000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of circles. Find the points of intersection of two graphs. Use mathematical models to model and solve real-life problems. The Graph of an Equation In Section., ou used a coordinate sstem to represent graphicall the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane (see Eample in Section.). Frequentl, a relationship between two quantities is epressed as an equation. For instance, degrees on the Fahrenheit scale are related to degrees on the Celsius scale b the equation F 9 5C. In this section, ou will stud some basic procedures for sketching the graphs of such equations. The graph of an equation is the set of all points that are solutions of the equation. EXAMPLE Sketch the graph of 7. 0 7 Sketching the Graph of an Equation SOLUTION The simplest wa to sketch the graph of an equation is the pointplotting method. With this method, ou construct a table of values that consists of several solution points of the equation, as shown in the table below. For instance, when 0 7 0 7 which implies that 0, 7 is a solution point of the graph. 7 5 From the table, it follows that 0, 7,,,,,,, and, 5 are solution points of the equation. After plotting these points, ou can see that the appear to lie on a line, as shown in Figure.. The graph of the equation is the line that passes through the five plotted points. 8 (0, 7) (, ) (, ) 8 (, ) (, 5) FIGURE. Solution Points for 7 TRY IT Sketch the graph of. STUDY TIP Even though we refer to the sketch shown in Figure. as the graph of 7, it actuall represents onl a portion of the graph. The entire graph is a line that would etend off the page.
000_00.qd /5/05 : AM Page CHAPTER Functions, Graphs, and Limits STUDY TIP The graph shown in Eample is a parabola. The graph of an second-degree equation of the form a b c, a 0 has a similar shape. If a > 0, the parabola opens upward, as in Figure.(b), and if a < 0, the parabola opens downward. EXAMPLE Sketch the graph of. SOLUTION Sketching the Graph of an Equation Begin b constructing a table of values, as shown below. 0 7 Net, plot the points given in the table, as shown in Figure.(a). Finall, connect the points with a smooth curve, as shown in Figure.(b). 8 (, 7) 8 = (, ) (, ) (, ) (, ) (0, ) (a) (b) FIGURE. TRY IT Sketch the graph of. The point-plotting technique demonstrated in Eamples and is eas to use, but it does have some shortcomings. With too few solution points, ou can badl misrepresent the graph of a given equation. For instance, how would ou connect the four points in Figure.5? Without further information, an one of the three graphs in Figure. would be reasonable. FIGURE.5 FIGURE.
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations Intercepts of a Graph It is often eas to determine the solution points that have zero as either the -coordinate or the -coordinate. These points are called intercepts because the are the points at which the graph intersects the - or -ais. Some tets denote the -intercept as the -coordinate of the point a, 0 rather than the point itself. Unless it is necessar to make a distinction, we will use the term intercept to mean either the point or the coordinate. A graph ma have no intercepts or several intercepts, as shown in Figure.7. ALGEBRA REVIEW Finding intercepts involves solving equations. For a review of some techniques for solving equations, see page 7. No -intercept One -intercept Three -intercepts One -intercept One -intercept Two -intercepts No intercepts FIGURE.7 Finding Intercepts. To find -intercepts, let be zero and solve the equation for.. To find -intercepts, let be zero and solve the equation for. = EXAMPLE Finding - and -Intercepts Find the - and -intercepts of the graph of each equation. (a) (b) SOLUTION (a) Let 0. Then 0 has solutions 0 and ±. Let 0. Then 0 0 0. -intercepts: 0, 0,, 0,, 0 -intercept: 0, 0 See Figure.8. (b) Let 0. Then 0. Let 0. Then 0 has solutions ±. -intercept:, 0 -intercepts: 0,, 0, See Figure.9. TRY IT Find the - and -intercepts of the graph of each equation. (a) (b) (, 0) (0, 0) (, 0) FIGURE.8 (, 0) = FIGURE.9 (0, ) (0, )
000_00.qd /5/05 : AM Page CHAPTER Functions, Graphs, and Limits TECHNOLOGY STUDY TIP Some graphing utilities have a built-in program that can find the -intercepts of a graph. If our graphing utilit has this feature, tr using it to find the -intercept of the graph shown on the left. (Your calculator ma call this the root or zero feature.) Zooming in to Find Intercepts You can use the zoom feature of a graphing utilit to approimate the -intercepts of a graph. Suppose ou want to approimate the -intercept(s) of the graph of. Begin b graphing the equation, as shown below in part (a). From the viewing window shown, the graph appears to have onl one -intercept. This intercept lies between and. B zooming in on the intercept, ou can improve the approimation, as shown in part (b). To three decimal places, the solution is.7. = + = + 0..8.7 (a) (b) 0. Here are some suggestions for using the zoom feature.. With each successive zoom-in, adjust the -scale so that the viewing window shows at least one tick mark on each side of the -intercept.. The error in our approimation will be less than the distance between two scale marks.. The trace feature can usuall be used to add one more decimal place of accurac without changing the viewing window. Part (a) below shows the graph of 5. Parts (b) and (c) show zoom-in views of the two intercepts. From these views, ou can approimate the -intercepts to be 0.97 and.0. 0 0.0 0.0 0 0 0.8 0.7.9. 0 = 5 + 0.0 = 5 + 0.0 = 5 + (a) (b) (c)
000_00.qd /5/05 : AM Page 5 SECTION. Graphs of Equations 5 Circles Throughout this course, ou will learn to recognize several tpes of graphs from their equations. For instance, ou should recognize that the graph of a seconddegree equation of the form a b c, a 0 Center: (h, k) is a parabola (see Eample ). Another easil recognized graph is that of a circle. Consider the circle shown in Figure.0. A point, is on the circle if and onl if its distance from the center h, k is r. B the Distance Formula, h k r. Radius: r Point on circle: (, ) B squaring both sides of this equation, ou obtain the standard form of the equation of a circle. FIGURE.0 Standard Form of the Equation of a Circle The point, lies on the circle of radius r and center h, k if and onl if h k r. From this result, ou can see that the standard form of the equation of a circle with its center at the origin, h, k 0, 0, is simpl r. Circle with center at origin EXAMPLE Finding the Equation of a Circle The point, lies on a circle whose center is at,, as shown in Figure.. Find the standard form of the equation of this circle. SOLUTION The radius of the circle is the distance between, and,. r Distance Formula Simplif. 0 Radius Using h, k,, the standard form of the equation of the circle is h k r 0 Substitute for h, k, and r. 0. Write in standard form. 8 (, ) (, ) ( + ) + ( ) = 0 FIGURE. TRY IT The point, 5 lies on a circle whose center is at,. Find the standard form of the equation of this circle.
000_00.qd /5/05 :7 AM Page CHAPTER Functions, Graphs, and Limits To graph a circle on a graphing utilit, ou can solve its equation for and graph the top and bottom halves of the circle separatel. For instance, ou can graph the circle 0 b graphing the following equations. If ou want the result to appear circular, ou need to use a square setting, as shown below. 0 TECHNOLOGY 0 0 0 0 General Form of the Equation of a Circle A A D E F 0, A 0 To change from general form to standard form, ou can use a process called completing the square, as demonstrated in Eample 5. EXAMPLE 5 Completing the Square Sketch the graph of the circle whose general equation is 0 7 0. SOLUTION First divide b so that the coefficients of and are both. 0 7 0 5 7 0 5 7 5 5 7 5 Write original equation. Divide each side b. Group terms. Complete the square. Standard setting 0 9 Half Half 5 From the standard form, ou can see that the circle is centered at a radius of, as shown in Figure.. Write in standard form. 5, and has + 5 + ( ) = 9 9 r = Square setting 5, FIGURE. TRY IT 5 Write the equation of the circle 0 in standard form and sketch its graph. The general equation A A D E F 0 ma not alwas represent a circle. In fact, such an equation will have no solution points if the procedure of completing the square ields the impossible result h k negative number. No solution points Further assistance with the calculus and algebra used in this eample is available on the CD that accompanies this tet.
000_00.qd /5/05 :7 AM Page 7 SECTION. Graphs of Equations 7 Points of Intersection A point of intersection of two graphs is an ordered pair that is a solution point of both graphs. For instance, Figure. shows that the graphs of and have two points of intersection:, and,. To find the points analticall, set the two -values equal to each other and solve the equation for. A common business application that involves points of intersection is breakeven analsis. The marketing of a new product tpicall requires an initial investment. When sufficient units have been sold so that the total revenue has offset the total cost, the sale of the product has reached the break-even point. The total cost of producing units of a product is denoted b C, and the total revenue from the sale of units of the product is denoted b R. So, ou can find the break-even point b setting the cost C equal to the revenue R, and solving for. EXAMPLE Finding a Break-Even Point A business manufactures a product at a cost of $0.5 per unit and sells the product for $.0 per unit. The compan s initial investment to produce the product was $0,000. How man units must the compan sell to break even? SOLUTION The total cost of producing units of the product is given b C 0.5 0,000. Cost equation The total revenue from the sale of units is given b R.. Revenue equation To find the break-even point, set the cost equal to the revenue and solve for. R C Set revenue equal to cost.. 0.5 0,000 Substitute for R and C. 0.55 0,000 Subtract 0.5 from each side. 0,000 Divide each side b 0.55. 0.55 8,8 Use a calculator. So, the compan must sell 8,8 units before it breaks even. This result is shown graphicall in Figure.. TRY IT How man units must the compan in Eample sell to break even if the selling price is $.5 per unit? FIGURE. Sales (in dollars) 50,000 5,000 0,000 5,000 0,000 5,000 0,000 5,000 0,000 5,000 Break-Even Analsis C = 0.5 + 0,000 Loss FIGURE. STUDY TIP The Technolog note on page describes how to use a graphing utilit to find the -intercepts of a graph. A similar procedure can be used to find the points of intersection of two graphs. (Your calculator ma call this the intersect feature.) Break-even point: 8,8 units R =. 0,000 0,000 Number of units Profit
000_00.qd /5/05 :7 AM Page 8 8 CHAPTER Functions, Graphs, and Limits p FIGURE.5 p Suppl Curve Two tpes of applications that economists use to analze a market are suppl and demand equations. A suppl equation shows the relationship between the unit price p of a product and the quantit supplied. The graph of a suppl equation is called a suppl curve. (See Figure.5.) A tpical suppl curve rises because producers of a product want to sell more units if the unit price is higher. A demand equation shows the relationship between the unit price p of a product and the quantit demanded. The graph of a demand equation is called a demand curve. (See Figure..) A tpical demand curve tends to show a decrease in the quantit demanded with each increase in price. In an ideal situation, with no other factors present to influence the market, the production level should stabilize at the point of intersection of the graphs of the suppl and demand equations. This point is called the equilibrium point. The -coordinate of the equilibrium point is called the equilibrium quantit and the p-coordinate is called the equilibrium price. (See Figure.7.) You can find the equilibrium point b setting the demand equation equal to the suppl equation and solving for. FIGURE. Equilibrium price p 0 p FIGURE.7 Price per unit (in dollars) 50 00 50 00 50 p FIGURE.8 Demand Demand Curve Suppl Equilibrium point ( 0, p 0 ) 0 Equilibrium quantit Equilibrium Point Equilibrium Point (5, ) Suppl Demand 5 7 8 9 Number of units (in millions) EXAMPLE 7 Finding the Equilibrium Point The demand and suppl equations for a DVD plaer are given b p 95 5.8 Demand equation p 50. Suppl equation where p is the price in dollars and represents the number of units in millions. Find the equilibrium point for this market. SOLUTION Begin b setting the demand equation equal to the suppl equation. 95 5.8 50. Set equations equal to each other. 5 5.8. Subtract 50 from each side. 5 9 Add 5.8 to each side. 5 Divide each side b 9. So, the equilibrium point occurs when the demand and suppl are each five million units. (See Figure.8.) The price that corresponds to this -value is obtained b substituting 5 into either of the original equations. For instance, substituting into the demand equation produces p 95 5.8 5 95 9 $. Substitute 5 into the suppl equation to see that ou obtain the same price. TRY IT 7 The demand and suppl equations for a calculator are p.5 and p.5, respectivel, where p is the price in dollars and represents the number of units in millions. Find the equilibrium point for this market.
000_00.qd /5/05 :7 AM Page 9 Mathematical Models In this tet, ou will see man eamples of the use of equations as mathematical models of real-life phenomena. In developing a mathematical model to represent actual data, ou should strive for two (often conflicting) goals accurac and simplicit. SECTION. Graphs of Equations 9 EXAMPLE 8 Using Mathematical Models The table shows the annual sales (in millions of dollars) for Dillard s and Kohl s for 999 through 00. In the spring of 00, the publication Value Line listed the projected 00 sales for the companies as $770 million and $,975 million, respectivel. How do ou think these projections were obtained? (Source: Dillard s Inc. and Kohl s Corp.) ALGEBRA REVIEW For help in evaluating the epressions in Eample 8, see the review of order of operations on page 7. Year 999 000 00 00 00 t 9 0 Dillard s 877 857 855 79 7599 Kohl s 557 5 789 90 0,8 SOLUTION The projections were obtained b using past sales to predict future sales. The past sales were modeled b equations that were found b a statistical procedure called least squares regression analsis. S.8t 89.7t 99, S 0.8t 0.7t,0, 9 t 9 t Using t to represent 00, ou can predict the 00 sales to be S.8 89.7 99 70 S 0.8 0.7,0,559. Dillard s Kohl s Dillard s Kohl s These two projections are close to those projected b Value Line. The graphs of the two models are shown in Figure.9. TRY IT 8 The table shows the annual sales (in millions of dollars) for Dollar General for 995 through 00. In the winter of 00, the publication Value Line listed projected 00 sales for the compan as $7800 million. How does this projection compare with the projection obtained using the model below? (Source: Dollar General Corp.) S.t 78.t 50.9, 5 t Year 995 99 997 998 999 000 00 00 t 5 7 8 9 0 Sales 7.. 7..0 888.0 550. 5.9 00. Annual sales (in millions of dollars),000 0,000 9,000 8,000 7,000,000 5,000,000 FIGURE.9 Annual Sales S Dillard's Kohl's 9 0 Year (9 999) STUDY TIP To test the accurac of a model, ou can compare the actual data with the values given b the model. For instance, the table below compares the actual Kohl s sales with those given b the model. Year 999 000 00 Actual 557 5 789 Model 55. 9 70. Year 00 00 Actual 90 0,8 Model 900. 0, t
000_00.qd /5/05 :7 AM Page 0 0 CHAPTER Functions, Graphs, and Limits Much of our stud of calculus will center around the behavior of the graphs of mathematical models. Figure.0 shows the graphs of si basic algebraic equations. Familiarit with these graphs will help ou in the creation and use of mathematical models. = = = (a) Linear model (b) Quadratic model (c) Cubic model = = = (d) Square root model FIGURE.0 (e) Absolute value model (f) Rational model TAKE ANOTHER LOOK Graphical, Numerical, and Analtic Solutions Most problems in calculus can be solved in a variet of was. Often, ou can solve a problem graphicall, numericall (using a table), and analticall. For instance, Eample compares graphical and analtic approaches to finding points of intersection. In Eample 8, suppose ou were asked to find the point in time at which Kohl s sales eceeded Dillard s sales. Eplain how to use each of the three approaches to answer the question. For this question, which approach do ou think is best? Eplain. Suppose ou answered the question and obtained t.. What date does this represent April 00 or April 00? Eplain.
000_00.qd /5/05 :7 AM Page SECTION. Graphs of Equations 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, solve for.. 5... 5. 9. 5 0 5 8 In Eercises 7 0, complete the square to write the epression as a perfect square trinomial. 7. 8. 9. 5 0. In Eercises, factor the epression... 5. 9. 7 9 EXERCISES. In Eercises, determine whether the points are solution points of the given equation.. 0 (a), (b), (c), 5. 7 0 (a), 9 (b) 5, 0 (c), 5 8. (a), (b) (c), 7, 5 0 0, 5. (a) (b), (c), 5. (a) 0, (b), (c),. 5 (a) 7, 5 (b), (c), 5 In Eercises 7, match the equation with its graph. Use a graphing utilit, set for a square setting, to confirm our result. [The graphs are labeled (a) (f ).] 7. 8. 9. 0. 9.. (a) (c) (e) In Eercises, find the - and -intercepts of the graph of the equation.. 0. 5 0 5.. 7. 9 8. (b) (d) (f) 5
000_00.qd /5/05 :7 AM Page CHAPTER Functions, Graphs, and Limits 9. 0.. 0. 8 In Eercises 8, sketch the graph of the equation and label the intercepts. Use a graphing utilit to verif our results... 5.. 7. 8. 5 9. 0..... 5.. 7. 8. In Eercises 9, write the general form of the equation of the circle. 9. Center: 0, 0 ; radius: 0. Center: 0, 0 ; radius: 5. Center:, ; radius:. Center:, ; radius:. Center:, ; solution point: 0, 0. Center:, ; solution point:, 5. Endpoints of a diameter:,,,. Endpoints of a diameter:,,, In Eercises 7 5, complete the square to write the equation of the circle in standard form. Then use a graphing utilit to graph the circle. 7. 0 8. 5 0 9. 0 50. 0 5. 0 5. 0 5. 0 7 0 5. 0 In Eercises 55, find the points of intersection (if an) of the graphs of the equations. Use a graphing utilit to check our results. 55., 5. 7, 57. 58. 5, 0, 59., 0.,.,.,. Break-Even Analsis You are setting up a part-time business with an initial investment of $5,000. The unit cost of the product is $.80, and the selling price is $9.0. (a) Find equations for the total cost C and total revenue R for units. (b) Find the break-even point b finding the point of intersection of the cost and revenue equations. (c) How man units would ield a profit of $000?. Break-Even Analsis A 00 Chevrolet Malibu costs $0,90 with a gasoline engine. A 00 Toota Prius costs $,05 with a hbrid engine. The Malibu gets miles per gallon of gasoline and the Prius gets 5 miles per gallon of gasoline. Assume that the price of gasoline is $.759. (Source: Adapted from Consumer Reports, Ma 00) (a) Show that the cost C g of driving the Chevrolet Malibu miles is C g 0,90.759 and the cost C h of driving the Toota Prius miles is C h,05.759 5. (b) Find the break-even point. That is, find the mileage at which the hbrid-powered Toota Prius becomes more economical than the gasoline-powered Chevrolet Malibu. Break-Even Analsis In Eercises 5 8, find the sales necessar to break even for the given cost and revenue equations. (Round our answer up to the nearest whole unit.) Use a graphing utilit to graph the equations and then find the break-even point. 5. C 0.85 5,000, R.55. C 500,000, R 5 7. C 850 50,000, R 9950 8. C 5.5 0,000, R.9 9. Suppl and Demand The demand and suppl equations for an electronic organizer are given b p 80 Demand equation p 75 Suppl equation where p is the price in dollars and represents the number of units, in thousands. Find the equilibrium point for this market. 70. Suppl and Demand The demand and suppl equations for a portable CD plaer are given b p 90 5 Demand equation p 75 8 Suppl equation where p is the price in dollars and represents the number of units, in hundreds of thousands. Find the equilibrium point for this market.
000_00.qd /5/05 :7 AM Page SECTION. Graphs of Equations 7. Consumer Trends The amounts of mone (in millions of dollars) spent on college tetbooks in the United States in the ears 995 to 00 are shown in the table. (Source: Book Industr Stud Group, Inc.) Year 995 99 997 998 Epense 708 90 0 5 Year 999 000 00 00 Epense 77 905 87 70 A mathematical model for the data is given b.77t.99t 97.t 985 where t represents the ear, with t 5 corresponding to 995. (a) Compare the actual epenses with those given b the model. How good is the model? Eplain our reasoning. (b) Use the model to predict the epenses in 00. 7. Farm Work Force The numbers of workers in farm work force in the United States for selected ears from 955 to 000, as percents of the total work force, are shown in the table. (Source: Department of Commerce) Year 955 90 95 970 975 Percent 9.9 7.8 5.9.. Year 980 985 990 995 000 Percent..8...7 A mathematical model for the data is given b where represents the percent and t represents the ear, with t 55 corresponding to 955. (a) Compare the actual percents with those given b the model. How good is the model? (b) Use the model to predict the farm work force as a percent of the total work force in 00. (c) Discuss the validit of our prediction in part (b). 7. Weekl Salar A mathematical model for the average weekl salar of a person in finance, insurance, or real estate is given b.97 0.0t 0.05t 9.8 7.7t 0.0t where t represents the ear, with t 7 corresponding to 997. (Source: U.S. Bureau of Labor Statistics) (a) Use the model to complete the table. Year 997 998 999 000 00 00 Salar (b) This model was created using actual data from 997 through 00. How accurate do ou think the model is in predicting the 00 average weekl salar? Eplain our reasoning. (c) What does this model predict the average weekl salar to be in 00? Do ou think this prediction is valid? 7. Medicine A mathematical model for the numbers of kidne transplants performed in the United States in the ears 998 to 00 is given b 0.t 5.0t, where is the number of transplants and t is the time in ears, with t 8 corresponding to 998. (Source: United Network for Organ Sharing) (a) Enter the model into a graphing utilit and use it to complete the table. Year 998 999 000 00 00 Transplants (b) Use our school s librar, the Internet, or some other reference source to find the actual numbers of kidne transplants for the ears 998 to 00. Compare the actual numbers with those given b the model. How good is the model? Eplain our reasoning. (c) Using this model, what is the prediction for the number of transplants in the ear 008? How valid do ou think the prediction is? What factors could affect this model s accurac? 75. Use a graphing utilit to graph the equation c for c,,,, and 5. Then make a conjecture about the -coefficient and the graph of the equation. 7. Define the break-even point for a business marketing a new product. Give eamples of a linear cost equation and a linear revenue equation for which the break-even point is 0,000 units. In Eercises 77 8, use a graphing utilit to graph the equation. Use the graphing utilit to approimate the - and -intercepts of the graph. 77. 0.. 5. 78. 0.5 5..5 79. 0.. 5.7 80... 5. 0. 5. 8. 0. 8. 0.. 0. 5.