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Solution Manual for Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving, 6th Edition

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    A NSWERS TO THE W ORKSHEETS C HRISTINE V ERITY M ATHEMATICS IN A CTION : A LGEBRAIC , G RAPHICAL , AND T RIGONOMETRIC P ROBLEM S OLVING S IXTH E DITION The Consortium for Foundation Mathematics

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    283 Complete Answers Chapter 1 FUNCTION SENSE Activity 1.1 Key Terms 1. ordered pair 2. dependent 3. variable 4. independent 5. function Practice Exercises 6. The input is x . 7. The output is h ( x ) or y . 8. The function name is h . 9. y equals h of x . 10. The input is 7. 11. The output is 6.931. 12. The function name is g . 13. g of 7 equals 6.931. 14. The input is t . 15. The output is 762. 16. The function name is f . 17. 762 equals f of t . 18. The input is hours. 19. The output is salary. 20. The function name is s . 21. Salary is a function of hours, or salary equals s of hours. 22. ( ) price commission C = 23. (6000, 20) 24. Yes 25. Each input has only one output. 26. No 27. The input 2 has two different outputs. 28. The input 9 is paired with three different outputs. Concept Connections 29. Answers may vary. One example is the output is the wages received. 30. No. The input, number of hours worked, is paired with 4 different outputs, wages received. Activity 1.2 Key Terms 1. Practical range 2. Practical domain Practice Exercises 3. ( ) 3 9 f = 4. ( ) 2.7 19.5 f - = - 5. ( ) 5 6 f c c = - 6. ( ) 4 90.2 g = - 7. ( ) 5.1 226.7 g - = - 8. ( ) 2 8 6.2 13 g b b b = - + + 9. ( ) 6 11 h = 10. ( ) 14.7 11 h - = 11. ( ) 11 h d = 12. ( ) 2 2.5 p - = - 13. ( ) 0.5 10 p = 14. ( ) 5 p a a =

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    284 15. ( ) 7 20.1 r - = 16. ( ) 8.4 15.32 r = - 17. ( ) 4 2.3 r c c = - 18. The amount of reimbursement is 0.51 times the number of miles traveled. 19. miles 20. the amount of reimbursement 21. ( ) 0.51 R m m = 22. R 23. ( ) 74 $37.74 R = 24. Domain: all real numbers 25. Range: all real numbers 26. Practical domain may vary and is probably the real numbers from 0 to 100 miles. 27. Using the domain from Exercise #26, the range is $0 to $51. 28. Domain {–3, 9, 7, 4}; range: {6, 0, 4, 17} Concept Connections 29. The domain of the function is the collection of all possible replacement values for the independent variable. The practical domain is the collection of replacement values of the independent variable that makes practical sense in the context of the situation. 30. A real number is any rational or irrational number. Activity 1.3 Key Terms 1. discrete 2. continuous 3. graphically Practice Exercises 4. 2 0.5 y x = 5. 2 0.5 y x = - 6. 2 0.5 2 y x = + 7. 2 0.5 2 y x = -

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    285 8. 9. {1, 2, 3, 4, 5} 10. {44, 61, 59, 82, 98} 11. Yes 12. Since a particular day of the week is the input, the only input values that are defined are integers from 1 through 5. 13. No 14. The number of student make-up tests cannot be predicted. 15. The employee discount on an item of food is calculated by multiplying the price of the food item by 0.25. 16. If d represents the discount amount on an item priced at p dollars, then d = 0.25 p . 17. Answers may vary. Item price 4 8 12 16 20 Amount of discount 1 2 3 4 5 18. 19. No 20. Since the function is defined for all values between the input values in the table, it is appropriate to connect the data points with a smooth, continuous curve. 21. 2 0.0005 y x = (no graph appears on screen) 22. 3 2 1 0 1 2 3 0.0045 0.002 0.0005 0 0.0005 0.002 0.0045 x y - - - 23. The y values are very small. Therefore, you need to have Ymax = 0.01 and Ymin = –0.01.

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    286 24. 2 0.0005 y x = 25. 2 1000 y x = (no graph appears on screen) 26. 3 2 1 0 1 2 3 9000 4000 1000 0 1000 2000 90000 x y - - - 27. The y values are very large. Therefore, you need to have Ymax = 10,000 and Ymin = –10,000. 28. Concept Connections 29. A discrete function is defined only at isolated input values, and is not defined for input values between those values. A continuous function is defined for all input values, and there are no gaps between any consecutive input values. 30. Functions can be represented verbally, symbolically, numerically and graphically. Activity 1.4 Key Terms 1. increasing 2. mathematical model 3. constant 4. decreasing Practice Exercises 5. What is the value of the home after a certain number of years? 6. The value of the home and the number of years of ownership. 7. the value of the home 8. the number of years of ownership 9. Independent Variable 1 2 3 4 Dependent Variable 86, 250 87,500 88,750 90,000 10. The value of the home is obtained by adding the product of 1250 and the number of years to 85,000. 11. Let v represent the value of the home and t represent the number of years. 12. 85,000 1250 v t = + 13. ( ) 85,000 1250 8 $95,000 v = + =

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    287 14. 15. increasing 16. The graph goes up to the right. 17. 18. constant 19. The graph is horizontal. 20. 21. decreasing 22. function 23. not a function 24. not a function 25. function 26. function 27. not a function 28. function Concept Connections 29. A mathematical model can be used to predict output values for input values not in the table of actual data. 30. In the vertical line test, a graph defines a function if any vertical line drawn through the graph intersects the graph no more than once. Activity 1.5 Key Terms 1. minimum 2. maximum Practice Exercises 3. The year 4. The account’s return 5. The account’s return is decreasing. 6. The account’s return is increasing. 7. The account’s return is constant. 8. The account’s return is decreasing. 9. The account’s return is increasing. 10. The account’s return is a maximum value. 11. The account’s return is a minimum value. 12. The account’s return increases most rapidly. 13. The number of months after origination 14. The delinquency rate as a percent 15. The percent of car-loan delinquencies increases with time from the origination date. 16. The graph rises to the right. 17. The time in years 18. The percent of the population living on less than $1.25/day

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    288 19. The population of people living on less than $1.25/day is decreasing with time. 20. The graph falls to the right. 21. The time in years 22. The number of internet users in millions 23. The number of internet users in China increases with time. 24. The graph rises to the right. 25. The time in years 26. The number of billions of pounds spent on defense 27. The spending decreases until the year 1997; then it increases. 28. First the graph falls, and then it rises to the right. Concept Connections 29. It is a minimum since the line is decreasing from 1989 to 1997 and increasing from 1997 to 2009. 30. The maximum occurs at 2010. The line is increasing until that year. It has the greatest output value. Activity 1.6 Key Terms 1. average rate of change 2. scatterplot Practice Exercises 3. 0.789 4. The national debt is increasing at an average rate of 0.789 trillion dollars/year. 5. 0.315 6. None 7. The national debt is never decreasing over the 30-year time period. 8. From 2005 to 2010 9. 1.126 10. During the period from 2005 to 2010, the national debt increased by 1.126 trillion dollars/year. 11. No 12. The national debt is not constant during the 30-year time period. 13. 0.422 14. Over the 30-year period, the national debt is increasing at an average rate of 0.422 trillion dollars/year. 15. The graph would rise to the right. 16. 2.8 17. The population is increasing at an average rate of 2.8 million people/year. 18. 0.9 19. None 20. The national population is never decreasing over the 100-year time period. 21. From 1990 to 2000 22. 3.2 23. During the period from 1990 to 2000, the national population increased by 3.2 million people per year. 24. No 25. The national population is not constant during the 100-year time period. 26. 2.17 27. Over the 100-year period, the national population is increasing at an average rate of 2.17 million people/year. 28. The graph would rise to the right.

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    289 Concept Connections 29. The graph for that period remains constant. The graph would be horizontal. 30. 2 1 x x x D = - and represents the change in x . 2 1 y y y D = - and represents the change in y . Activity 1.7 Key Terms 1. slope 2. linear function Practice Exercises 3. Yes 4. 3 7 m = - 5. No 6. Answers will vary. For example, 1 0 1 5 1 3 x y - - 7. 4 m = - 8. (0, 1) 9. 1 , 0 4 æ ö ÷ ç ÷ ç ÷ ç è ø 10. (0, 4) 11. 2 m = - 12. ( ) 2 4 f x x = - + 13. ( ) 2, 0 14. ( ) 2, 0 15. 3 m = 16. ( ) 0, 3 - 17. ( ) 1, 0 18. ( ) 1, 0 19. 1 2 2 y x = - 20. ( ) 4, 0 21. ( ) 4, 0 22. 1 m = - 23. ( ) 0, 1 24. ( ) 1 f x x = - + 25. ( ) 1, 0 26. The slopes are the same, 1 3 m = - , and the vertical intercepts are different. 27. The slopes are different; the vertical intercepts are the same, b = 3. 28. The slopes are different; the vertical intercepts are the origin (0, 0). Concept Connections 29. The vertical intercept (0, b ) of a graph is the point where the graph crosses the vertical axis. The horizontal intercept ( a , 0) of a graph is the point where the graph crosses the horizontal axis. 30. If the slope of a linear function is positive, the graph of the function rises to the right. If the slope of a linear function is negative, the graph of the function falls to the right.

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    290 Activity 1.8 Key Terms 1. slope-intercept 2. point-slope Practice Exercises 3. 2 13 3 y x = - + 4. 8 7 y x = - + 5. 4 8 5 y x = + 6. 3 3 4 y x = - + 7. 2 3 y x = - - 8. 0.5 5 y x = - 9. 5 39 y x = + 10. 6 54 y x = - 11. 5 27 y x = - + 12. 0.7 1 y x = + 13. 7 3 3 y x = + 14. 4 5 7 y x = - + 15. 9 75 y x = - - 16. 15 914 y x = + 17. 1 12 5 y x = - 18. 2 1 y x = - 19. 4 25 y x = - 20. 32 75 y x = + 21. 1 58 8 y x = + 22. 2 y x = - - 23. 1.2 y x = - - 24. 13 176 y x = + 25. 10 3.6 y x = + 26. 7 y x = - 27. 12 13 y x = - 28. 300 135 y x = + Concept Connections 29. Parallel lines have the same slopes with different y -intercepts. 30. For a linear function, the average rate of change is constant. Activity 1.9 Key Terms 1. vertical 2. horizontal Practice Exercises 3. 4 4 5 y x = - 4. 4 5 m = 5. (0, –4) 6. (5, 0) 7. 2 2 3 y x = + 8. 2 3 m = 9. (0, 2) 10. (–3, 0) 11. 0 5 y x = - 12. 0 m = 13. (0, –5) 14. none

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    291 15. 6 y = - 16. 0 m = 17. (0, –6) 18. none 19. yes 20. For each x -value input, there is exactly one y -value output. 21. 8 x = 22. The slope is undefined. 23. none 24. (8, 0) 25. no 26. It does not pass the vertical line test. 27. Answers will vary. (0, 4), (4, 4), (–3, 4) 28. Answers will vary. (–2, 0), (–2, 2), (–2, 5) Concept Connections 29. Standard form of a linear equation is A x By C + = , where A , B , and C are constants, and A and B are not both zero. 30. A horizontal line has general form ( ) or y c f x c = = , where c is a constant. A vertical line has general form x a = , where a is a constant. Activity 1.10 Practice Exercises 1. 2. Yes, the points are very close to the line. 3. ( ) 3.763 9.414 f x x = - 4. 0.999998 r = 5. 3.763 m = 6. The regression equation yields 47.03. 7. The regression equation yields 9.40. 8. ( ) 5 f is probably more accurate since 5 is within the given data and 15 is not. ( ) 5 f uses interpolation; ( ) 15 f uses extrapolation. 9. 2.50 x = 10. 11. Yes, the points are very close to the line. 12. ( ) 4.21 16.44 f x x = - + 13. 0.999995 r = - 14. 4.21 m = -

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    292 15. The regression equation yields –4.63. 16. The regression equation yields 37.51. 17. ( ) 5 f 18. 3.90 x = 19. 20. Yes, the points are very close to the line. 21. ( ) 14.322 25.464 f x x = - 22. 0.98726 r = 23. 14.322 m = 24. The regression equation yields 3.2. 25. The regression equation yields 332.6. 26. ( ) 2 f 27. 1.78 x = 28. ( ) 2 f is probably more accurate. Since 2 is within the given data and 25 is not. ( ) 2 f uses interpolation; ( ) 25 f uses extrapolation. Concept Connections 29. Interpolation uses a regression model to predict an output within the boundaries of the input values of the given data. Extrapolation uses a regression model to predict an output outside the boundaries of the input values of the given data. 30. A correlation coefficient, r , measures how strongly two related variables follow a linear pattern. The value of r ranges between –1 and 1. When the value of r is closer to zero, you would conclude that there is little or no linear correlation. The closer r is to either 1 or –1, the stronger the linear correlation between the two variables. If r < 0, a negative correlation, then the linear pattern follows a negative slope. If r > 0, a positive correlation, then the linear patters follows a positive slope. Activity 1.11 Key Terms 1. inconsistent 2. consistent 3. dependent

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    293 Practice Exercises 4. The answer is (1, 3). 5. The answer is (3, 4). 1 2 2 3 6 1 1 5 0 1 4 1 3 3 2 5 2 3 7 1 x y y - - - - 1 2 2 1 14 1 0 12 0 1 10 1 2 8 2 3 6 3 4 4 x y y - - - 6. The answer is (–1, 1). 7. The answer is (0, 5). 1 2 2 1 0 1 1 1 0 3 2 1 5 3 2 7 4 3 9 5 x y y - - - 1 2 2 13 3 1 9 4 0 5 5 1 1 6 2 3 7 3 7 8 x y y - - - - 8. The answer is (3, 12). 9. No solution 1 2 2 2 3 1 4 0 0 6 3 1 8 6 2 10 9 3 12 12 x y y - - - 1 2 2 4 1 1 3 2 0 2 3 1 1 4 2 0 5 3 1 6 x y y - - - - - - - - - 10. (2, –1) 11. (–2, 5) 12. No solution 13. (3, 1) 14. (–1, –1) 15. (4, 0) 16. (3, –5) 17. (–3, –4.5) 18. No solution 19. (–2, 5) 20. (8, 110) 21. (5, –20.5) 22. x = 1 23. x = –3 24. n = 60 25. x = 2.9 26. x = 0.5 27. x = –8 28. x = 7 Concept Connections 29. If the system is consistent, there is at least one solution, the points of intersection of the graphs. If the system is inconsistent, there is no solution and the lines are parallel. If the

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    294 system is dependent, there are infinitely many solutions and the equations represent the same line. 30. Solving the system algebraically is most accurate since there is an exact answer. If the answer is not an integer, graphing by hand produces only an answer that is close. Using the graphing calculator is more accurate. Activity 1.12 Practice Exercises 1. (2, 3) 2. (8, 2) 3. (1, 1) 4. (4, 1) 5. (6, 11) 6. (–7, 1) 7. (1.4, 0.8) 8. (3, –4) 9. (–1, 2) 10. (–8, –5) 11. (0, 9) 12. (1.5, 1.5), or 3 3 , 2 2 æ ö ÷ ç ÷ ç ÷ ç è ø 13. (0, –2) 14. (–2, 5) 15. (1, –1) 16. (–4, 3) 17. (6, 30) 18. (7, 4) 19. (11, 5) 20. (–3, –12) 21. (1, 2) 22. (3, –4) 23. (2, 7) 24. (–1, 3) 25. (40, –120) 26. (1, –6) 27. (–3, 2) 28. (–5, 3) Concept Connections 29. Solve one or both equations for a variable. Substitute the expression that represents the variable in one equation for that variable in the other equation. Solve the resulting equation for the remaining variable. Substitute the value from the previous step into one of the original equations, and solve for the other variable. 30. Set up the equation in standard form. Multiply one equation or both equations by the number (s) that will make the coefficients of one of the variables opposites. Add the two equations to eliminate one variable and solve the resulting equation. Substitute the value from the previous step into one of the original equations, and solve for the other variable. Activity 1.13 Key Terms 1. inconsistent 2. dependent Practice Exercises 3. (3, 0, 1) 4. (–1, 3, –2) 5. (5, 1, –1) 6. (4, –1, –2) 7. (2, 0, 1) 8. (–2, –1, 4) 9. (3, –5, 8) 10. (2, –3, 0)

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    295 11. (1, 2, –1) 12. (3, 1, 2) 13. (–2, 4, 1) 14. (4, –5, 3) 15. (7, –3, –4) 16. (3, –2, 1) 17. inconsistent 18. dependent 19. inconsistent 20. dependent 21. yes 22. no 23. yes 24. no 25. no 26. no 27. no 28. yes Concept Connections 29. A 3 3 ´ system of linear equations consists of three equations with a total of three variables. 30. A linear equation in three variables, such as x , y , and z is of the form , A x By Cz D + + = where A , B , C , and D are any constants. Activity 1.14 Key Terms 1. Elementary row operations 2. augmented matrix 3. reduced row echelon form 4. matrix Practice Exercises 5. 8 4 3 2 3 1 é ù ê ú ê ú - ë û 6. 1 6 0 3 1 5 é ù - ê ú ê ú - ë û 7. 4 3 1 1 2 2 1 5 8 1 1 5 é ù - - ê ú ê ú ê ú ê ú - ë û 8. 2 2 3 3 1 2 1 5 1 4 2 0 é ù ê ú ê ú - - - ê ú ê ú - ë û 9. 7 3 5 14 0 1 2 2 1 0 3 6 é ù - - ê ú ê ú ê ú ê ú - ë û 10. 1 1 1 4 1 4 0 8 1 0 2 1 é ù - ê ú ê ú ê ú ê ú - - ë û 11. 1 7 1 8 0 1 6 2 0 0 1 1 é ù - ê ú ê ú - ê ú ê ú ë û 12. 1 3 4 1 0 1 2 2 0 0 1 3 é ù - - - ê ú ê ú - ê ú ê ú - ë û 13. 2 8 5 3 1 x y x y - = + = - 14. 3 2 5 0 x y x y + = - - = 15. 4 3 2 2 1 2 5 10 x y z x y z x y z - + = + - = - - = 16. 2 2 6 3 3 8 2 9 2 5 x y z x y z x y z + - = - - = + + = -

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    296 17. 4 2 0 4 5 3 0 x y z y z x z + - = + = - - = 18. 2 3 4 5 6 8 x y z y z z - + = - - = = - 19. ( ) ( ) , , 4, 3, 2 x y z = - - 20. ( ) ( ) , , 1, 0, 3 x y z = 21. 1 0 0 2 0 1 0 1 0 0 1 4 é ù ê ú ê ú - ê ú ê ú ë û 22. 1 0 0 2 0 1 0 2 0 0 1 1 é ù ê ú ê ú - ê ú ê ú ë û 23. ( ) ( ) , , 1, 1, 2 x y z = - 24. ( ) ( ) , , 3, 2, 1 x y z = - 25. ( ) ( ) , , 1, 3, 2 x y z = 26. ( ) ( ) , , 2, 6, 5 x y z = - 27. ( ) ( ) , , 1, 2, 2 x y z = - - 28. ( ) ( ) , 3, 2 x y = - Concept Connections 29. The price of a double cheeseburger is $2.65. The price of the fries is $1.35. The price of the large soda is $1.20. 30. Yes. Devon has enough money. The total cost for the lunch is $38.90. Activity 1.15 Practice Exercises 1. ( ) , 14 -¥ 2. [ ) 5, - ¥ 3. ( 2.4, 13] - 4. [ ] 100, 100 - 5. 5 9 x - £ < 6. 6 x > 7. 2 x £ 8. 8.2 4 x - < <- 9. 7 x > 10. 6 x <- 11. 5 x £- 12. 4 x £ 13. 3 x > 14. 2 x > - 15. 5 x ³ 16. 4 x > 17. 1 4 x < < 18. 21 15 x - < < 19. 1 6 x £ £ 20. 6 19 x - ³ ³ - 21. 0.2 x £ - 22. 2.7 x £ 23. 1.5 x > - 24. 1.8 x ³

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