5g on a rectangular coordinate system.

Transcription

1 Chapter : Functions and Graphs Section.1 A Two-Dimensional Coordinate Sstem and Graphs Cartesian Coordinate Sstems 1. Plot the point C b, 5 on a rectanular coordinate sstem. C C C C. Identif the point at 5,. b M P O N O M P N. Determine the quadrant in which b, is located so that the conditions are satisfied. 0 and 5 Quadrant I Quadrant II Quadrant III Quadrant IV. Determine the quadrant in which the point b, 1 is located, or the ais on which it is located. Quadrant I Quadrant II -ais -ais Copriht Houhton Mifflin Compan. All rihts reserved. 61

2 Chapter Functions and Graphs The Distance and Midpoint Formulas 5. Find the distance between the two plotted points. b 7, b8, 1 6. Find the possible values of if the points 8, 11 b and b 7, are a distance d 9 apart., 15, 15 9, 1 9, 1 7. Find the midpoint of QR. b b, 1 b, 1 b, 1, 1 Q R 8. The midsement of a trapezoid connects the midpoints of the two non-parallel sides. Find the midpoint of the midsement of the trapezoid if A e, f,,,, 0. A B b B b f and C b h (0, 0) C b bf + + h, f be+ + h, f b+ h, f e+, f +f 6 Copriht Houhton Mifflin Compan. All rihts reserved.

3 Section.1 A Two-Dimensional Coordinate Sstem and Graphs Graph of an Equation 9. Which is the raph of 7? Graph:. 9 Copriht Houhton Mifflin Compan. All rihts reserved. 6

4 Chapter Functions and Graphs Graph: Identif the raph of the quadratic function and list an -intercepts. f 1, 0,, 0 b b b, 0 b, 0 1, 0,, 0 b b 6 Copriht Houhton Mifflin Compan. All rihts reserved.

5 Section.1 A Two-Dimensional Coordinate Sstem and Graphs Intercepts 1. Identif the raph of the function and find the -intercept. f intercept: 0, 5 b -intercept: 0, 5 b -intercept: 0, 5 b -intercept: 0, 5 b 1. Identif the - and -intercepts. -intercept: 0, b b 0 b, 0 b6 0 b, b 0 b, 0 b6 0 -intercepts:,,,, -intercept: 0 -intercepts:,,,, b b b b, b 0 b, 0 b6 0 b, -intercepts:, 0,, 0, 6, 0 -intercept: 0 -intercepts:,,,, -intercept: 0 Copriht Houhton Mifflin Compan. All rihts reserved. 65

6 Chapter Functions and Graphs 15. Identif the - and -intercepts of the raph of the equation. 6 b b, b b b b b,, b,, b, -intercepts: 0, 0, 6, 0, 6, 0, 8, 0, 8, 0 -intercept: 0 0 -intercepts: intercepts: none b b b b, b, b8, 0, b 8, 0 b, -intercepts: 0, 0, 6, 0, 6, 0 -intercept: 0 0 -intercepts: 0, 0 -intercept: Find the - and -intercepts of the raph of the equation. 1 6, none 0, 6 none,, 0 Circles, Their Equations, and Their Graphs 17. Find the standard form of the equation of the specified circle. Center:, 5 ; b Radius: 5 b b 7 b b 5 b b 9 None of these 18. Find the center and radius of the circle with the iven equation b b b, ; 9 b, ; b, ; b, ; b and D b, are endpoints of a diameter. b b b 1 b 6 b b none of these 19. Find the equation of a circle where C 6, 5 0. Find the standard form of the equation of the specified circle. Center:, ; b Radius: b b b b b b b b 66 Copriht Houhton Mifflin Compan. All rihts reserved.

7 Section. Introduction to Functions Section. Introduction to Functions Relations 1. Choose a table that represents each relation shown Choose a table that represents each relation shown Copriht Houhton Mifflin Compan. All rihts reserved. 67

8 Chapter Functions and Graphs b b b b. Construct a model of the relation m 8, 6, 7, 1, 8,,, 9r. Determine whether the relation is a function Function Function Not a function Not a function. Find the domain and rane of the relation l l l l D 0, 7, 7, 5, 5 R, 0,, 8 D, 0,, 8 q q q q R 0, 7, 7, 5, 5 l l l l D, 0, R 0, 7, 5, 5 D 0, 7, 5, 5 R, 0, q q q q Functions b represents as a function of? mb, b, b, b r mb,, b,, b, r l q mb,, b,, b, r 5. Which set of ordered pairs,,,,,,,, 68 Copriht Houhton Mifflin Compan. All rihts reserved.

9 Section. Introduction to Functions 6. Which of the followin relations is not a function? mb b b b r,,, 8, 5, 7, 7, 6 mb b b b r,, 6,, 5, 7, 7, 7. Which of the followin relations is not a function? mb b b b r 7, 8, 1,, 9,,, Which of the followin relations is also a function? Functional Notation Evaluate the function at the specified value(s) of the independent variable and simplif. 9. f ; f b 6 0. f 9 ; f b ; n 5 b n n n 5 n n 5 n 8 n 0 n n n 0 Copriht Houhton Mifflin Compan. All rihts reserved. 69

10 Chapter Functions and Graphs. Evaluate the function and simplif the results. R f if < 1 S 7 8 if 1 T b 1 fb0 b 5 fb 05. a. f b. c. f d. a. 15 b. 0 a. b. 0 c. d. 11 c. 1 d. a. 15 b. 7 a. b. 7 c. d c. d. Identifin Functions. Which equation defines as a function of? 6. Identif the raph that does not represent a function. 5. Which of the followin is not a polnomial function? f 5 5 f 5 5 f 1 5 f 70 Copriht Houhton Mifflin Compan. All rihts reserved.

11 Section. Introduction to Functions 6. Determine which relation is a function with the correct domain and rane l l domain: rane: 9 7 7, 6, 5, 9, l l domain: rane: q 9 7 q q, 9, 7 6, 7 q mb b b b r domain: l6, 8, 1q rane: l7,, 1, 9q 6, 7, 6,, 8, 1, 1, l l domain: rane: 9 8, 7 q, 9, q Graphs of Functions 7. Determine the intervals over which the function is increasin, decreasin or constant. f b b b 0 b0 b b b b b 0 b0 b b increasin on, ; decreasin on, increasin on, and, ; decreasin on, and, increasin on, and, ; decreasin on, and, increasin on, ; decreasin on, 1 Copriht Houhton Mifflin Compan. All rihts reserved. 71

12 Chapter Functions and Graphs 8. Use the Vertical Line Test to determine which raph does not represent as a function of. 9. Graph: f R S T 1, 0, 0 7 Copriht Houhton Mifflin Compan. All rihts reserved.

13 Section. Introduction to Functions 0. Determine which of the followin are one-to-one functions. i. ii. iii. iv. i and iv onl i, ii, and iv onl iv onl ii and iv onl The Greatest Inteer Function 1. Identif the raph of the function. (The smbol represents the reatest inteer function.) L N O Q L f N M 1 M O Q PP Copriht Houhton Mifflin Compan. All rihts reserved. 7

14 Chapter Functions and Graphs. A team-buildin incentive proram for workers in an athletic shoe factor ives them a $,000 bonus to share for ever 5,000 pairs of shoes completed. The basic production cost of a pair of hih-qualit shoes is $5 before bonuses. Usin to denote the reatest inteer function, find the equation that ives the cost, C b, for producin hih-qualit shoes. C C L L NM NM L L NM NM,000 5,000, ,000 OO QP P Q OO QP P Q P 5 C L L NM NM 5,000,000 P 5 C 5,000 OO QP P Q L L NM NM P 5, 999 5,000. A tai compan chares $5.5 for a ride of up to miles, plus $1.0 for each additional mile or fraction thereof. Usin to denote the reatest inteer function, find the equation for the cost, C that miles. C C C C OO QP QP b, of a ride of miles, iven. A wireless telephone compan that does not distinuish between local and lon-distance calls chares $6.95 per month and a per minute chare for use beond 0 minutes. Fractions of a minute count as whole minutes. Usin to denote the reatest inteer function, the cost, C R C S T 6. 95, , 0 Find the cost for a person who uses 7.7 minutes in one month. b, for minutes of use can be calculated usin the followin: $ $ $ $ Applications 5. A music club membership costs $1.00 to join and $6 per CD. Find an equation that shows, the cost for joinin and buin CDs b 7 6b 1 7 6b 1 b Debbie is 6 ears old, which is 1 less than times Minnie s ae. Which equation could be used to solve for Minnie s ae? How old is Minnie? 1 6; 6 r 1 6; 1 r 1 6;. r 1 6;. 7 r 7. Ms. Espinoza put awa $69 for a part fund. Each time her class did well on a bi project, $5 was deducted from the fund so that she could bu supplies for a class part. Choose the equation that represents the value remainin in her account after withdrawals from the fund. Find the value remainin in the fund after 9 parties. V 69 5; $ V 5 69; $ V 69 5; $ 15 V 69 5; $ 10 7 Copriht Houhton Mifflin Compan. All rihts reserved.

15 Section. Introduction to Functions 8. A sweatshirt printin business spends $50 on equipment and supplies for each new batch of sweatshirts. In addition to these one time chares, the cost of printin each sweatshirt is $.5. Let be the number of sweatshirts printed. a. Write the total cost C as a function of the total number of sweatshirts printed. b. Write the averae cost per unit, C C, as a function of. c. Find the averae cost of printin 9000 sweatshirts. a. C b. C c. $.50 a. C b. C c. $.0 a. C b. C c. $5.0 a. C b. C c. $.50 Section. Linear Functions Slopes of Lines 9. Find the slope of the line passin throuh the pair of points., 8, 8 8 b b, Undefined 50. Find the slope of the line passin throuh the iven points. F 1 I F, and HG KJ I, 1 HG 5 KJ Which is the raph of the line with a -intercept of and slope of? 5. Determine the value of r so the line that passes throuh each pair of points has the iven slope. b,,, r ; b m (e) None of these Copriht Houhton Mifflin Compan. All rihts reserved. 75

16 Chapter Functions and Graphs Find the Equation of a Line 5. Which shows the equation of a line, in slope-intercept form, that passes throuh the point, 5 b with slope? Determine the equation of the line that has the iven slope and contains the iven point. m 1, 5, b Determine the slope-intercept form of the equation of the line Given f 6 5, find the value of which ives f 9., 8. 9, 18, 18., Copriht Houhton Mifflin Compan. All rihts reserved.

17 Section. Linear Functions Applications 57. A sloth travels at a rate of 0. meter per minute. Let t be the number of minutes spent travelin. a. Find the distance d traveled b the sloth as a function of t. b. Find the distance traveled b the sloth in minutes. c. Construct a raph illustratin the distance traveled b the sloth versus time. 0. a. dt t b. 0.5 m c. d a. dt b. 0.9 m c. d 0. t 1 0 t 0 t a. dt 0. b. 0.8 m c. d t a. dt 0. b. 0.6 m c. d d 0 t 0 t 58. Case McDonald plans to decorate vases to sell at a crafts fair. The decorations for all of the vases cost $7.50 and the vases cost $8.50 each. a. Write a function epressin the cost, C b of the project in terms of the number of vases decorated,. b. Determine the cost of decoratin 60 vases. c. How man vases can be decorated with a budet of $6.50? a. C b. $ c. 70 a. C 850. b. $5.00 c. 71 a. C b. $ c. 7 a. C b. $56.00 c A monthl phone bill, b, in dollars, consists of a $ service fee plus $0.1 per minute, m, of lon distance calls. Find an equation that relates the total cost to the number of minutes used. Then plot the total costs for usin 8, 8, 5, and 58 minutes. b 01. m b 01. m b 0. 1 m 0. 1 bm Copriht Houhton Mifflin Compan. All rihts reserved. 77

18 Chapter Functions and Graphs 60. In 199 the averae price of a house in a small town was $97,000. B 1998, the averae price of a home was $9,000. Find a linear model for the price of such a house, P, if t 0 represents the ear 199. P 000t 97,000 P 9, t P 9,000 1,000t P 1,000t 97,000 (e) None of these Parallel and Perpendicular Lines 61. Determine whether the raphs of the equations are parallel, perpendicular, or neither Parallel Perpendicular Neither 6. Which best describes the relationship between the lines with equations 7 5 and 8 1 0? Perpendicular Parallel Same line Neither parallel nor perpendicular 6. A woman standin in the middle of a lare field was spinnin a rock attached at the end of a rope, counterclockwise above her head. When she let o of the rope, the rock traveled alon a linear path perpendicular to the radius created b the woman s position and the point of the rock s release. The rock traveled directl over a wall in the field. If the field is represented on a coordinate ais sstem with the woman at the oriin, the point of release at b6,, and the wall represented b the line 8, at what point was the ball directl over the wall? F HG I F HG 7 1, 5, KJ 5, KJ 1 I F HG I KJ F HG 7 1, 6. A bo is plain with a ball suspended from a strin, spinnin it counterclockwise about the oriin. He accidentall lets o of the ball and it hits the trunk of a tree in the adjoinin backard, after travellin alon a path perpendicular to its radius of oscillation and the point at which it was released. If the point at which the ball was released is P b, 8, and the equation representin the tree is 6, what is the point of contact of the ball and the trunk of the tree? F HG I KJ, 0 F HG 8, Section. Quadratic Functions Verte of a Parabola 65. Find the verte of the raph of the function. f I KJ F HG, 1 I KJ F HG I KJ 9 8, b b b b, b, b,, I KJ Which is the standard form of the function f 6 15? b b f 6 b b f b b f 6 9 b b f Copriht Houhton Mifflin Compan. All rihts reserved.

19 Section. Quadratic Functions 67. Write the equation of the quadratic function in standard form and find the verte of the raph. b b Verte: b 5, 71 b Verte: b71, 5 f 0 f 5 71 f 5 71 b b b b b b f 5 71 Verte: 5, 71 f 5 71 Verte: 71, Write the equation in standard form. Then raph the equation. b 1 5 b 1 b 1 b 1 5 Maimum and Minimum of a Quadratic Function 69. Find the domain and rane of the function. f 5 b b b b Domain: b, ; Rane: b Domain: ; Rane:, Domain: ; Rane:, Domain:, ; Rane: State whether the function has a minimum or a maimum value and find the value. 70. fb minimum: maimum: maimum: minimum: 71. f maimum: 1 minimum: minimum: 1 maimum: Copriht Houhton Mifflin Compan. All rihts reserved. 79

20 Chapter Functions and Graphs 7. Use a calculator to find the maimum or minimum value of the parabola. Applications 5 min at 5 ma at 5 ma at 5 ma at 6 (e) min at 5 7. Find the maimum revenue for the revenue function R ,01, , Moe estimates that the cost, C, to his compan (in thousands of dollars) of producin boes of his computer chips can be modeled b C 60, How man boes should Moe s compan produce to pa the minimum cost per bo? , (e) None of these 75. The area of a rectanle with a perimeter of 6 units is iven b A b where represents the width of the rectanle. A farmer has available 6 ards of fencin and wishes to enclose a rectanular area. Use a raphin utilit to raph the equation for the area of the rectanle and find the value of that ives the larest area. d 1 d 16 d 11 d 76. Macro Manufacturin estimates that its weekl profit, P, in hundreds of dollars, can be approimated b the formula P 5 0, where is the number of units produced per week, in thousands. How man units should the compan produce per week to earn the maimum profit? Find the maimum weekl profit. 00 units, $ units, $800 units, $800 units, $7800 Section.5 Properties of Graphs Smmetr 77. Use smmetr tests to verif an of the three smmetries (-ais, -ais, or oriin) the raph suests. The raph is smmetrical with respect to the oriin. The raph is smmetrical with respect to the -ais, the -ais, and the oriin. The raph is smmetrical with respect to the -ais. The raph has none of the three smmetries. 78. Identif the equation whose raph is smmetric with respect to the the oriin Copriht Houhton Mifflin Compan. All rihts reserved.

21 Section.5 Properties of Graphs Which smmetries does the raph of 6 5 possess? -ais, -ais, and the oriin oriin onl -ais onl -ais onl 80. Find the equation whose raph is smmetric with respect to the the oriin Even and Odd Functions 81. Use the raph to determine if the function is even, odd, or neither. Even Odd Neither Copriht Houhton Mifflin Compan. All rihts reserved. 81

22 Chapter Functions and Graphs 8. Which is the raph of a function that is neither even nor odd? 8. One of the followin functions is neither odd nor even. Which one? Which of the followin functions is even? Fb 5 5 fb h Translations of Graphs 85. Use the raph of f to identif the raph of f. 8 Copriht Houhton Mifflin Compan. All rihts reserved.

23 Section.5 Properties of Graphs b. 86. Use transformations of the raph of to determine the raph of 1 9 Copriht Houhton Mifflin Compan. All rihts reserved. 8

24 Chapter Functions and Graphs 87. Usin the raph of f as a uide, raph the function b b Without usin raphin technolo, sketch the parent raph and translate it to obtain the raph of. 8 Copriht Houhton Mifflin Compan. All rihts reserved.

25 Section.5 Properties of Graphs Reflections of Graphs 89. The raph of the function f the reflection of fb in the -ais. 1 is shown below. Find the equation and the raph of the function b which is b F bh G 1I K J 1 Copriht Houhton Mifflin Compan. All rihts reserved. 85

26 Chapter Functions and Graphs 90. Use the raph of fb to identif the raph of fb. 91. Which of these functions is the reflection of f across the -ais? b b b 86 Copriht Houhton Mifflin Compan. All rihts reserved.

27 Section.5 Properties of Graphs 9. The raph of the function f reflected over the -ais. Find the equation of the raph on the riht. is shown on the left below. The raph on the riht is the same raph 0 0 f f b fb f Compressin and Stretchin Graphs 9. Given is the raph of a function l b. Identif the raph of lb. Copriht Houhton Mifflin Compan. All rihts reserved. 87

28 Chapter Functions and Graphs 9. Given f, consider a dilation b a factor of. Find the dilated function and its raph. F H G I K J f 1 F H G I K J f b F H G I K J f 1 f F HG I KJ b b 95. Find the raph of f Copriht Houhton Mifflin Compan. All rihts reserved.

29 Section.5 Properties of Graphs 96. Find an equation and raph for the function that is described b the iven characteristics. The shape of f, with a vertical shrink of b5 5 b 1 5 Section.6 The Alebra of Functions Operations on Functions 97. Given f 7 5 and 9 7,find b f b b 5b 7,,, b 5b 7,, 16, 1 b b. 98. If f and, find f Find f b b for f 1 and Given f 81,and 9, find f b Copriht Houhton Mifflin Compan. All rihts reserved. 89

30 Chapter Functions and Graphs The Difference Quotient 1. Which is the difference quotient of the function fb? b b b h b b h b b h b h h h. Find the difference quotient f b b for the function. f f Find the difference quotient for the function fb. 8 h 8h h h 8h h 8. The position of a movin bod is iven b the formula s t 5t where s is measured in meters and t in seconds. Find the averae velocit for the time interval from t tot 6 seconds. m sec 8 m sec 0 m sec 65 m sec Composition of Functions 5. Find f b b for the pair of functions. f and, 1 b b and bf b for f and 1. b b 1 f 5 bf 5 bf b b 1 f bf 1 bf 5 and, find bf b and fb. 6. Find f f f 7. If f For the pair of functions, evaluate f b. b 8 f, b Copriht Houhton Mifflin Compan. All rihts reserved.

31 Section.7 Modelin Data Usin Reression Section.7 Modelin Data Usin Reression Linear Reression Models 9. Use the reression feature of a raphin utilit to find the least squares reression line that fits these data Earl in the 1900s, an airplane manufacturer was able to increase the time its planes could sta aloft b constantl refinin its techniques. Use the reression feature of a raphin utilit to find the least squares reression line that fits these data. Assume t is the time aloft and is the number of ears after 19. Years after Time aloft (hrs) t t t t A restaurant owner wants to know the demand for the famil special as a function of price. The dail sales for four different prices are listed in the table. Price, Demand, $9. 00 $9. 75 $. 5 $ Use the least squares reression line to estimate the demand for the famil special at $ An auto manufacturer wants to determine the relation between averae speed and miles per allon for her new econom vehicle. The table shows the results from si averae speed tests. Averae Speed (mph), Miles Per Gallon, Use the least squares reression line to estimate the miles per allon for 1 mph Correlation Coefficient and the Coefficient of Determination 11. A radio disc jocke kept track of the number of requests for sons b a certain artist and the time of da the request calls were made. The data is displaed below. requests time of da p.m. p.m. p.m. 5 p.m. 6 p.m. 7 p.m. 8 p.m. 9 p.m. If the data were displaed on a scatter plot, could the correlation of the data be represented b a linear model? If there is a linear correlation, is there a positive or neative correlation? It is a linear model with a positive correlation. It is a linear model with a neative correlation. The data have no reliable correlation. It is not a linear model. Copriht Houhton Mifflin Compan. All rihts reserved. 91

32 Chapter Functions and Graphs 11. A real estate aent believes that the sellin price of a house for a particular town is directl related to the number of square feet the house has. In order to justif his assumption, he took a sample of 8 homes that had been recentl sold, and recorded the price and size. Usin the data collected, compute r, the correlation coefficient. House Size Sellin Price e0 ft j bin thousands Which of the followin raphs has a coefficient of determination close to 1? Copriht Houhton Mifflin Compan. All rihts reserved.

33 Section.7 Modelin Data Usin Reression 116. Do the results of the linear reression shown below seem to indicate a linear correlation between the variables? If the do, is it stron or moderate, positive or neative? If raphed, would the points tend to rise or fall as the value of the variable increases? LinRe a b a b r r There seems to be a moderatel stron positive linear relationship between the variables. There is a stron positive linear relationship between the variables. There is a stron neative linear relationship between the variables. A linear model ma not be appropriate for the data. Quadratic Reression Models Use a raphin utilit with a least squares reression proram to find the least squares reression quadratic for the iven points. b, b,, b, 117., 5 f f b, b0,, b1, 9, b, 8, b, , f f f f 6 1 f f Copriht Houhton Mifflin Compan. All rihts reserved. 9

34 Chapter Functions and Graphs 119. A small manufacturin firm collected the followin data on advertisin ependitures (in thousands of dollars) and total revenue (in thousands of dollars). Advertisin b Total Revenue b Find the quadratic function that will best model the data A bioloist took a count of the the number of miratin waterfowl at a particular lake, and recounted the lake s population of waterfowl on each of the net si weeks. Use the reression capabilities of our raphin utilit to find a quadratic model for the data which epresses the population as a function of, the number of weeks. Week Population P P P P Copriht Houhton Mifflin Compan. All rihts reserved.

35 Chapter : Functions and Graphs (Answer Ke) Section.1 A Two-Dimensional Coordinate Sstem and Graphs Cartesian Coordinate Sstems [1] [] Section. Introduction to Functions Relations [1] [] [] [] Applications [5] [] [] [6] [] [] [7] The Distance and Midpoint Formulas [5] [6] [7] [8] Graph of an Equation [9] [] [11] [1] Intercepts [1] [1] [15] [16] Circles, Their Equations, and Their Graphs [17] [18] [19] [0] Functions [5] [6] [7] [8] Functional Notation [9] [0] [1] [] Identifin Functions [] [] [5] [6] Graphs of Functions [7] [8] [9] [0] The Greatest Inteer Function [1] [8] Section. Linear Functions Slopes of Lines [9] [50] [51] [5] Find the Equation of a Line [5] [5] [55] [56] Applications [57] [58] [59] [60] Parallel and Perpendicular Lines [61] [6] [6] [6] [] Copriht Houhton Mifflin Compan. All rihts reserved. 95

36 Chapter Functions and Graphs Section. Quadratic Functions Verte of a Parabola [65] [66] [67] [68] Maimum and Minimum of a Quadratic Function [69] [70] [71] [7] Applications [7] [7] [75] [76] Section.5 Properties of Graphs Smmetr [77] [78] [79] [80] [87] [88] Reflections of Graphs [89] [90] [91] [9] Compressin and Stretchin Graphs [9] [9] [95] [96] Section.6 The Alebra of Functions Operations on Functions [97] [98] [99] [0] The Difference Quotient [1] [] Section.7 Modelin Data Usin Reression Linear Reression Models [9] [1] [111] [11] Correlation Coefficient and the Coefficient of Determination [11] [11] [115] [116] Quadratic Reression Models [117] [118] [119] [10] Even and Odd Functions [81] [8] [8] [8] [] [] Composition of Functions [5] [6] Translations of Graphs [85] [86] [7] [8] 96 Copriht Houhton Mifflin Compan. All rihts reserved.