Comparing linear and exponential growth

Transcription

1 Januar 16, 2009 Comparing Linear and Exponential Growth page 1 Comparing linear and exponential growth How does exponential growth, which we ve been studing this week, compare to linear growth, which we ve studied previousl? The purpose of toda s assignment is to develop comparisons between these two tpes of growth functions. In terms of function formulas, we alread know a general formula for each kind of function: = mx + b for linear growth, = a b x for exponential growth. However, if ou want to take the NOW-NEXT approach to describing growth, we ve never talked about linear growth in those terms. So that s the main thing we need to learn about. NOW-NEXT equations for linear functions Example Here s a linear function and its table: = 3x If ou think of x = 0 as being the start of the process, then the output at the start is 5. What is the NOW-NEXT rule that gets us from one value to the next? You tr it 1. Fill in the NOW-NEXT statement and the starting value for these functions. a. = 4x b. = 3x c. = x Complete the tables, then fill in the NOW-NEXT equation and the starting value.

2 Januar 16, 2009 Comparing Linear and Exponential Growth page 2 a. = 2x + 3 b. = 5x + 8 c. = 3x Look at our answers to problem 2. Notice how the numbers from the = mx + b equation alwas appear in the NOW-NEXT equation and the starting value. Tr to fill in these NOW-NEXT equations and starting values without having to write out the tables. a. = 3x + 4 b. = 5x + 30 c. = 8x

3 Januar 16, 2009 Comparing Linear and Exponential Growth page 3 Recap Here is the pattern in all of these linear growth problems: for the function equation = mx + b, the NOW-NEXT description will alwas be NEXT = NOW + m, starting from b. More problems 4. From these tables, fill in the function equation, the NOW-NEXT equation, and the starting value. a b c Here are some word problem situations involving linear functions. Fill in the function equation, the NOW-NEXT equation, and the starting value. a. A librar has 8000 books, and is adding 500 more books each ear. b. A gm s customers must pa $50 for a membership, plus $3 for each time the use the gm. c. At the start of a carnival, ou have 50 ride tickets. Each time ou ride the roller coaster, ou have to pa 6 tickets. d. A car takes a highwa trip at a speed of 50 miles per hour. (Here x stands for how man hours driven, stands for how man miles driven.)

4 Januar 16, 2009 Comparing Linear and Exponential Growth page 4 Comparison of linear and exponential growth Linear Growth Exponential Growth Growth pattern involves Growth pattern involves repeated addition repeated multiplication of the same number. b the same number. b is the starting value, m is the number repeatedl added (also called the rate or the slope) NEXT = NOW + m starting from b The repeated addition gives us multiplication in the function formula: = mx + b Graph is straight line with slope m; -intercept at (0, b). a is the starting value, b is the number repeatedl multiplied (also called the multiplier) NEXT = NOW b starting from a The repeated multiplication gives us an exponent in the function formula: = a b x Graph is curved, gets steeper and steeper; -intercept at (0, a). More problems 6. Here are various equations, tables, and word problems. Determine whether each of them involves a linear function or an exponential function. All ou need to do is answer linear or exponential but make sure ou could explain wh ou made our choice. a. = 6 2 x b. = 2x + 6 c. NEXT = NOW + 4, starting from 10 d. NEXT = NOW 4, starting from 10 e. NEXT = NOW 4, starting from 10 f. g h. The population of a town grows b 5,000 people ever decade. i. The population of a town doubles ever decade. j. The population of a town decreases b 1,000 people ever decade.

5 Januar 16, 2009 Comparing Linear and Exponential Growth page 5 7. From these tables, fill in the function equation, the NOW-NEXT equation, and the starting value. The first thing ou need to decide is whether the table is linear or exponential. If the output row shows a repeated addition pattern, it s linear. If the output row shows a repeated multiplication pattern, it s exponential. If ou need a reminder of what the equations should look like, see the chart on top of page 4. a b c d e f g