLOGY On most graphing utilities, the displa screen is twothirds as high as it is wide. On such screens, ou can obtain a graph with a true geometric perspective b using a square setting one in which Y ma Y min X ma X min. One such setting is shown below. Notice that the and tick marks are equall spaced on a square setting, but not on a standard setting. Using Slope The simplest mathematical model for relating two variables is the linear equation m b. The equation is called linear because its graph is a line. (In this tet, the term line is used to mean straight line.) B letting 0, ou can see that the line crosses the -ais at b, as shown in Figure.. In other words, the -intercept is 0, b. The steepness or slope of the line is m. m b Slope The slope of a line is the number of units the line rises (or falls) verticall for each unit of horizontal change from left to right, as shown in Figure.. -intercept (0, b) -intercept = m + b unit m units, m > 0 (0, b) -intercept = m + b unit m units, m < 0 FIGURE. Positive slope, line rises. Negative slope, line falls. DISCOVERY Use a graphing utilit to compare the slopes of the lines m, where m 0.,,, and. Which line rises most quickl? Now, let m 0.,,, and. Which line falls most quickl? Let m 0.0, 0.00, and What is the slope of a horizontal line? Use a square setting to obtain a true geometric perspective. A linear equation that is written in the form m b written in slope-intercept form. The Slope-Intercept Form of the Equation of a Line The graph of the equation m b is a line whose slope is m and whose -intercept is 0, b. is said to be

2 000_00.qd //0 : AM Page SECTION. Lines in the Plane and Slope Once ou have determined the slope and the -intercept of a line, it is a relativel simple matter to sketch its graph. In the following eample, note that none of the lines is vertical. A vertical line has an equation of the form a. Vertical line Because such an equation cannot be written in the form m b, it follows that the slope of a vertical line is undefined, as indicated in Figure.. EXAMPLE Graphing a Linear Equation Sketch the graph of each linear equation. (a) (b) (c) = FIGURE. When the line is vertical, the slope is undefined. SOLUTION (a) Because b, the -intercept is 0,. Moreover, because the slope is m, the line rises two units for each unit the line moves to the right, as shown in Figure.(a). (b) B writing this equation in the form 0, ou can see that the -intercept is 0, and the slope is zero. A zero slope implies that the line is horizontal that is, it doesn t rise or fall, as shown in Figure.(b). (c) B writing this equation in slope-intercept form Write original equation. Subtract from each side. Write in slope-intercept form. ou can see that the -intercept is 0,. Moreover, because the slope is m, this line falls one unit for each unit the line moves to the right, as shown in Figure.(c). TRY IT Sketch the graph of each linear equation. (a) (b) = + = = + (0, ) m = (0, ) m = 0 (0, ) m = (a) When m is positive, the line rises. FIGURE. (b) When m is zero, the line is horizontal. (c) When m is negative, the line falls.

3 000_00.qd //0 : AM Page CHAPTER Functions, Graphs, and Limits In real-life problems, the slope of a line can be interpreted as either a ratio or a rate. If the -ais and -ais have the same unit of measure, then the slope has no units and is a ratio. If the -ais and -ais have different units of measure, then the slope is a rate or rate of change. EXAMPLE Using Slope as a Ratio The maimum recommended slope of a wheelchair ramp is A business is installing a wheelchair ramp that rises inches over a horizontal length of feet, as shown in Figure.. Is the ramp steeper than recommended? (Source: American Disabilities Act Handbook) SOLUTION The horizontal length of the ramp is feet or 88 inches. So, the slope of the ramp is Slope vertical change horizontal change in. 88 in So, the slope is not steeper than recommended. TRY IT If the ramp in Eample rises inches over a horizontal length of feet, is it steeper than recommended? in. FIGURE. ft Cost (in dollars) 0,000 9,000 8,000,000,000,000,000,000,000,000 C FIGURE. Production Cost Marginal cost: m = $ C = + 00 Fied cost: $ Number of units EXAMPLE Using Slope as a Rate of Change A manufacturing compan determines that the total cost in dollars of producing units of a product is C 00. Describe the practical significance of the -intercept and slope of the line given b this equation. SOLUTION The -intercept 0, 00 tells ou that the cost of producing zero units is $00. This is the fied cost of production it includes costs that must be paid regardless of the number of units produced. The slope of m tells ou that the cost of producing each unit is $, as shown in Figure.. Economists call the cost per unit the marginal cost. If the production increases b one unit, then the margin or etra amount of cost is $. TRY IT A small business purchases a copier and determines that the value of the copier t ears after its purchase is V t 8. Describe the practical significance of the -intercept and slope of the line given b this equation.

4 000_00.qd //0 : AM Page SECTION. Lines in the Plane and Slope Finding the Slope of a Line Given an equation of a nonvertical line, ou can find its slope b writing the equation in slope-intercept form. If ou are not given an equation, ou can still find the slope of a line. For instance, suppose ou want to find the slope of the line passing through the points, and,, as shown in Figure.. As ou move from left to right along this line, a change of units in the vertical direction corresponds to a change of units in the horizontal direction. These two changes are denoted b the smbols (, ) (, ) the change in and the change in. FIGURE. (The smbol is the Greek capital letter delta, and the smbols and are read as delta and delta. ) The ratio of to represents the slope of the line that passes through the points, and,. Slope Be sure ou see that represents a single number, not the product of two numbers ( and ). The same is true for. The Slope of a Line Passing Through Two Points The slope m of the line passing through, and, is m where. When this formula is used for slope, the order of subtraction is important. Given two points on a line, ou are free to label either one of them as, and the other as,. However, once ou have done this, ou must form the numerator and denominator using the same order of subtraction. m m m Correct Correct Incorrect For instance, the slope of the line passing through the points, and, can be calculated as m or m.

5 000_00.qd //0 : AM Page 8 8 CHAPTER Functions, Graphs, and Limits DISCOVERY The line in Eample (b) is a horizontal line. Find an equation for this line. The line in Eample (d) is a vertical line. Find an equation for this line. EXAMPLE Finding the Slope of a Line Find the slope of the line passing through each pair of points. (a), 0 and, (b), and, (c) 0, and, (d), and, SOLUTION (a) Letting,, 0 and,,, ou obtain a slope of m 0 Difference in -values Difference in -values as shown in Figure.(a). (b) The slope of the line passing through, and, is m See Figure.(b) (c) The slope of the line passing through 0, and, is m 0. See Figure.(c). (d) The slope of the vertical line passing through, and, is not defined because division b zero is undefined. [See Figure.(d).] (, 0) m = (, ) (, ) m = 0 (, ) (a) Positive slope; line rises. (b) Zero slope; line is horizontal. TRY IT Find the slope of the line passing through each pair of points. (a), and, 8 (b), and, (0, ) m = (, ) (c) Negative slope; line falls. FIGURE. m is undefined. (, ) (, ) (d) Vertical line; undefined slope.

6 000_00.qd //0 : AM Page 9 Writing Linear Equations If, is a point ling on a nonvertical line of slope m and, is an other point on the line, then m. This equation, involving the variables and, can be rewritten in the form m, which is the point-slope form of the equation of a line. SECTION. Lines in the Plane and Slope 9 Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through the point, is m. The point-slope form is most useful for finding the equation of a nonvertical line. You should remember this formula it is used throughout the tet. EXAMPLE Using the Point-Slope Form Find the equation of the line that has a slope of and passes through the point,. SOLUTION Use the point-slope form with m and,,. = (, ) = m Point-slope form Substitute for m,, and. Simplif. Write in slope-intercept form. The slope-intercept form of the equation of the line is. The graph of this line is shown in Figure.8. FIGURE.8 = TRY IT Find the equation of the line that has a slope of and passes through the point,. The point-slope form can be used to find an equation of the line passing through points, and,. To do this, first find the slope of the line m, and then use the point-slope form to obtain the equation. Two-point form STUDY TIP The two-point form of a line is similar to the slope-intercept form. What is the slope of a line given in two-point form? This is sometimes called the two-point form of the equation of a line.

7 000_00.qd //0 : AM Page 0 0 CHAPTER Functions, Graphs, and Limits Cash flow per share (in dollars) Cash Flow. = 0.t +.8 FIGURE.9 (,.9) (,.) (0,.8) Year (0 00) t EXAMPLE Predicting Cash Flow Per Share The cash flow per share for Rub Tuesda, Inc. was $.8 in 00 and $. in 00. Using onl this information, write a linear equation that gives the cash flow per share in terms of the ear. Then predict the cash flow for 00. (Source: Rub Tuesda, Inc.) SOLUTION Let t 0 represent 00. Then the two given values are represented b the data points 0,.8 and,.. The slope of the line through these points is m Using the point-slope form, ou can find the equation that relates the cash flow and the ear t to be 0.t.8. According to this equation, the cash flow in 00 was $.9, as shown in Figure.9. (In this case, the prediction is fairl good the actual cash flow in 00 was $.09.) Given points Etrapolated point TRY IT The sales per share for Cloro Compan were $.9 in 00 and $8. in 00. Write a linear equation that gives the sales per share in terms of the ear. Let t 0 represent 00. Then predict the sales per share for 00. (Source: Cloro Compan) (a) Linear Etrapolation Given points The prediction method illustrated in Eample is called linear etrapolation. Note in Figure.0(a) that an etrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in Figure.0(b), the procedure is called linear interpolation. Because the slope of a vertical line is not defined, its equation cannot be written in slope-intercept form. However, ever line has an equation that can be written in the general form A B C 0 General form Interpolated point (b) Linear Interpolation FIGURE.0 where A and B are not both zero. For instance, the vertical line given b a can be represented b the general form a 0. The five most common forms of equations of lines are summarized below. Equations of Lines. General form: A B C 0. Vertical line: a. Horizontal line: b. Slope-intercept form: m b. Point-slope form: m

8 000_00.qd //0 : AM Page SECTION. Lines in the Plane and Slope Parallel and Perpendicular Lines Slope can be used to decide whether two nonvertical lines in a plane are parallel, perpendicular, or neither. Parallel and Perpendicular Lines. Two distinct nonvertical lines are parallel if and onl if their slopes are equal. That is,. Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other. That is, m m. EXAMPLE Finding Parallel and Perpendicular Lines Find equations of the lines that pass through the point, and are (a) parallel to the line. (b) perpendicular to the line. SOLUTION B writing the given equation in slope-intercept form Write original equation. Subtract from each side. Write in slope-intercept form. ou can see that it has a slope of m, as shown in Figure.. (a) An line parallel to the given line must also have a slope of. So, the line through, that is parallel to the given line has the following equation. Write in point-slope form. Multipl each side b. Distributive Propert 0 Write in general form. Write in slope-intercept form. (b) An line perpendicular to the given line must have a slope of m m. or. So, the line through, that is perpendicular to the given line has the following equation. 0 Write in point-slope form. Multipl each side b. Distributive Propert Write in general form. Write in slope-intercept form. TECHNOLOGY On a graphing utilit, lines will not appear to have the correct slope unless ou use a viewing window that has a square setting. For instance, tr graphing the lines in Eample using the standard setting 0 0 and 0 0. Then reset the viewing window with the square setting 9 9 and. On which setting do the lines and appear to be perpendicular? = + FIGURE. TRY IT = (, ) = Find equations of the lines that pass through the point, and are (a) parallel to the line. (b) perpendicular to the line.

9 000_00.qd //0 : AM Page Enter CHAPTER Functions, Graphs, and Limits Etended Application: Linear Depreciation Most business epenses can be deducted the same ear the occur. One eception to this is the cost of propert that has a useful life of more than ear, such as buildings, cars, or equipment. Such costs must be depreciated over the useful life of the propert. If the same amount is depreciated each ear, the procedure is called linear depreciation or straight-line depreciation. The book value is the difference between the original value and the total amount of depreciation accumulated to date. Nondepreciated value (in dollars) TRY IT 8 Write a linear equation for the machine in Eample 8 if the salvage value at the end of 8 ears is $000. Straight-Line Depreciation V,000,000 0,000 9,000 8,000,000,000,000,000,000,000,000 FIGURE. (0,,000) V = 0t +,000 (8, 000) Number of ears t EXAMPLE 8 Depreciating Equipment Your compan has purchased a $,000 machine that has a useful life of 8 ears. The salvage value at the end of 8 ears is $000. Write a linear equation that describes the book value of the machine each ear. SOLUTION Let V represent the value of the machine at the end of ear t. You can represent the initial value of the machine b the ordered pair 0,,000 and the salvage value of the machine b the ordered pair 8, 000. The slope of the line is m 000, $0 which represents the annual depreciation in dollars per ear. Using the pointslope form, ou can write the equation of the line as shown. V,000 0 t 0 Write in point-slope form. V 0t,000 Write in slope-intercept form. The table shows the book value of the machine at the end of each ear. The graph of this equation is shown in Figure.. m t t t 0 8 V,000 0, T AKE ANOTHER LOOK Comparing Different Tpes of Depreciation The Internal Revenue Service allows businesses to choose different tpes of depreciation. Another tpe is Uniform Declining Balances: V,000 n.0 n 8. n t, Construct a table that compares this tpe of depreciation with linear depreciation. What are the advantages of each tpe?

10 000_00.qd //0 : AM Page SECTION. Lines in the Plane and Slope PREREQUISITE REVIEW. The following warm-up eercises involve skills that were covered in earlier sections. You will use these skills in the eercise set for this section. In Eercises and, simplif the epression Evaluate when m.. Evaluate when m m m. In Eercises 0, solve for in terms of EXERCISES. In Eercises, estimate the slope of the line..... In Eercises, plot the points and find the slope of the line passing through the pair of points..,,,.,,,.,,, ,, 8, 0.,,,.,,,.,,,.,, 8,.,,,.,,,. 8,,,,,, 0 In Eercises, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are man correct answers.) Point Slope Point Slope., m 0 8., m 0 9., m 0., m., m. 0, m. 8, m is undefined.., m is undefined. In Eercises, find the slope and -intercept (if possible) of the equation of the line In Eercises, write an equation of the line that passes through the points. Then use the equation to sketch the line..,, 0,.,,,. 0, 0,, 8.,,, 9.,,, 0.,, 0,.,,,.,,, 0.,,,. 8,,,.,,, 8.,,,

11 000_00.qd //0 : AM Page CHAPTER Functions, Graphs, and Limits In Eercises, write an equation of the line that passes through the given point and has the given slope. Then use a graphing utilit to graph the line. Point Slope Point Slope. 0, m 8. 0, 0 m 9., m is undefined. 0. 0, m is undefined.., m 0., m 0. 0, m., m. 0, m. 0, m In Eercises and 8, eplain how to use the concept of slope to determine whether the three points are collinear. Then eplain how to use the Distance Formula to determine whether the points are collinear.. 8.,,, 0,, 0,,,,, 9. Write an equation of the vertical line with -intercept at. 0. Write an equation of the horizontal line through 0,.. Write an equation of the line with -intercept at 0 and parallel to all horizontal lines.. Write an equation of the line with -intercept at and parallel to all vertical lines. In Eercises 0, write the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Then use a graphing utilit to graph all three equations in the same viewing window. Point Line.,.,., 8. 8, 0., , 0 9., 0 0., 0 In Eercises 8, sketch the graph of the equation. Use a graphing utilit to verif our result Population The resident population of South Carolina (in thousands) was 80 in 99 and 0 in 00. Assume that the relationship between the population and the ear t is linear. Let t represent 99. (Source: U.S. Census Bureau) (a) Write a linear model for the data. What is the slope and what does it tell ou about the population? (b) Estimate the population in 999. (c) Use our model to estimate the population in 00. (d) Use our school s librar, the Internet, or some other reference source to find the actual populations in 999 and 00. How close were our estimates? (e) Do ou think our model could be used to predict the population in 00? Eplain. 80. Annual Salar Your annual salar was $,00 in 00 and $9,00 in 00. Assume our salar can be modeled b a linear equation. (a) Write a linear equation giving our salar S in terms of the ear. Let t represent 00. (b) Use the linear model to predict our salar in Temperature Conversion Write a linear equation that epresses the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. Use the fact that water freezes at 0 C ( F) and boils at 00 C ( F). 8. Chemistr Use the result of Eercise 8 to answer the following: (a) A person has a temperature of 0. F. What is this temperature on the Celsius scale? (b) If the temperature in a room is F, what is this temperature on the Celsius scale? (Source: Adapted from Zumdahl, Chemistr, Sith Edition) 8. Reimbursed Epenses A compan reimburses its sales representatives $0 per da for lodging and meals, plus $0. per mile driven. Write a linear equation giving the dail cost C in terms of, the number of miles driven. 8. Union Negotiation You are on a negotiating panel in a union hearing for a large corporation. The union is asking for a base pa of $9. per hour plus an additional piecework rate of $0.80 per unit produced. The corporation is offering a base pa of $.8 per hour plus a piecework rate of $.. (a) Write a linear equation for the hourl wages W in terms of, the number of units produced per hour, for each pa schedule. (b) Use a graphing utilit to graph each linear equation and find the point of intersection. (c) Interpret the meaning of the point of intersection of the graphs. How would ou use this information to advise the corporation and the union?

12 000_00.qd //0 : AM Page SECTION. Lines in the Plane and Slope 8. Chemistr Ethlene glcol is the main component in automobile antifreeze. To monitor the temperature of an auto cooling sstem, ou intend to use a meter that reads from 0 to 00. You devise a new temperature scale A based on the approimate melting and boiling points of a tpical antifreeze solution C and C. You wish these points to correspond to 0 A and 00 A, respectivel. (a) Derive an epression for converting between A and C. (b) Derive an epression for converting between F and A. (c) At what temperature would our thermometer and a Celsius thermometer give the same numerical reading? (d) Your thermometer reads 8 A. What is the temperature in C and in F? (e) What is a temperature of C in A? (Source: Zumdahl, Chemistr, Sith Edition) 8. Linear Depreciation A compan constructs a warehouse for $8,000. The warehouse has an estimated useful life of ears, after which its value is epected to be $,000. Write a linear equation giving the value of the warehouse during its ears of useful life. (Let t represent the time in ears.) 8. Linear Depreciation A small business purchases a piece of equipment for $0. After ears the equipment will be outdated, having no value. (a) Write a linear equation giving the value of the equipment in terms of the time t in ears, 0 t. (b) Use a graphing utilit to graph the equation. (c) Move the cursor along the graph and estimate (to twodecimal-place accurac) the value of the equipment when t. (d) Move the cursor along the graph and estimate (to twodecimal-place accurac) the time when the value of the equipment will be $ College Enrollment A small college had students in 00 and 0 students in 00. If the enrollment follows a linear growth pattern, how man students will the college have in 008? 89. Consumer Awareness A real estate office handles an apartment comple with 0 units. When the rent is $80 per month, all 0 units are occupied. When the rent is $, however, the average number of occupied units drops to. Assume that the relationship between the monthl rent p and the demand is linear. (The term demand refers to the number of occupied units.) (a) Write a linear equation epressing in terms of p. (b) Linear Etrapolation Predict the number of occupied units when the rent is set at $. (c) Linear Interpolation Predict the number of occupied units when the rent is set at $ Profit You are a contractor and have purchased a piece of equipment for $,00. The equipment costs an average of $. per hour for fuel and maintenance, and the operator is paid $9.0 per hour. (a) Write a linear equation giving the total cost C of operating the equipment for t hours. (b) You charge our customers $ per hour of machine use. Write an equation for the revenue R derived from t hours of use. (c) Use the formula for profit, P R C, to write an equation for the profit derived from t hours of use. (d) Find the number of hours ou must operate the equipment before ou break even. 9. Personal Income Personal income (in billions of dollars) in the United States was 9 in 99 and 88 in 00. Assume that the relationship between the personal income Y and the time t (in ears) is linear. Let t 0 correspond to 990. (Source: U.S. Bureau of Economic Analsis) (a) Write a linear model for the data. (b) Linear Interpolation Estimate the personal income in 999. (c) Linear Etrapolation Estimate the personal income in 00. (d) Use our school s librar, the Internet, or some other reference source to find the actual personal income in 999 and 00. How close were our estimates? 9. Sales Commission As a salesperson, ou receive a monthl salar of $000, plus a commission of % of sales. You are offered a new job at $00 per month, plus a commission of % of sales. (a) Write a linear equation for our current monthl wage W in terms of our monthl sales S. (b) Write a linear equation for the monthl wage W of our job offer in terms of the monthl sales S. (c) Use a graphing utilit to graph both equations in the same viewing window. Find the point of intersection. What does it signif? (d) You think ou can sell $0,000 worth of a product per month. Should ou change jobs? Eplain. In Eercises 9 0, use a graphing utilit to graph the cost function. Determine the maimum production level,given that the cost C cannot eceed $00, C, C 0, C 8, 0 9. C, C, C 8,0 99. C, C, C 0, C,00.0