8. y = 5 - X2. x y = 16-4X2 10. y = (x +' 3) y = 5x y = 8-3x 13. y =.Jx+Li. 14.y=~

Transcription

1 Chapter Function and Their Graphs [ ;) Eercises VOCABULARY CHECK: Fill in the blanks.. An ordered pair (a, b) is a of an equation in and if the equation is true when a is substituted for and b is substituted for.. The set of all solution points of an equation is the of the equation. 3. The points at which a graph intersects or touches an ais are called the of the graph.. A graph is smmetric with respect to the if, whenever (, ) is on the graph, (-, ) is also on the graph. S. The equation ( - h) + ( - k) = r is the standard form of the equation of a with center -'-- and radius _ 6. When ou construct and use a table to solve a problem, ou are using a approach. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises -, determine whether each point lies on the graph of the equation. Equation. =.JX+. Y = X = - I -. = t 3 - (a) (0,) (a) (,0) (a) (,5) (a) (, -.If) Points (b) (5,3) (b) (,8) (b) (6,0) (b) (-3,9) In Eercises 5-8, complete the table. Use the resulting solution points to sketch the graph of the equation. 5. = = 5 - X - 0 (,) In Eercises 90, find the - and -intercepts of the graph 'of the equation. 9. = 6 - X 0. = ( +' 3) - 0 (,) 5 ' 0 6. = ~ (X, ) 7. = X Y (,). = 5-6. = =.J+Li.=~ 5. = = -I = 3 - X 8. = X = 6-0. = X +

2 Section. Graphs of Equations 3 In Eercises, assume that the graph has the indicated tpe of smmetr. Sketch the complete graph of the equation. To print an enlarged cop of the graph, go to the website ais srrunetr -+-+-H--~H-+-~ - - Origin srrunetr ais srrunetr ais srrunetr 53. = J = I = (6 - )J 56. = - Il In Eercises 57-6, write the standard form of the equation of the circle with the given characteristics. 57. Center: (0,0); radius: 58. Center: (0,0); 59. Center: (, -); radius: 60. Center: (-7, -); radius: 7 6. Center: (-,); solution point: (0,0) 6. Center: (3, - ); solution point: (-, ) 63. Endpoints of a diameter: (0,0), (6, 8) 6. Endpoints of a diameter: (-, -), (, ) radius: 5 In Eercises 65-70, find the center and radius of the circle, and sketch its graph. 65. X + = = ( - ) + ( + 3) = X + ( - ) = 69. ( - ) + ( - ) = ~ 70. ( - ) + ( + 3) = ~ In Eercises 5~3, use the algebraic tests to check for smmetr with respect to both aes and the origin. 5. X - Y = 0 7. = 3 9. = X = = 0 8. = X - X = + 3. = In Eercises 33-, use smmetr to sketch the graph of the equation. 33. = Y = Y = = J""=3. = I = - I 3. Y == = - X = 3-0.=~. Y = - Il. = - 5 ~ In Eercises 5-56, use a graphing utilit to graph the equation. Use a standard setting. Approimate an intercepts. 5. = 3 - ~ 6. Y = ~ - 7. Y = X Y = X + X = -I 50. Y = X + 5. Y =.f 5. Y = + 'I The smbol ~ 7. Depreciation A manufacturing plant purchases a new molding machine for $5,000. The depreciated value (reduced value) after t ears is given b = 5,000-0,000t, 0 ~ t ~ 8. Sketch the graph of the equation. 7. Consumerism,You purchase a jet ski for $800. The depreciated value after t ears is given b = t, 0 ~ t ~ 6. Sketch the graph of the equation. 73. Geometr A regulation NFL plaing field (including the end zones) of length and width has a perimeter of or -3- ards. 3 (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) S how that the Width. 0f thee rectang rectanzlee IS i Y = d.. (50) an Its area IS A = 3 -. ffi (c) Use a graphing utilit to graph the area equation. Be sure to adjust our window settings. ~ (d) From the graph in part (c), estimate the dimensions of the rectangle that ield a maimum area. (e) Use our school's librar, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL plaing field and compare our findings with the results of part (d). indicates an eercise or a part of an eercise in which ou are instructed to use a graphing utilit.

3 Chapter Function and Their Graphs 7. Geometr A soccer plaing field of length and width has a perimeter of 360 meters. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is w = 80 - and its area is A = (80 - ). ~~ ~ (c) Use a.graphing utilit to graph the area equation. Be sure to adjust our window settings. ~ (d) From the graph in part (c), estimate the dimensions of the rectangle that ield a maimum area. (e) Use our school's librar, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare our findings with the results of parted). Model It. 75. Population Statistics The table shows the life epectancies of a child (at birth) in the United States for selected ears from 90 to 000. (Source: U.S. National Center for Health Statistics) 76. Electronics The resistance (in ohms) of 000 feet of solid copper wire at 68 degrees Fahrenheit can be approi- 0,770 mated b the model = , 5 ~ ~ 00 where is the diameter of the wire in mils (0.00 inch). (Source: American Wire Gage) (a) Complete the table (b) Use the table of values in part (a) to sketch a graph of the model. Then use our graph to estimate the resistance when = (c) Use the model to confirm algebraicall the estimate ou found in part (b). (d) What can ou conclude in general about the relationship between the diameter of the copper wire and the resistance? Snthesis A model for the life epectanc during this period is = t t +.5, 0 ~ t ~ 00 where represents the life epectanc and t is the time in ears, with t = 0 corresponding to 90. (a) Sketch a scatter plot of the data. (b) Graph the model for the data and compare the scatter plot and the graph. (c) Determine the life epectanc in 98 both graphicall and algebraicall. (d) Use the graph of the model to estimate the life epectancies of a child for the ears 005 and 00. (e) Do ou think this model can be used to predict the life epectanc of a child 50 ears from now? Eplain.!'rue or False? the statement In Eercises 77 and 78, determine whether is true or false. Justif our answer. 77. A graph is smmetric with respect to the -ais if, whenever (, ) is on the graph, (-, ) is also on the graph. 78. A graph of an equation can have more than one -intercept, m 79. Think About It Suppose ou correctl enter an epression for the variable on a graphing utilit. However, no graph appears on the displa when ou graph the equation. Give a possible eplanation and the steps ou could take to remed the problem. lllustrate our eplanation with an eample. 80. Think About It Find a and b if the graph of = a + b 3 is smmetric with respect to (a) the -ais and (b) the origin. (There are man correct answers.) Skills Review 8. Identif the terms: 9 s Rewrite the epression using eponential notation. -( ) In Eercises 83-88, simplif the epression. 83..fi8 -.JX 8. W r;:;-: 86. ;;:;r; -J'l..,; W 88. fl

4 3 Chapter Functions and Their Graphs [.3] Eercises J VOCABULARY In Eercises CHECK: -6, fill in the blanks.. The simplest mathematical model for relating two variables is the equation in two variables =.X + b.. For a line', the ratio of the change in to the change in is called the of the line. 3. Two lines are if and onl if their slopes are equal.. Two lines are if and onl if their slopes are negative reciprocals of each other. S. When the -ais and -ais have different units of measure, the slope can be interpreted as a _ 6. The prediction method is the method used to estimate a point on a line that does not lie between the given points. 7. Match each equation of a line with its form. (a) A + B + C = 0 (i) Vertical line (b) = a (ii) Slope-intercept form (c) = b (iii) General form (d) = m + b (iv) Point-slope form (e) - YI = m( - Xl) (v) Horizontal line PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises and, identif the line that has each slope.. (a) m = ~. (a) m = 0 7. Y 8+ ;..', ;.. ;..,..,.., 8. Y (b) m is undefined. (b) m = - ~ (c)m= (c)m=l +t+-r+-t-+-+t+-..~.. -H-+-t "\~ ----~~--~ ~~~---- In Eercises 3 and, sketch the lines through the point with the indicated slopes on the same set of coordinate aes. 3. (,3) Point. (-, I) Slopes (a) 0 (b) I (c) (d) -3 (a) 3 (b) - 3 (c)! (d) Undefined In Eercises 5-8, estimate the slope of the line. 5. -t- '..., " h t... '.~. -t-'! "' '..' + + ' ;/ ;,+ ;..-l-...;..+,...,...."...,...,...;..+...!. +~"'"!'L l $ -it..t ~ ++--I---+-f In Eercises 90, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. 9. = = -! = = = = 0 0. = - 0. = -~ = = 9 8. Y + = = 0 In Eercises 8, plot the points and find the slope of the line passing through the pair of points.. (-3, ), (,6) 3. (-6,-),(-6,) 5. (l{-,-~), (-~, -t) 7. (.8,3.), (-5.,.6) 8. (-.75, - 8.3), (.5, -.6). (,), (, -). (0, -0), (-,0) 6. G, ~),(~,-i)

5 Section.3 Linear Equations in Two Variables 35 In Eercises 9-38, 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 9. (. ) m = 0 Slope 30. (-,) m is undefined. 3. (5, -6) m = 3. (0,-6) m =- 33. (-8,) m is undefined. 3. (-3, -) m = (-5.) m = 36. (0, -9) m = 37. (7. ) m = 38. (-, -6) m = In Eercises 39-50, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Point 39. (0, ) m = 3 0. (0, 0) m =-. (-3,6) m =. (0,0) m = 3. (,0) Slope m =-}. (, -5) m = ~ 5. (6, -) m is undefined. 6. (-0,) m is undefined. 7. (,~) m = 0 8. (-~,~) m = 0 9. (-5.,.8) m = (.3,- 8.5) m = In Eercises 5-6, find the slope-intercept form of the equation of the line passing through the points. Sketch the line. 5. (5, -), (- 5, 5) 53. (-8,), (-8,7) 55. (, ~), (~, ~) 57. (-fa, -~), (-&, -~) 59. (,0.6), (-, -0.6) 60. (- 8,0.6), (, -.) 6. (, -), (t, -) 6. (t. ), (-6, ) 63. G, -8), G, ) 6. n.s, ), (i.s, 0.) 5 5. (,3), (-, -) 5. (-,), (6,) 56. (,), (6, -~) 58. G,~),(-. n In Eercises 65-68, determine whether the lines L, and L passing through the pairs of points are parallel, perpendicular, or neither. 65. L : (0, -), (5, 9) L : (0,3), (, ) 67. L : (3,6), (-6,0) t.; (0, -), (5, i) 66. L : (, -), (,5) L : (,3), (5, - 5) 68. L : (, 8), (-, ) Lz: (3, - 5), (-, t) In Eercises 69-78, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. 69. (, ) Point 70. (-3,) 7. (-~, n 7. G,~) 73. (-,0) 7. (, ) 75. (,5) 76. (-5,) 77. (.5,6.8) 78. (-3.9, -.) Line - = 3 +=7 3 + = = 0 = -3 = = = -= 6 + = 9 In Eercises 79-8, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts (a, 0) and (0, b) is a + b =, a "* 0, b "* O intercept: (,0) -intercept: (0, 3) 8. -intercept: (-~, 0) -intercept: (0, -n 83. Point on line: (,) -intercept: (c, 0) -intercept: (0, c), c*-o 80. -intercept: (- 3, 0) -intercept: (0, ).. () 8. -mtercept: 3' 0 -intercept: (0, ) 8. Point on line: (- 3, ) -intercept: (d, 0) -intercept: (0, d), d*-o '"~ Graphical Interpretation In Eercises 85-88, identif an relationships that eist among the lines, and then use a graphing utilit to graph the three equations in the same viewing window. Adjust the viewing window so that the. slope appears visuall correct-that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. 85. (a) = 86. (a) = ~ (b) = (b) = -~ (c) = ~ (c) = ~ +

6 36 Chapter Functions and Their Graphs 87. (a) = -~ 88. (a) = - 8 (b) = -! + 3 (b) = + (c) = - (c) = Net Profit The graph shows the net profits (in millions) for Applebee's International, Inc. for the ears 99 through 003. (Source: Applebee's International, Inc.) In Eercises 89-9, find a relationship between and such that (, ) is equidistant (the same distance) from the two points. 89. (, -), (,3) 90. (6.5), (, - 8) 9. (3, ~), (-7, ) 9. (-t -), G,~) 93. Sales The following are the slopes of lines representing annual sales in terms of time in ears. Use the slopes to interpret an change in annual sales for a one-ear increase in time.. (a) The line has a slope of rn = 35. (b) The line has a slope of m = O. (c) The line has a slope of rn = Revenue The following are the slopes of lines representing dail revenues in terms of time in das. Use the slopes to interpret an change in dail revenues for a one-da increase in time. (a) The line has a slope of m = 00. (b) The line has a slope of m = 00. (c) The line has a slope of rn = O. 95. Average Salar The graph shows the average salaries for senior high school principals from 990 through 00. (Source: Educational Research Service) _... "t...;'; Year ( H 99) (a) Use the slopes to determine the ears in which the net profit showed the greatest increase and the least increase. (b) Find the slope of the line segment connecting 99 and 003. the ears (c) Interpret the meaning of the slope in part (b) in the contet of the problem. 97. Road Grade You are driving on a road that has a 6% uphill grade (see figure). This means that the slope of the road is jzo' Approimate the amount of vertical change in our position if ou drive 00 feet. 85,000 ~ 80,000 '"~ = 75,000 "0, 70,000 ;>, 65,000 ~ 60,000 CIl 55, ; (.83,9)- '.~m( ),mm """"', ~ (8, 7.380), '.... ( ). (.6,768).-- (0, 55,7)... (.6.993) (a) Use the slopes to determine the average salar increased Year (0 H 990) the time periods in which the greatest and the least. (b) Find the slope of the line segment connecting 990 and 00. the ears (c) Interpret the meaning of the slope in part (b) in the contet of the problem. 98. Road Grade From the top of a mountain road, a surveor takes several horizontal measurements and several vertical measurements, as shown in the table ( and are measured in feet) (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that ou think best fits the data. (c) Find an equation for the'line ou sketched in part (b). (d) Interpret the meaning of the slope of the line in part (c) in the contet of the problem. (e) The surveor needs to put up a road sign that indicates the steepness of the road. For instance, a surveor would put up a sign that states "8% grade" on a road with a downhill grade that has a slope of -,&. What should the sign state for the road in this problem?

7 Section.3 Linear Equations intwo Variables 37 Rate of Change In Eercises 99 and 00, ou are given the dollar value of a product in 005 and the rate at which the value of the product is epected to change during the net 5 ears. Use this information to write a linear equation that gives the dollar value V of the product in terms of the ear t. (Let t :;;:5 represent 005.) 005 Value Rate 99. $ $56 $5 decrease per ear $.50 increase per ear Graphical Interpretation In Eercises 0-0, match the description of the situation with its graph. Also determine the slope and -intercept of each graph and interpret the slope and -intercept in the contet of the situation. [The graphs are labeled (a), (b), (c),and (d).] (a) (b) (c) 0 30 L if-+++-h H-+++-t-H~ if-+++-h- +t ~ (d) A person is paing $0 per week to a friend to repa a $00 loan. 0. An emploee is paid $8.50 per hour plus $ for each unit produced per hour. 03. A sales representative receives $30 per da for food plus $0.3 for each mile traveled. 0. A computer that was purchased for $750 depreciates $00 per ear. 05. Cash Flow per Share The cash flow per share for the Timberland Co. was $0.8 in 995 and $.0 in 003. Write a linear equation that gives the cash flow per share in terms of the ear. Let t = 5 represent 995. Then predict the cash flows for the ears 008 and 00. (Source: The Timberland Co.) 06. Number of Stores In 999 there were 076 J.e. Penne stores and in 003 there were 078 stores. Write a linear equation that gives the number of stores in terms of the ear. Let t = 9 represent 999. Then predict the numbers of stores for the ears 008 and 00. Are our answers reasonable? Eplain. (Source: J.C. Penne Co.) 07. Depreciation A sub shop purchases a used pizza oven for $875. After 5 ears, the oven will have to be replaced. Write a linear equation giving the value V of the equipment during the 5 ears it will be in use. 08. Depredation A school district purchases a high-volume printer, copier, and scanner for $5,000. After 0 ears, the equipment will have to be replaced. Its value at that time is epected to be $000. Write a linear equation giving the value V of the equipment during the 0 ears it will be in use. 09. College Enrollment The Pennslvania State Universit had enrollments of 0,57 students in 000 and,89 students in 00 at its main campus in Universit Park, Pennslvania. (Source: Penn State Fact Book) (a) Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the ear t, where t = 0 corresponds to 000. the enroll- (b) Use our model from part (a) to predict ments in 008 and 00. (c) What is the slope of our model? Eplain its meaning in the contet of the situation. no. College Enrollment The Universit of Florida had enrollments of 36,53 students in 990 and 8,673 students in 003. (Source: Universit of Florida) (a) What was the average annual change in enrollment from 990 to 003? (b) Use the average annual change in enrollment to estimate the enrollments in 99, 998, and 00. (c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the contet of the problem. (d) Using the results of parts (a)-(c), write a short paragraph discussing the concepts of slope and average rate of change.. Sales A discount outlet is offering a 5% discount on all items. Write a linear equation giving the sale price S for an item with a list price L.. Hourl Wage A microchip manufacturer pas its assembl line workers $.50 per hour. In addition, workers receive a piecework rate of $0.75 per unit produced. Write a linear equation for the hourl wage W in terms of the number of units produced per hour. 3. Cost, Revenue, and Profit A roofing' contractor purchases a shingle deliver truck with a shingle elevator for $36,500. The vehicle requires an average ependiture of $5.5 per hour for fuel and maintenance, and the operator is paid $.50 per hour. (a) Write a linear equation giving the total cost C of operating this equipment for t hours. (Include the purchase cost of the equipment.)

8 38 Chapter Functions and Their Graphs (b) Assuming that customers are charged $7 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) Use the result of part (c) to find the break-even point-that is, the number of hours this equipment must be used to ield a profit of 0 dollars.. Rental Demand A real estate office handles an apartment comple with 50 units. When the rent per unit is $580 per month, all 50 units are occupied. However, when the rent is $65 per month, the average number of occupied units drops to 7. Assume that the relationship between the monthl rent p and the demand is linear. (a) Write the equation of the line giving the demand in terms of the rent p. (b) Use this equation to predict the number of units occupied when the rent is $655. (c) Predict the number of units occupied when the rent is $ Geometr The length and width of a rectangular garden are 5 meters and 0 meters, respectivel. A walkwa of width surrounds the garden. (a) Draw a diagram that gives a visual representation the problem. (b) Write the equation for the perimeter of the walkwa in terms of. GE (c) Use a graphing utilit to graph the equation for the perimeter. ~ (d) Determine the slope of the graph in part (c). For each additional one-meter increase in the width of the walkwa, determine the increase in its perimeter. 6. Monthl Salar A pharmaceutical salesperson receives a monthl salar of $500 plus a commission of 7% of sales. Write a linear equation for the salesperson's monthl wage. W in terms of monthl sales S. 7. Business Costs A sales representative of a compan using a personal car receives $0 per da for lodging and meals plus $0.38 per mile driven. Write a linear equation giving the dail cost C to the compan in terms of, the number of miles driven. 8. Sports The median salaries (in thousands of dollars) for plaers on the Los Angeles Dodgers from 996 to 003 are shown in the scatter plot. Find the equation of the line that ou think best fits these data. (Let represent the median salar and let t represent the ear, with t = 6 corresponding to 996.) (Source: USA TODAY) of FIGURE FOR , _. r~... ~~,\".... I-K _ _ Year (6 H 996) Model It 9. Data Analsis: Cell Phone Suscribers The numbers of cellular phone suscribers (in millions) in the United States from 990 through 00, where is the ear, are shown as data points (, ). (Source: Cellular Telecommunications & Internet Association) (990, (99, (99, (993, (99, (995, (996, (997, (998, (999, (000, (00, (00, (a) Sketch a scatter plot of the data. Let = 0 correspond to 990. (b) Use a straightedge to sketch the line that ou think best fits the data. (c) Find the equation of the line from part (b). Eplain the procedure ou used. (d) Write a short paragraph eplaining the meanings of the slope and -intercept of the line in terms of the data. (e) Compare the values obtained using our model with the actual values. (f) 5.3) 7.6).0) 6.0).) 33.8).0) 55.3) 69.) 86.0) 09.5) 8.) 0.8) Use our model to estimate the number of cellular phone suscribers in 008.

9 Section.3 Linear Equations in Two Variables Data Analsis: Average Scores An instructor gives regular 0-point quizzes and loo-point eams in an algebra course. Average scores for si students, given as data points (, ) where is the average quiz score and is the average test score, are (8,87), (0,55), (9,96), (6,79), (3,76), and (5,8). (Note: There are man correct answers for parts '(b)-(d).] (a) Sketch a scatter plot of the data. (b) Use a straightedge best fits the data. to sketch the line that ou think (c) Find an equation for the line ou sketched in part (b). (d) Use the equation in. part (c) to estimate the average test score for a person with an average quiz score of 7. (e) The instructor adds points to the average test score of each student in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. Snthesis True or False? In Eercises and, determine whether the statement is true or false. Justif our answer.. A line with a slope of - ~ is steeper than a line with a slope of -~.. The line through (- 8,) and (-,) and the line through (0, - ) and (-7, 7) are parallel. 3. Eplain how ou could show that the points A (, 3), B (, 9), and C (,3) are the vertices of a right triangle.. Eplain wh the slope of a vertical line is said to be undefined. 5. With the information shown in the graphs, is it possible to determine the slope of each line? Is it possible that the lines could have the same slope? Eplain. (a) (b) 8. Think About It Is it possible for two lines with positive slopes to be perpendicular? Eplain. Skills Review In Eercises 9-3, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (e) (d) Y = = 8 - J 3. =! = [ + - In Eercises 33-38, find all the solutions of the equation. Check our solution(s) in the original equation. ~33. -7(3 - ) = ( - ) = = X = ~ + 5 = v + 5 = The slopes of two lines are - and~. Which is steeper? Eplain. 7. The value V of a molding machine t ears after it is purchased is V = -000t + 58,500, 0:::; t :::; Make a Decision To work an etended application analzing the numbers of bachelor's degrees earned b women in the United States from 985 to 00, visit this tet's website at college.hmco.com. (Data Source: U.S. Census Bureau) Eplain what the V-intercept and slope measure.

10 8 Chapter Functions and Their Graphs [~'_i_.~.lj_e e_r_ci_s_e_s ~) VOCABULARY CHECK: Fill in the blanks.. A relation that assigns to each element from a set of inputs, or, eactl one element in a set of outputs, or, is called a _. Functions are commonl represented in four different was,, and _ 3. For an equation that represents as a function of, the set of all values taken on b the variable is the domain, and the set of all values taken on b the variable is the range.. The function given b j() = { -, <o X +, ~ is an eample.of a function. 5. If the domain of the function j is not given, then the set of values of the independent variable for which the epression is defined is called the _. j( + h) - j() 6. In calculus, one of the basic definitions is that of a, given b h,h * 0. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises -, is the relationship a function? 6.. Domain. Domain Range Range 5 -~~~ 8 3. Domain Range. Domain Range (Year) (Number of National <: Cubs North Atlantic League Pirates tropical storms Dodgers and hurricanes) American ~ League Orioles Yankees Twins In Eercises 5-8, does the table describe a function? Eplain our reasoning. 5. Input value - Output value -8 - a Input value Output value - Input value 0 7 Output value Input value. " Output value In Eercises 9 and 0, which sets of ordered pairs represent functions from A to B? Eplain. 9. A = {O,,, 3} and B = {, -,0,, } (a) {CO, ),(, ), (,0), (3, n (b) {CO, -), (,), (, ), (3,0), (, I)} (c) {CO,0), (,0), (, 0), (3, On (d) {CO, ), (3,0), (, l)} 0. A = {a, b, c} and B = {O,,, 3} (a) {(a, ), (c, ), (c, 3), (b; 3)} (b) {(a, ), (b, ), (c, 3)} (c) {(l, a), (0, a), (, c), (3, bp (d) {(e, 0), (b, 0), (a, 3)} I

11 Section. Functions 9 Circulation of Newspapers In Eercises and, use the graph, which shows the. circulation (in millions) of dail newspapers in the United States. (Source: Editor & Puhlisher sa c:.s p 0..S 30 '-" c:: 0 0 ~ '" OJ.. U 0 Compan) ~ GI. e e If Year. Is the circulation of morning newspapers a function of the ear? Is the circulation of evening newspapers a function of the ear? Eplain.. Let f() represent the circulation of evening newspapers in ear. Find f(998). In Eercises 3, determine whether the equation represents as a function of. 3. X + = 5. + = = 9. = X -. = - l 3. =. = 6. + = 8. ( - ) + = 0. = J+s. Il = -. = -75 In Eercises 5-38, evaluate the function at each specified value of the independent variable and simplif. 5. f() = - 3 (a) f(l) (b) f(-3) (c) f( - ) 6. g() = 7-3 (a) g(o) (b) gg) (c) g(s + ) 7. VCr) = 37Tr3 (a) V(3) (b) vg) (c) V(r) 8. h(t) = t - t (a) h() (b) h(.5) (c) he + ) 9. fe) = 3 -.J (a) f() (b) f(0.5) (c) f( ) 30. f() =.JX+8 + (a) f(-8) (b) f(l) (c)'f(-8) 3. q() = X _ 9 (a) q(o) (b) q(3) (c) q( + 3) t q(t) = - - t (a) q() (b) q(o) (c) q(-) 33. f() = M (a) f() (b) f() (c) f( - ) 3. f() = Il + (a) f() (b) f( ) (c) f( ) () {+l, <O 35. f = +, ~O (a) f(-) (b) f(o) (c) f() {X +, ~ 36. f() = +, > (a) f() (b) f(l) (c) f() r - I < f() =, -~~ r - S s X, > (a) f( ) (b) f( -~) (c) f(3) 38. f() = 0, ' < < X +, > (a) n-» (b) f() (c) fe-i) In Eercises 39 -, complete the table. 39. f() = fe) 0. g() = ~ g(). h(t) = ~It + 3 t h(t). f(s) = Is - s s 0 f(s)

12 50 Chapter Functions and Their Graphs :O:: 0 >o - 0 f() 9 - X,. f() = { ẋ - 3', < 3 ~ f() In Eercises 5-5, find all real values of such that () = O. 5. f() = f() = f() = X f() = 3 - X 6. f() = f() = ~ X 50. f() = X f() = 3 - Xl - + In Eercises 53-56, find the value(s) of for which () = g(). 53. f() = X + +, g() = f() = X - X, g() = 55. f() = $ +, g() = f() =.J -, g() = - In Eercises 57-70, find the domain of the function. 57. f() = h(t) = - t 6. g() = J-=--iO 63. f() = -Y - X g() = f(s) = -Js - s f() =.J 58. g() = s() = f(t) = t + 6. f() = -YX he) = -- X f() =.J f() = ~ <r : - 9 In Eercises 7-7, assume that the domain of f is the set A == {-, -,0,,}. Determine the set of ordered pairs that represents the function f. 7. I() = 7. I() = f() = Il + 7. f() = I + Eploration In Eercises 75-78, match the data with one of the following functions () = e, g() = e", he) = c.,jjt, and r() =. and determine the value of the constant e that will make the function fit the data in the table Y ' 0 ' Y -8-3 Undef ji In Eercises 79-86, find the difference quotient and simplif our answer., ( + h) - () 79. f() = X - X +, h' h '* f() = 5 - X, f(5 + h~ - f(5), h '* 0 f( + h) - f() 8. f() = 3 + 3, h' h '* 0 f( + h) - f() 8. f() = -, h' h '* 0 g() - g(3) 83. g() = ', '* 3-3. f(t) - f(l) 8. f(t) = --,, t '* t t- 85. f() =.JS;, f() - f(5) '*- -5, ) 86. f() = / 3 +, f(x; = {(8). '* Geometr Write the area A of a square as a function of its perimeter P. 88. Geometr Write the area A of a circle as a function of its circumference C. The smbol Ii indicates an eample or eercise that highlights algebraic techniques specificall used in calculus.

13 Section.5 Analzing Graphs of Functions ] Eercises VOCABULARY CHECK: Fill in the blanks.. The graph of a function f is the collection of or (,f()) such that is in the domain of f. The is used to determine whether the graph of an equation is a function of in terms of. 3. The of a function f are the values of for which f() = o.. A function f is on an interval if, for an Xl and ~ in the interval, Xl < X z implies f( l) > f() 5. A function value f(a) is a relative of f if there eists an interval (Xl' z) containing a such that Xl < X < Xz implies f(a) ~ f(). 6. The between an two points (l,f(j) and (Xz,f(z» is the slope of the line through the two points, and this line is called the line. 7. A function f is if for the each X in the domain of f, f( - ) = - f(). 8. A function f is if its graph is smmetric with respect to the -ais. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises -, use the graph of the function to find the domain and range of f. (b) f() (d) f() 8. (a) f() (c) f(3) (b) f(l) (d) f(-l).. 6 -t--h'r-i--l--'h IY=~ 0---'--- In Eercises 9-, use the Vertical Line Test to determine whether is a function of.to print an enlarged cop of the. graph, go to the website 9. Y = ~z 0. = * 3 In Eercises 5-8; use the graph of the function to find the indicated function values. (b) f(-i) (d) f(l) (b) f() (d) f(l) -t--hf---f' l z =. XZ + Y z = 5 +-t--t-tf-h-r t--hf-+-t-'h-t , -+-+-f-+-+-l-+-f---l-+- +t h

14 6 Chapter Functions and Their Graphs 3. X = -. = I hi"+-+-+-t-+-b- 33. f() = 3 -, f() = ~ A 6 (0,) Y b - -6 (, ) - In Eercises 5, find the zeros of the function algebraicall. 5. f() = X f() = 9 --;- 9. f() = ~3-0. f() = 3 - X f() = X + 6. f() = f() = -J -. f() =.J f() = f() = X - ~: + ~ In Eercises 5-30, (al use a graphing utilit to graph the function and find the zeros of the function and (b) verif our results from part (a) algebraicall f() = f() = ( - 7) 7. f() =.J + 8. f() =.J f() = f() = X In Eercises 3-38, determine the intervals over which the function is increasing, decreasing, or constant. 3. f() = ~ 3. f() = X - { + 3 :S;O 35. f() = 3, 0< :S; +, > 6! 36. f() = {~~~ L -+-t-+-+-t-+.h X:S; - > f() = I + + I f() = X + X ; + (0, ) - (7"\' -++--I-+--I-+--f--f--l

15 Section.5 Analzing Graphs of Functions 63 ~ In Eercises 39-8, (a) use a graphing utilit to graph the function and visuall determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verif whether the function is increasing, decreasing, or constant over the intervals ou identified in part (a). 39. j() = 3 S. g(s) = " 3. Jet) = - t 5. j() =.J"=X 7. j() = 3 / 0. g() =. he) = X -. j() = j() = J+3 8. j() = X / 3 ~ In Eercises 9-5, use a graphing utilit to graph the function and approimate (to two decimal places) an relative minimum or relative maimum values. 9. j() = ( - )( + ) 5. f() = - X j() = ( - )( + 3) 5. f() = X f() = f() = J In Eercises 77-80, write the height h of the rectangle as a function of h;:' 3 3 (,) :(3,), I, X X I X I 3 (8,) lfi In Eercises 8-8, write the length L of the rectangle as a function of. t«in Eercises 55-6, graph the function and determine the interval(s) for which (() ~ o. 55. f() = f() = X f() =.J'-=l 6. f() = -( + Il) 56. f() = f() = f() = -F+ 6. f() = H + Il) jji In Eercises 63-70, find the average rate of change of the function from Xl to Function 63. f() = f() = f() = X f() = j() = 3 - '3-68. f() = f() = - ~ f() = - -F+l + 3 -Values Xl = 0, = 3 Xl = 0, = 3 Xl =, = 5 Xl = l,' = 5 Xl =, = 3 Xl =, = 6 Xl = 3, = Xl = 3, = 8 In Eercises 7-76,determine whether the function is even, odd, or neither. Then describe the smmetr. 7. f() = g() = f(t) = t + t he) = f() = -JI=X 76. g(s) = s /3 -+~+-= \ _ 3 ~ 85. Electronics The number of lumens (time rate of flow of light) L from a fluorescent lamp can be approimated b the model L = , 0 ~ X ~ 90 where X is the wattage of the lamp. (a) Use a graphing utilit to graph the function. (b) Use the graph from part (a) to estimate necessar to obtain 000 lumens. the wattage