Lesson 3 - Practice Problems

Lesson 3 - Practice Problems Section 3.1: Average Rate of Change 1. The data below represent the number of times your friend s embarrassing YouTube video has been viewed per hour since you uploaded it. Time,!t, in hours 0 1 2 3 4 Views,! V(t), in thousands of views 0 6200 11200 18600 26600 For each of the following, show all of your work and be sure to include correct units in your answer. Write a sentence explaining the meaning of your answer. a) Determine the average rate of change from hours 0 to 1. b) Determine the average rate of change from hours 1 to 2. c) Determine the average rate of change from hours 0 to 4. 2. You decided to save up for a vacation to Europe by throwing all your loose change in a large coffee can. After a few months, you discover that the jar is 2 inches full and contains $124. a) Determine the average rate of change, in $/inch (dollars per inch), for the coffee can from when it was empty (0 inches) to when it was 2 inches deep. b) A month later, you check the can and find the change is 3 inches deep and adds up to $186. Find the average rate of change, in $/inch, for the coffee can from 0 inches to 3 inches. c) What is the meaning of the average rate of change in this situation? 117

3. A candy company has a machine that produces candy canes. The number of candy canes produced depends on the amount of time the machine has been operating. The machine produces 160 candy canes in five minutes. In twenty minutes, the machine can produce 640 candy canes. Determine the average rate of change in this situation. Write a sentence explaining its meaning. 4. The enrollment at a local charter has been decreasing linearly. In 2006, there were 857 students enrolled at a local charter school. By 2015, there were only 785 students enrolled. Determine the average rate of change of this school s enrollment during this time period. Write a sentence explaining its meaning. 5. In the year 1987, an investment was worth $30,200. In the year 1996, this investment was worth $43,700. Determine the average rate of change in this situation. Write a sentence explaining its meaning. 6. In the year 1998, the surface elevation of Lake Powell was 3,843 feet above sea level. In the year 2001, the surface elevation of Lake Powell was 3,609 feet above sea level. Determine the average rate of change in this situation. Write a sentence explaining its meaning. 118

7. The graph of A m! ( ) below shows the amount of water in a play pool after!m minutes. Use the graph to determine the rate of change over the indicated time intervals. Show all of your work and include correct units in your answers. Write a sentence explaining the meaning of each answer. a)! [0,1] (This notation denotes the interval of time from 0 minutes to 1 minute) b)! [2,4] c)! [5,10] d)! [0,12] e) What do you notice about the values above? What do these values tell you about A m! ( )? 119

8. The graph of D t! ( )below shows the distance Sally is from home (in miles) after!t minutes. Use the graph to determine the rate of change over the indicated time intervals. Show all of your work and include correct units in your answers. Write a sentence explaining the meaning of each answer. a)! [0,8] b)! [8,16] c)! [16,25] d) What do you notice about the values above? What do these values tell you about D t! ( )? 120

9. The graph of H t! ( )below shows the height (in feet) of a toy rocket t seconds after being launched into the air. Use the graph to determine the rate of change over the indicated time intervals. Show all of your work and include correct units in your answers. Write a sentence explaining the meaning of each answer. a)! [0,1] b)! [1,3.5] c)! [3.5,6] d)! [6,7] e)! [1,6] f)! [0,7] g) What do you notice about the values above? What do these values tell you about H t! ( )? h) Talk about the values you obtained for e) and f). Can you explain why the results make sense? 121

10. The graph of P t! ( ) below shows the population (in thousands of people) of a town t years after 2008. Use the graph to determine the rate of change over the indicated time intervals. Show all of your work and include correct units in your answers. Write a sentence explaining the meaning of each answer. a) 2008 to 2017 b) 2008 to 2010 c) 2015 to 2017 d) What do you notice about the values above? What do these values tell you about P t! ( )? 122

Section 3.2: Linear Functions 11. Determine the slope, behavior (increasing, decreasing, constant, or vertical), and vertical intercept (as an ordered pair) of each of the following. Write DNE if a value does not exist. a) b) c) d) e) f) g) h) i) Equation Slope Behavior Vertical Intercept! y = x 2! f ( a ) = 6 4a! P ( n )! =!3n! y!=!4! x!= 7! y = 3 5 x 4! y = x! B ( x ) = 8 x! V ( t ) = 70 123

12. Determine the horizontal intercepts for each of the following. Write DNE if there is no horizontal intercept. a)! y = x 2 b)! f ( a ) = 6 4a c) P n! ( )! =!3n d)! y!=!4 e)!x!= 7 f)! y = 3 5 x 4 g)! y = x h) B x! ( ) = 8 x 13. The function P n! ( ) = 455n 1820 represents a computer manufacturer s profit when!n computers are sold. a) Identify the slope. Interpret its meaning in a complete sentence. b) Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. c) Determine the horizontal intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. 124

14. John is a door-to-door vacuum cleaner salesman. His weekly salary is given by the linear function! S v ( ) = 200! +!50v, where!v represents the number of vacuums sold. a) Identify the slope, and interpret its meaning in a complete sentence. b) Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. c) Determine the horizontal intercept. Write it as an ordered pair. Discuss whether or not the horizontal intercept might have meaning in the context of this problem. 15. The linear function V n! ( ) =!221.4! +!4.25n! gives the value, in thousands of dollars, of an investment after!n years. a) Identify the slope, and interpret its meaning in a complete sentence. b) Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. c) Determine the horizontal intercept. Write it as an ordered pair. Discuss whether or not the horizontal intercept might have meaning in the context of this problem. 125

16. For each of the following, find the equation of the line that meets the indicated criteria. a) Slope Point Equation of Line in Slope-Intercept Form!m = 2 (0, 3) b) c)!m = 4! m = 3 8 0, 2 3! (0, 5) d)!m = 2.37 (0, 6.35) 17. For each of the following, find the equation of the line that meets the indicated criteria. Slope Point Find Vertical Intercept Equation of Line in Slope-Intercept Form a)!m = 2 (2, 3) b)!m = 4 (3, 4) c)! m = 5 16 ( 8, 5) d)!m = 1.4 (2, 2.34) 126

18. For each of the following, complete the table below. Two Points Find Slope Find Vertical Intercept Equation of Line in Slope-Intercept Form a) (2, 3), (4, 7) b) ( 3, 6), (3, 12) c) (5, 5), ( 1, 3) d) (2, 4.2), (6, 9.4) e) (4, 5) and (2, 3) f) ( 3, 2) and (1, 8) g) (2, 4) and (7, 4) 127

19. For each of the following, complete the table below. Parallel to: Find Slope Find Vertical Intercept Equation of Line in Slope-Intercept Form a) Parallel to: ( ) = 2x + 1! f x Goes through: (0,! 3) b) Parallel to: ( ) = 3x 5! f x Goes through: (1, 2) c) Parallel to: ( ) = 4x! f x Goes through: (5, 7) d) Parallel to: ( ) = 1 2 x + 1! f x Goes through: (0, 5) e) Parallel to: ( ) = 2 5 x + 4! f x Goes through: (! 5, 1) f) Parallel to:!x = 6 Goes through: (! 2, 5) g) Parallel to: ( ) = 5! f x Goes through: (2, 4) 128

20. For each of the following, complete the table below. Perpendicular to: Find Slope Find Vertical Intercept Equation of Line in Slope-Intercept Form a) Perpendicular to: ( ) = 2x + 1! f x Goes through: (0,! 3) b) Perpendicular to: ( ) = 1 3 x 5! f x Goes through: (1, 2) c) Perpendicular to: ( ) = 1 4 x! f x Goes through: (5, 7) d) Perpendicular to: ( ) = 1 2 x + 1! f x Goes through: (0, 5) e) Perpendicular to: ( ) = 2 5 x + 4! f x Goes through: (! 5, 1) f) Perpendicular to:!x = 6 Goes through: (! 2, 5) g) Perpendicular to: ( ) = 5! f x Goes through: (2, 4) 129

21. Find the equation of the linear function that generates the following table of values. Write your answer in slope-intercept form.!x! f ( x ) 5 91 2 67 1 43 4 19 9 21 22. Find the equation of the linear function that generates the following table of values. Write your answer in slope-intercept form.!t! C ( t ) 5 1250 15 900 20 725 35 200 45 150 23. Find the equation of the linear function that generates the following graph. Write your answer in slope-intercept form. 130

24. Find the equation of the linear function that generates the following graph. Write your answer in slope-intercept form. 25. Find the equation of the linear function that generates the following graph. Write your answer in slope-intercept form. 26. Find the equation of the linear function that generates the following graph. Write your answer in slope-intercept form. 131

27. Find the equation of the linear function that generates the following graph. Write your answer in slope-intercept form. 28. Find the equation of the linear function that generates the following graph. Write your answer in slope-intercept form. 29. Give the equation of the horizontal line passing through the point ( 6, 11). 30. Give the equation of the vertical line passing through the point (4, 7). 31. Give the equation of the!x -axis. 32. Give the equation of the! y -axis. 132

Section 3.3: Modeling with Linear Functions 33. A candy company has a machine that produces candy canes. The number of candy canes produced depends on the amount of time the machine has been operating. The machine produces 160 candy canes in five minutes. In twenty minutes, the machine can produce 640 candy canes. a) Determine the equation of the linear function that represents this situation. Let C x! ( ) represent the number of candy canes produced in!x minutes. Write your answer in function notation. b) Determine C 10! ( ). Write a sentence explaining the meaning of your answer. c) What is the practical meaning of the slope of this linear function? Include units. d) Determine the horizontal intercept of this linear function. Write it as an ordered pair and interpret its meaning. e) How many candy canes will this machine produce in 1 hour? 133

34. Your workplace is 20 miles from your house. The graph below shows the distance you are from your house if you leave work and drive in the opposite direction. Distance from House (Miles) 400 350 300 250 200 150 100 50 0 6, 380 5, 320 4, 260 3, 200 2, 140 1, 80 0, 20 0 1 2 3 4 5 6 7 Time (Hours) a) Determine the equation of the linear function that represents this situation. Let D t! ( )represent your distance from home after!t hours. Write your answer in function notation. b) Use the equation from part a) to determine how long it would take for you to be 500 miles from your house. Express your answer in hours and minutes. c) How far from your house would you be after 12 hours? d) What is the practical meaning of the slope of this linear function? Include units. 134

35. A local carpet cleaning company charges $10 for each room plus a reservation fee of $25. They can clean a maximum of 12 rooms in one trip. Also, they have the policy that once a reservation is made, if you cancel, the reservation fee is non-refundable. a) Determine the equation of the linear function, C n! ( ), that represents the total cost for cleaning!n rooms. b) Complete the table below. Graph the results and decide if it would make sense to connect the data points on the graph.!n 0! C ( n ) 1 2 3 6 12 c) Determine the practical domain of C n! ( ) : d) Determine the practical range of C n! ( ) : 135

36. Water is leaking out of a tank at a constant rate of 1 gallon every 2 minutes. The tank initially held 30 gallons of water. a) Determine the equation of the linear function, A t! ( ), that represents the amount of water (in gallons) remaining in the tank after!t minutes. b) Complete the table below. Graph the results, and decide if it would make sense to connect the data points on the graph.!t 0! A ( t ) 1 2 3 5 10 60 c) Determine the practical domain of A t! ( ): d) Determine the practical range of! A(t) : 136

37. With good credit and a $5000 down payment, you can finance a new 2012 Chevrolet Camaro convertible for 60 months for $615.17 per month. a) Determine the equation of the linear function,! T(n), that represents the total amount paid for this car after!n months. b) Use the equation from part a) to determine the total payment over the 60-month time period. c) A new 2012 Chevrolet Camaro convertible has a base MSRP of $35,080. Why is this value lower than your answer in part b)? 38. You adopted an adult cat four years ago. The data below represent your cat s weight for the four years she has lived with you. The data are exactly linear. Time,!t, in years 0 1 2 3 4 Weight,! W(t), in pounds 6 7.25 8.5 9.75 11 a) Identify the vertical intercept and average rate of change for the data. b) Use your results from part a) to write the linear function that represents the data table. Use the indicated variables and proper function notation. c) Use your function to determine how much the cat will weigh in year 8. Write your final result as a complete sentence. d) Use your function to determine how many years it would take for your cat to reach 20 pounds. Round to the nearest whole year. 137

39. You starting saving mad money in a cookie jar under the kitchen sink. When you started saving, you put $100 in the jar. Write a formula for the function, S t! ( ), that models the savings in the jar after!t months, assuming that the amount you put in the jar: a) Increases by $20 per month b) Decreases by $40 per month c) Increases by $100 students every 4 months d) Decreases by $60 students every 3 months e) Remains constant (does not change) 138

Section 3.4: Is the Function Linear? 40. For each of the following functions, determine if the function is linear. If linear, give the slope and write the equation. a)!x 3 2 1 0 1 2 3! f x ( ) 2 4 8 16 32 64 128 b)!x 3 2 1 0 1 2 3! g x ( ) 2 0 2 4 6 8 10 c)!t 4 1 2 5 8 11 14! s t ( ) 28 19 10 1 8 17 26 d)!x 4 2 0 2 4 6 8! h x ( ) 5 6 7 8 9 10 11 e)!n 4 1 5 6 8 9 12! p ( n ) 4 2 0 2 4 6 8 139

41. Data below represent how many pushups Tim can do in a minute at the start of a 5-week exercise program and each week thereafter. Time,!t, in weeks 0 1 2 3 4 5 Number of Pushups,! N(t), in a 2 6 10 14 18 20 minute a) Compute the average rate of change for weeks 0 through 3. Be sure to include units. b) Compute the average rate of change for weeks 1 through 4. Be sure to include units. c) Compute the average rate of change for the enire 5-week period (weeks 0 through 5). Be sure to include units. d) What is the meaning of the average rate of change in this situation? e) Do the data points in the table define a perfectly linear function? Why or why not? 42. Cora decided to go on a diet. On the day she started, she weighed 200 pounds. For the next 8 weeks, she consistently lost 2 pounds a week. Construct a linear function model for this situation and provide a GOOD graph of this function on the grid below. Clearly indicate what your variables represent. 140

43. Mark is at Home Depot and needs to buy 200 pounds of roofing nails for a project. He decides to purchase the loose nails because nails are cheaper if bought by the pound. He looked around and could not find someone to help him right away. He then noticed a one-cup measuring scoop next to the bin of nails. He poured one cup of nails into a bucket and found that it weighed 2.3 pounds. He then poured 4 more cups of nails into the bucket and found that it weighed 9.5 pounds. He figured if he used the points (1, 2.3) and (5, 9.5), he could figure out a formula (i.e. equation) and calculate how many cups he would need. a) Construct a linear function model for this situation. Clearly indicate what your variables represent. b) Use your function from part a) to determine how many cups of roofing nails Mark needs for his project. c) Provide a GOOD graph of this function on the grid below. d) Challenge question. The formula you found above does not go through the origin. Why do you think 0 cups of nails actually weighs MORE than 0 pounds in Mark s equation? 141

Section 3.5: Scatterplots on the Graphing Calculator 44. Use your graphing calculator to create a scatterplot of the data set shown below. Be sure to use an appropriate viewing window.!x 1 3 4 6 7 9 10! y 437 886 1097 1558 1768 2217 2437 In the space below, sketch what you see on your calculator screen and write down the viewing window you used. Xmin = Xmax = Ymin = Ymax = 45. Use your graphing calculator to create of scatterplot of the data set shown below. Be sure to use an appropriate viewing window.!x 2 9 14 23 33 42! y 60.2 130.1 243.7 328.9 580.5 643.8 In the space below, sketch what you see on your calculator screen and write down the viewing window you used. Xmin = Xmax = Ymin = Ymax = 142

Section 3.6: Linear Regression 46. Consider the data set shown below:!n 2 9 14 23 33 42! A n ( ) 160 130 243 328 580 643 a) Use your graphing calculator to find the equation of the regression line for this data set (round to three decimal places as needed). Use the indicated variables and proper function notation. b) Use your graphing calculator to generate a scatterplot of the data and regression line on the same screen. You must choose an appropriate viewing window. In the space below, sketch what you see on your calculator screen and write down the viewing window you used. Xmin = Xmax = Ymin = Ymax = 47. The following table shows the number of newspaper subscriptions in Middletown, USA where!t represents the number of years since 2002 (t = 0 in 2002).!t 0 2 4 6 8! S ( t ) (Subscriptions in 1000 s) 448 372 198 145 45 a) Use your calculator to determine the equation of the regression line to two decimal places. Write the regression equation in S t! ( )! =!at! +!b form and record it in the space below. b) Use your linear regression equation to estimate the total number of subscriptions in 2004. How does this value compare to the data value in the table? Why might they be different? c) Use your linear regression equation to estimate the year in which the circulation will be 100,000. Round to the closest whole year. (Hint:! S t ( ) = 100 ).! S t ( ) is measured in thousands so solve 143

48. Merry hiked the Appalachian Trail from Georgia to Maine. The distance of her hike was 2200 miles. It took Merry 123 days to complete the hike. The data below represent the distance,! D t ( ), she had hiked!t days after the start of her trip.!t (days hiking) 0 32 47 73 99 123! D ( t ) (distance in miles) 0 590 912 1212 1876 2200 a) Use your calculator to determine the equation of the regression line. (Round to 2 decimal places). Write the regression equation in! D t ( ) = at! +!b form and record it in the space below. b) What is the slope of your linear regression equation for D t! ( ) and what is its meaning in the context of this problem? Include appropriate units in your response. c) Use your linear regression equation to estimate the total number of miles Merry hiked in 50 days. Show your computations here and your final result. d) Use your linear regression equation to determine how long it took Merry to hike 1000 miles. 144

49. The table below shows the total sales, S t! ( ) in millions of dollars, as a function of time,!t, in years since 1980.! S ( t ) =Total Sales (in millions of dollars) 0 1.19 5 1.40 10 1.91 15 1.88 21 2.01 25 2.12 26 2.38!t =Years Since 1980 a) Use the table to determine the total sales in 2005. Write your answer in a complete sentence. b) Use your calculator to determine a linear regression equation for this data set. Use function notation to write your response and round to four decimal places as needed. c) Use the regression equation you found in part b) to determine sales in 2005. Round your answer to the nearest hundredth. Write your answer in a complete sentence. d) Your answers for parts a) and c) should be different. Why is this the case? Write your answer in a complete sentence. e) Use the regression equation you found in part b) to determine the year in which sales should reach $3,000,000. Write your answer in a complete sentence. f) Interpret the meaning of the statement S 30! ( )! =!2.44. g) Explain the meaning of the slope of S t! ( ). Be sure to include appropriate units. 145