S14 Exponential Growth and Decay (Graphing Calculator or App Needed)

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1 1010 Homework Name S14 Exponential Growth and Decay (Graphing Calculator or App Needed) 1. Without graphing, classify each of the following as increasing or decreasing and find f (0). a. f (x) = 1.5(0.75) x b. f (x) = 0.6(1.03) x c. f (x) = 3(1/2) x 2. a. Given the following table, do you believe that the data can be approximately modeled by an exponential function or a linear function? Explain. x y b. If the function is exponential, what is the constant ratio between the output values or if the function is linear, what is the constant rate of change? c. What is the vertical intercept? d. Write the equation using function notation showing the relationship between the variables.

2 e. Is the function increasing or decreasing? Explain how you know whether the function is increasing or decreasing from the equation that you wrote in part d. 3. You are planning to purchase a new car and you have your eye on a specific model. You know that new car prices are projected to increase at a rate of 6% per year for the next few years. a. Write an equation that represents the projected cost, C, of your dream car t years in the future given that it costs $17,000 today. b. What is the growth factor? What is the growth rate? c. Use your equation in part a to project the cost of your car three years from now. d. If the price continues to increase at 6% per year, use your graphing calculator to approximate how many years it will take for your dream car to cost $23,000? e. Use the table or trace feature on your graphing calculator to determine how long it will take this car to double in cost? f. Sketch a graph of the function for the next 8 years using t = 0 for the present year and C to represent the cost.

3 4. a. Use your graphing calculator to explore the properties of each of the exponential functions. Function What is the domain? f (x) = 4(1.32) x g(x) = 2(0.6) x What is the range? Find the vertical intercept. What is the horizontal intercept? Is the graph of the function increasing or decreasing? What is the growth or decay factor? Determine the growth or decay rate. Write the equation that represents the horizontal asymptote. b. Sketch a graph of each of the functions in your table on the same axis. Use the properties that you discovered to help you with your sketch. Make sure you label each of the functions and label each axis. c. Using either the function f(x) or g(x), write an application problem that could be used to model the function.

4 5. a. Determine which of the following data sets are linear and which are exponential. b. For the linear sets, determine the slope. For the exponential sets, determine the growth factor or the decay factor. c. Write the equation that shows the relationship between the variables for each data set. a. x y b. c. a. x y b. c. 6. At a local college the total student body enrollment (in thousands) beginning in 2005 can be modeled by the function S (t) = 12.4(0.95) t, where S(t) represents the total enrollment in thousands and t is the number of years since a. Is the function increasing or decreasing? Explain how you can determine this from the function equation. b. From the equation, determine the vertical intercept. What practical meaning does this intercept have in this situation? c. Determine the growth or decay rate. What is the practical meaning of this rate in this situation? d. How many students will be enrolled at the college in the year 2010?

5 e. After the year 2010, the total enrollment is expected to remain the same as in the year 2010 for the next five years. Write a piecewise function g(t) to represent the enrollment from 2005 to Make sure you include the domain(s) for your function. f. Sketch a graph representing the enrollment at the college from 2005 to Use appropriate scaling and label each axis. 7. A small computer chip company has increased sales by 30% per year for five years. In 2010, the sales totaled 0.3 million dollars. a. Complete the table of values for this function using 0 to represent the year Year Total Sales (in millions) b. Explain why this is not a linear function. c. Explain why this is an exponential function. d. What is the growth factor? e. What is the vertical intercept? What is the practical meaning of this intercept in this situation?

6 f. Write the equation of the form f x = ab ' which represents your data, where f(x) is the total sales and x is the number of years since g. Determine f( -2 ). Does this value have any practical meaning in this situation? Explain. h. Determine f (5). Explain the meaning of f(5). i. Use your graphing calculator and estimate the number of years it will take for the total sales to double. j. Sketch a graph of the function. Use appropriate scaling and label each axis. 8. Your sporting goods company has just been given you an award salesperson of the year! A $2000 bonus is the amount of money you will receive. You are saving for a down payment on a camper. a. Being conservative you choose to invest the money at 4.25% compounded quarterly. How much will you have after 5 years?

7 b. You decide that $3200 will be enough money for the down payment. Using your graphing calculator, determine in how many years you will have this down payment money. c. Your friend who would prefer to take a few risks tells you to invest the $2000 in the stock market and gives you a tip on a stock that should grow 12% annually for the next five years. How much will you have after five years if the stock does perform as expected? d. If you choose to invest in the stock, using your graphing calculator, approximate in how many years you will have the down payment. 9. When drugs are administered into the blood stream, the amount present decreases continuously at a constant rate. The amount of a certain drug in the bloodstream is modeled by the function y = y + e -+./01 where y + is the amount of drug injected (in milligrams) and t is time (in hours). Suppose that 15 milligrams are injected at 8:00 a.m. a. How much of the drug is still in the bloodstream at 11:30 a.m.? b. If another dose needs to be administered when there is approximately 1.3 milligrams of the drug present in the bloodstream, approximately when should the next dose be given?

Answers to Exponential Growth and Decay HW 1. a. decreasing, f(0) = 1.5 b. increasing, f(0) = 0.6 c. decreasing, f(0) = 3 2. a. Exponential function, because there is a growth factor or constant ratio.

8 Answers to Exponential Growth and Decay HW 1. a. decreasing, f(0) = 1.5 b. increasing, f(0) = 0.6 c. decreasing, f(0) = 3 2. a. Exponential function, because there is a growth factor or constant ratio. b. The constant ratio is 2.5. c. (0,2) d. f (x) = 2(2.5) x e. Increasing; the base is greater than a C(t) = 17000(1.06) t b. growth factor: 1.06 growth rate; 0.06 or 6% c. $20, d. 5.2 years e years f. Cost Time 8 4. a. f(x) = 4(1.32) x g(x) = 2(0.6) x All real numbers All real numbers y > 0 y > 0 (0, 4) (0, 2) none none increasing decreasing Growth 32% Decay 40% y = 0 y = 0 b. c. Answers will vary. 5. Data set 1: a. Exponential b. 2.6 = growth factor c. Data set 2: a. Linear b. 2.6 = slope c. f (x) = 5(2.6) x f (x) = 2.6x + 5

6. a. decreasing; b = 0.95 When 0 < b < 1, it is a decreasing function. b. (0, 12.4) In 2005, the enrollment was 12,400 students. c. 5% The student population is decreasing at a rate of 5% per year.

9 6. a. decreasing; b = 0.95 When 0 < b < 1, it is a decreasing function. b. (0, 12.4) In 2005, the enrollment was 12,400 students. c. 5% The student population is decreasing at a rate of 5% per year. d students e. g(t) = 12.4(0.95)t, 0 ² t ² < t ² 10 f. Number of students 2005 time a. b. The rate of change is not constant. Year (2010 = 0) Total Sales (in millions) c. There is a growth factor. d e. (0, 0.3) Total sales in 2010, $0.3 million. f. f (x) = 0.3(1.30) x g Yes, it could be the amount of sales in 2008 if the growth rate is the same during that time. h. $1.11 million In 2015 the sales were $1.11 million i years j. Sales Ti m e 8. a. $ b. approximately 11 years c. $ d. 4.1 yrs. 9. a. 4.4 milligrams b. 3 p.m.

3.1 Exponential Functions and Their Graphs Date: Exponential Function

3.1 Exponential Functions and Their Graphs Date: Exponential Function Exponential Function: A function of the form f(x) = b x, where the b is a positive constant other than, and the exponent, x, is a variable.