2-6 Analyzing Functions with Successive Differences
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1 Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 1. ( 2, 8), ( 1, 5), (0, 2), (1, 1) linear 3. ( 3, 8), ( 2, 4), ( 1, 2), (0, 1), (1, 0.5) exponential Look for a pattern in each table of values to determine which kind of model best describes the data. 5. quadratic 7. exponential esolutions Manual - Powered by Cognero Page 1
2 Determine which kind of model best describes the data. Then write an equation for the function that models the data. 9. exponential; y = 3 3 x 11. linear; 13. PLANTS The table shows the height of a plant for four consecutive weeks. Determine which kind of function best models the height. Then write a function that models the data. linear; y = 0.5x + 3 Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 15. (1, 2.75), (2, 2.5), (3, 2.25), (4, 2) linear esolutions Manual - Powered by Cognero Page 2
3 17. ( 3, 11), ( 2, 5), ( 1, 3), (0, 5) quadratic 19. ( 1, 8), (0, 2), (1, 0.5), (2, 0.125) exponential Look for a pattern in each table of values to determine which kind of model best describes the data. Then write an equation for the function that models the data. 21. quadratic; y = 2.5x exponential; y = x esolutions Manual - Powered by Cognero Page 3
4 25. linear; y = 5x LONG DISTANCE The cost of a long-distance telephone call depends on the length of the call. The table shows the cost for up to 6 minutes. a. Graph the data and determine which kind of function best models the data. b. Write an equation for the function that models the data. c. Use your equation to determine how much a 10-minute call would cost. a. linear b. y = 0.12x c. $ BACTERIA A scientist estimates that a bacteria culture with an initial population of 12 will triple every hour. a. Make a table to show the bacteria population for the first 4 hours. b. Which kind of model best represents the data? c. Write a function that models the data. d. How many bacteria will there be after 8 hours? a. b. exponential c. b = 12 3 t d. 78,732 esolutions Manual - Powered by Cognero Page 4
5 31. CHALLENGE Write a function that has constant second differences, first differences that are not constant, a y- intercept of 5, and contains the point (2, 3). Sample answer: y = 2x OPEN ENDED Write a linear function that has a constant first difference of 4. y = 4x WRITING IN MATH How can you determine whether a given set of data should be modeled by a linear function, a quadratic function, or an exponential function? Sample answer: The data can be graphed to determine which function best models the data. You can also find the differences in ratios of the y-values. If the first differences are constant, the data can be modeled by a linear function. If the second differences are constant but the first differences are not, the data can be modeled by a quadratic function. If the ratios are constant, then the data can be modeled by an exponential function. 37. What is the equation of the line below? A B C D A esolutions Manual - Powered by Cognero Page 5
6 39. GEOMETRY The rectangle has an area of 220 square feet. Find the length l. A 8 feet B 10 feet C 22 feet D 34 feet B Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary x x = 10 5, 0.5 Solve each equation by taking the square root of each side. Round to the nearest tenth if necessary. 43. x 2 = 25 ±5 45. x 2 14x + 49 = , 10.9 Write an equation for the nth term of each geometric sequence, and find the seventh term of each sequence , 2, 4, 8, a n = 1(2) n 1 ; , 12, 36, a n = 4( 3) n 1 ; , 44, 88, a n = 22(2) n 1 ; 1408 esolutions Manual - Powered by Cognero Page 6
7 53. CANOE RENTAL To rent a canoe, you must pay a daily rate plus $10 per hour. Ilia and her friends rented a canoe for 3 hours and paid $45. Write a linear equation for the cost C of renting the canoe for h hours, and determine how much it cost to rent the canoe for 8 hours. C = 10h + 15; $95 Determine whether each equation is a linear equation. If so, write the equation in standard form y = 2x yes; 2x + y = y + 5 = 0 yes; y = y = 4x 2 1 no Evaluate each expression if x = 3, y = 1, and z = esolutions Manual - Powered by Cognero Page 7
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