Which Function will Have the Steepest Graph

4-5 Compare Properties of Functions 1. For the first leg of the Ramirez family’s trip, their speed averages 68 miles per hour. The second leg is shown in the graph. Compare the speeds for each part of their trip. SOLUTION: To compare their speeds, compare the rates of change. For the first leg of the trip, the rate was 68 miles per hour. To find the rate of change for the second leg, use the two points on the line and find the rate of change between them. The speed for the second leg of the trip was 55 miles per hour. The speed for the first leg is greater by 13 miles per hour. ANSWER: First part: 68 miles per hour; Second part: 55 miles per hour. The speed for the first leg is greater by 13 miles per hour. Page 1 4-5 Compare Properties of Functions 2. The late fees for a school library are represented by the function c = 0.25d, where c is the total cost and d is the number of days a book is late. The fees charged by a city library are shown in the table. a. Compare the functions’ y-intercepts and rates of change. b. Shamar checks out one book at each library and returns both books 3 days late. What are the late fees for each library? SOLUTION: a. Compare the y-intercepts. At 0 days late, there is no charge. So, the y-intercepts are the same, 0. Compare the rates of change. The daily charge for the school library is $0.25 per day. Use the table to find the daily charge for the city library. The rate of change for the city library is or $0.35 per day. b. Find the late fees for 3 days for the school library. c = 0.25d c = 0.25(3) c = 0.75 The late fees for 3 days for the school library are $0.75. As shown in the table, the late fees for 3 days for the city library are $1.05. ANSWER: a. Both have the same y-intercept of 0. Rates of change: school library: $0.25 per day; city library: $0.35 per day b. school library: $0.75; city library: $1.05 3. Matt and Seth purchase baseball cards each week. The amount of cards they each have in their collection is shown in the graph and table. Who will have more cards in Week 20? Justify your response. Page 2 4-5 Compare Properties of Functions SOLUTION: Find the number of cards that Seth will have by examining the table. He collects four cards per week, so after 20 weeks he will have 4(20) or 80 cards. Find the number of cards Matt will have by examining the graph. Matt began with 20 cards and collects two additional cards per week. After 20 weeks, Matt will have 2(20) + 20 or 60 cards. So, Seth will have more cards. ANSWER: Seth; Seth will have 4(20) or 80 cards and Matt will have 2(20) + 20 or 60 cards. Page 3 4-5 Compare Properties of Functions 4. The Shaw family is building a patio. One person can place the flagstone at a rate of 4.5 per hour. The equation s = 11h represents the number of stones s that two people can place in h hours. How many more flagstones can 2 people place in 3 hours than one person? Explain. SOLUTION: In 3 hours, one person can lay 4.5(3) or 13.5 stones. Two people can lay 11(3) or 33 stones. 33 – 13.5 = 19.5 So, two people can lay 19.5 more stones in 3 hours than one person can lay in the same amount of time. ANSWER: 19.5 stones; Sample answer: One person can lay 4.5(3) or 13.5 stones. Two people can lay 11(3) or 33 stones. 33 – 13.5 = 19.5 5. Reason Abstractly Refer to the conversions in the tables below. a. Write a function for each table. b. If you graph the points, the graph for which function would have the steepest slope? Justify your response. c. Which function has the least rate of change? Explain. SOLUTION: a. Since 0 cups, pints or quarts all have 0 ounces, the y-intercepts are all zero. Find the rates of change. Then write the function. Page 4 4-5 Compare Properties of Functions The rate of change is The rate of change is The rate of change is or 8. The function is z = 8c. or 16. The function is z = 16 p. or 16. The function is z = 32q. b. The steepest slope would have the greatest rate of change, so the graph of the quart equation would have the steepest slope. c. The first function has the least rate of change because 8 is less than 16 and 32. ANSWER: a. z = 8c ; z = 16 p ; z = 32q b. the quart equation; Sample answer: The greater the rate of change, the steeper the slope of the graph. c. The first function has the least rate of change because 8 is less than 16 and 32. 6. Model with Mathematics Write a real-world problem where you would want to compare rates of change for two different functions. SOLUTION: See students' work. ANSWER: See students' work. Page 5 4-5 Compare Properties of Functions 7. Persevere with Problems Explain why the graph of the function y = 3x + 40 will never intersect the graph of the function y = 3x + 35. SOLUTION: Sample answer: Both functions have the same rate of change but because they have different y-intercepts, they are parallel lines and parallel lines will never intersect. ANSWER: Sample answer: Both functions have the same rate of change but because they have different y-intercepts, they are parallel lines and parallel lines will never intersect. 8. Reason Inductively The exchange rate to convert U.S. dollars to British pounds is represented by the function p = 0.61d where p is the amount in pounds and d is the number of dollars. One U.S. dollar can also be exchanged for 0.69 European euro. If you exchange $250 for pounds and $250 for euros, which of the following is true? I You will receive 152.5 pounds and 172.5 euros. II You will receive about 410 pounds and about 362 euros. III You will receive 250 dollars. IV You will receive the same amount of pounds and euros. SOLUTION: Convert $250 to pounds and $250 to euros. dollars to pounds p = 0.61d p = 0.61(250) p = 152.50 There are 152.5 pounds in $250. dollars to euros 1 dollar = 0.69 euro 250 dollars = 0.69(250) or 172.5 euros There are 172.5 euros in $250. The correct answer is I. ANSWER: I Page 6 4-5 Compare Properties of Functions 9. A fabric store sells cotton for $7.00 a yard. The price of special occasion fabric is shown in the graph. Compare the functions’ rates of change. SOLUTION: The rate for the cotton fabric is $7.00 per yard The rate for the special occasion fabric is or $9.00 per yard. The special occasion fabric has the greater rate of change. ANSWER: Cotton fabric: $7.00 per yard Special occasion fabric: or $9.00 per yard. The special occasion fabric has the greater rate of change. Page 7 4-5 Compare Properties of Functions 10. Two players played a game. The first player’s score is represented by the function p = 5c – 3, where p is the number of points scored and c is the number of correct answers. The second player’s score is shown in the table. a. Compare the functions by comparing their y-intercepts and rates of change. b. How many points will the first player have if he or she correctly answers 30 questions? SOLUTION: a. Find the rates of change. The slope or rate of change for Player 1 is 5, so Player 1 scores 5 points per question. Examine the table to find the rate of change for player 2 Player 2 scores 5 points per question. So, both have the same rate of change. The function for Player 1 has a yintercept of -3 and the function for Player 2 has a y-intercept of 0. b. Use the equation to find how many points Player 1 will have if he or she correctly answers 30 questions. Replace c with 30. p = 5c – 3 p = 5(30) – 3 p = 150 – 3 p = 147 Player 1 would have a score of 147 points if he or she correctly answered 30 questions. ANSWER: a. Player 1: 5 points per question; Player 2: 5 points per question. Both have the same rate of change, but the function for Player 1 has a y-intercept of –3 and the function for Player 2 has a y-intercept of 0. b. 147 points Page 8 4-5 Compare Properties of Functions 11. Justify Conclusions Jesse and Juan each open savings accounts. The amounts in Jesse's account are shown in the table. Juan saves $5 per week. Who will have saved more in 8 weeks? Explain. SOLUTION: Because Juan saves $5 per week, in 8 weeks Juan will have 5(8) or $40. Extend the table to find the amount of money Jesse will have in 8 weeks. Jesse will have $37. So, Juan will have saved more money. ANSWER: Juan; Sample answer: In 8 weeks Juan will have 5(8) or $40. Jesse will have saved $37. Page 9 4-5 Compare Properties of Functions 12. Canada Olympic Park features sports training and entertainment facilities. The Monster zip line produces average speeds of 120 kilometers per hour. A smaller line produces speeds represented by the function d = 50h where d is the distance in kilometers after h hours. How much farther could you travel on the Monster zip line in 0.25 hours? SOLUTION: The distance traveled on the Monster Zip line is 120(0.25) or 30 kilometers. The distance traveled on the smaller line is 50(0.25) or 12.5 kilometers. Since 30 – 12.5 = 17.5, you travel 17.5 miles more on the Monster zip line than on the smaller line. ANSWER: 17.5 km 13. Raj gets a 1.5 mile head start and runs at a rate of 4.5 miles per hour. Jacinda’s progress is represented by a graph that goes through the points (1, 10), (2, 20), and (3, 30). How long will Jacinda need to run to catch up with Raj? SOLUTION: Write an expression that represents the distance each person can run in t hours. Raj gets a 1.5 mile head start and then runs 4.5 miles per hour: 1.5 + 4.5t. Jacinda does not get a head start and runs 10 miles per hour: 10t. Set the expressions equal to one another and solve for t. So, in hour, Jacinda will catch up with Raj. ANSWER: h 14. A museum charges $12.50 per adult ticket. The price of a student ticket is represented in the table. Graph both functions on the coordinate plane. Use circles to represent student ticket prices and use squares to represent adult ticket prices. Page 10 4-5 Compare Properties of Functions Compare the rates of change and y-intercepts of the linear functions. Explain what each of these represent. SOLUTION: Find the rates of change and y-intercepts for the functions that represent each ticket type. Student Tickets: y-intercept = 0, rate of change = 8.5 Adult Tickets: y-intercept = 0, rate of change = 12.5 Using this information graph the two functions on the coordinate plane. Sample answer: The rates of change represent the cost per ticket for adult and student tickets. The adult ticket function has a higher rate of change because the cost per ticket is higher. Both functions have a y-intercept at the origin because buying 0 tickets costs $0. ANSWER: Page 11 4-5 Compare Properties of Functions Sample answer: The rates of change represent the cost per ticket for adult and student tickets. The adult ticket function has a higher rate of change because the cost per ticket is higher. Both functions have a y-intercept at the origin because buying 0 tickets costs $0. 15. Julie swam, biked, and ran in a 22.2 mile triathlon. She completed the race in 2.15 hours. The function m = 13.8h represents the miles m Julie biked in h hours. Was her average speed biking less than or greater than her average speed for the entire race? Justify your answer. Round to the nearest tenth if necessary. SOLUTION: Sample answer: Her average speed biking was 13.8 mph, and her average speed for the entire race was about 22.2 ÷ 2.15 or about 10.3 mph. So, her average speed biking was greater than her average speed for the entire race. ANSWER: Sample answer: Her average speed biking was 13.8 mph, and her average speed for the entire race was about 22.2 ÷ 2.15 or about 10.3 mph. So, her average speed biking was greater than her average speed for the entire race. Write an equation in slope-intercept form for the table of values. 16. SOLUTION: First find the slope. Choose any two points from the table. The slope is 4. When x is 0, y is –3, so the y-intercept is –3. Replace m and b in the equation y = mx + b with the correct values. y = 4x – 3 ANSWER: y = 4x – 3 Page 12 4-5 Compare Properties of Functions 17. SOLUTION: First find the slope. Choose any two points from the table. The slope is –1. To find the y-intercept, use the slope and the coordinates of one point in the equation y = mx + b. Replace m and b in the equation y = mx + b with the correct values. y = –x + 4 ANSWER: y = –x + 4 Page 13