1) Now there are 4 bacteria in a dish. Every day we have two more bacteria than on the preceding day.

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1 Math 093 and 117A Linear Functions and Eponential Functions Pages 1, 2, and 3 are due the class after eam 1 Your Name If you need help go to the Math Science Center in MT 02 For each of problems 1-4, do the following: First: Identify the dependent and independent variables and corresponding units. Label them on the table and on the graph Second: Write the coordinates of a point that describes the initial condition in the story (0,..) Third: Read the problem again to discover the pattern and write the coordinates of four more points that satisfy the pattern Fourth: Plot the points on the attached grid and show the graph. Fifth: Two of the relationships are linear; write the equation of the line for these two. We will eplore the non-linear relationships in class. 1) Now there are 4 bacteria in a dish. Every day we have two more bacteria than on the preceding day y ) There are 4 bacteria in a dish. Every day the number of bacteria doubles (multiplies by 2)......y

2 3) There are 100 bacteria in a dish. Every day, after a bactericide was applied, the number decreases by 10 every day y ) There are 100 bacteria in a dish. Every day, after a bactericide was applied, the number of bacteria in the dish is half of the amount on the day before y

3 For each of the following problems given below, do the following: (only this part is due) - First: Identify the variables and corresponding units. Construct a table of values including at least 4 points. Second: We will construct the functions in class 5) There are 8000 bacteria in a dish when we apply an antibacterial solution. a) Every 2 hours the number of bacteria decreases by y b) Every 2 hours one-half as many bacteria remain in the dish......y 6) I have 200 cents in the piggy bank. a) Every four days the amount of money in the piggy bank increases by 50 cents......y b) Every four days the amount of money doubles.....y 3

4 MA 093 and 117A Eponents and Eponential Functions - Summary Properties of Eponents A. Properties of Eponents, for a>0, b>0 positive bases, m n m+ n 1. b b = b 2. a a m n = a m n 3. ( m m ab ) = a b m m m a a 4. = m b b 5. ( m n mn a ) = a n 1 6. b = n b 7. a 0 = 1 m/ n n m n 8. a = a = ( a) m B. The General Eponential Function f ( ) = ab eponent y-intercept base E1 Identify whether it is a linear or eponential function a. b. c. y y y

5 Graphs of Eponential functions of the form: E1 For f ( ) = (3) 2 y y = ab Identify: a. the y-intercept; b. the base; c. increasing or decreasing E2 Graph f() = 2 1 = () 2 y Identify: a. the y-intercept; b. the base; c. increasing or decreasing Summary: In the function y = ab If the base b > 1 then the function is If the base is 0 < b < 1, then, the function is a. C. Solving Power Equations 2 2 b = 4 b. b = 8 c. 5 b = 18 d. 15 b = 32 D. Finding the Eponential Equation Find the equation in the form f ( ) = ab, given that the curve contains the points a. (0, 3) and (4, 72) b. (0, 5) and (7, 63) 5

6 MA 093 and 117A Section 4.4 Constructing Eponential functions y = a b HINT: For each problem construct a table of values that satisfy the pattern; label variables in contet. Remember that a is the y-intercept and b is the multiplier per unit of. If you have not been given the multiplier per unit of, then, use algebra to find b. 7) A rumor is spreading across a college campus that there will be no final eams for any classes this semester. At 8 a.m. today, 7 people have heard the rumor. Assume that after each hour, 3 times as many students have heard the rumor. Let f(t) represent the number of people who have heard the rumor t hours after 8 a.m. Find an equation for f. * 8) Here is a variation to the last problem. A rumor is spreading across a college campus that there will be no final eams for any classes this semester. At 8 a.m. today, 7 people have heard the rumor. Assume that after FIVE HOURS, 3 times as many students have heard the rumor. Let f(t) represent the number of people who have heard the rumor t hours after 8 a.m. Find an equation for f. 9) Today (time 0), ten students come down with the flu. Assume that each day, the number of students who come down with the flu doubles. Let f(t) represent the number of students who come down with the flu at t days after today. 10) Here is a variation to the last problem: Today (time 0), ten students come down with the flu. Assume that EVERY TEN days, the number of students who come down with the flu doubles. Let f(t) represent the number of students who come down with the flu at t days after today. 6

7 11) Sales of portable MP3 players grew approimately eponentially from $0.06 billion in 1998 to $0.60 billion in Write an eponential function for the number of MP3 players, in billions, at t years since 1998; round to three decimal places. 12) A growing number of thieves are using keylogging programs to steal passwords and other personal information from internet users. The number of keylogging programs reported grew approimately eponentially from 0.2 thousand programs in 2000 to 13.0 thousand programs in Write an equation for the number of keylogging programs, in thousands, at te years since

8 MA093 and 117A - Eponential Models 13) Bald Eagle Pairs - Let NN be the number of bald eagle pairs (in thousands) in the U. S. at tt years since Year Number of Bald Eagle Pairs (thousands) a) By hand, create a scattergram of the given data on the given aes. b) What function describes the data best? Linear or eponential? c) Use the graphing calculator to determine the equation of the best model that fits the data at t years since Round to three decimal places and use N and t for your variables. d) Interpret in contet the coefficient a and the base b of the model equation. e) Predict the number of bald eagle pairs in How confident are you in this result? f) Graphical approach to solving eponential equations: Use the INTERSECTION method in the calculator to determine the year in which there were 5,500 eagle pairs in the U.S. Is this interpolation or etrapolation? Label functions on graph. g) Use algebra and the first and last ordered pairs to write an eponential model, yy = aabb, for the data; round to three decimal places. How does it compare to the regression equation from part (c)? 8

9 14) Number of Lawsuits - The number of lawsuits filed against tobacco companies is shown in the table below for various years. Let y be the number of lawsuits filed at years since Number of Lawsuits Filed Against Tobacco Companies Year Number of Lawsuits a) Label aes in contet. b) Construct a scatter diagram by hand. What function fits the data best; linear or eponential? c) Use the regression feature of your calculator to construct an eponential model, yy = aabb, with being years since Write the answer using function notation y = f(). d) Interpret in contet the coefficient a and the base b of the model equation. e) Use your model to predict the number of lawsuits that were filed in Write your answer in a complete sentence with correct units. f) Find f(2) and interpret within contet in a complete sentence with correct units. g) Graphical approach to solving eponential equations: Use the INTERSECTION method in the calculator to predict when the number of lawsuits will be Write your answer in a complete sentence with correct units. h) Find when f() = Interpret within contet using a complete sentence with correct units. i) Use algebra and the first and last points to construct an eponential model, yy = aabb, with being years since Write the answer using function notation y = f(). 9

10 MA 093 and Math 117A More Eponential Models - Compound Interest Using A= P(1 + r) t 15) A person invests $7000 at 10% interest compounded annually. That is, each year, the amount in the account is times the amount of the previous year. Find an equation for the value of the investment after t years. 16) On the day you were born, your grandparents set a college fund for you. They deposited $10,000 in an account that paid 8% compounded annually. How much will you have available for college when you turn 18? 17) The population of a country was 2.5 million people in the year 2000 and since then it has been increasing at a rate of 2% annually. That is, each year, the population is about times the previous year s population. Write an equation for the population of the country (in millions) at t years since the year ) The revenue from music downloads was $1.98 billion in 2007 and has grown by about 86% per year since then. That is, each year the revenue is about times the previous year s revenue. a) Find an equation for the revenue, R(t) (in billions of dollars) in the year that is t years since b) Find R(7) and interpret in contet with correct units. 10

11 Mied practice: 19) A person invests $10,000 AT 5% interest compounded annually. Find an equation for the value of the investment after t years. How much money will there be in the account after 15 years? 20) The population of a country is growing eponentially. In the year 2000 there were 3.5 million people and the population doubles every five years. Produce an equation to model the population of this country in terms of years since ) The number of bacteria in is growing eponentially. Assume that now there are 2000 bacteria and the number triple every hour. Write an equation to model the number of bacteria in the dish in terms of the number of hours from now. 22) The number of bacteria in is growing eponentially. Assume that now there are 2000 bacteria and the number triple every 10 hours. Write an equation to model the number of bacteria in the dish in terms of the number of hours from now. 11

12 Math 117A and Math 093 Interpreting the coefficient a and the base b of an eponential function y = a* b For each of the following problems a) Interpret in contet the meaning of the coefficient b) Interpret in contet the meaning of the base: as a multiplier and as a rate. c) Use your knowledge of eponential functions and sketch the graph. Label the y-intercept and the aes in contet. 23) Eample: The function Pt ( ) = 200(1.12) t represents the population of a country P, in millions of people at t years since We want to interpret in contet the meaning of the coefficient a = 200 and the base b = 1.12 that are given in the function. Recall that: The coefficient a = 200 represents the y-intercept. The point (0, 200) = (0 years from 1995, 200 million people) The base b represents the multiplier per year and also, if you compare the two formats of the function: y = a* b and A= P(1 + r) t, then b = 1 + r. since b = 1.12 = , then, the rate is 0.12 or 12% The interpretation follows: In the year 1995 there were 200 million people in the country. Every year after that, the population has been increasing by multiplying by 1.12 which implies a rate of 12%per year. 24) The number of web pages contained in Google s inde, in millions, at t years since 2012 can be modeled with the function Gt ( ) = 30(1.95) t. 25) The average number of cable channels per household N, at t years since 2010 can be modeled with the function ( ) 135(1.08) t Nt = 12

13 26) The number of TiVo subscribers who got TiVo through DIRECTV, in millions, at t years since 2007 can be modeled with the eponential function ( ) 25(1.50) t St = 27) Let N(t) represent the population of a country in millions at t years since Nt ( ) = 2.5(1.25) t 28) The price of a car, in thousands, at t years since bought is given by the model Pt ( ) = 25(0.87) t 13

14 MA 093 and Math 117A More Eponential Models - Eponential Decay 29) A storage tank contains a radioactive element. Let p = f(t) be the amount (in grams) of the element that remains at t years after today. The graph for f is shown below: a) Use the graph to determine the initial amount of radioactive substance in the tank. Use proper units. b) Use the graph to determine the half life of the element? Use proper units. c) Use the graph to estimate the amount remaining 70 years from today. Use proper units. d) Use the graph to estimate when the amount remaining will be 20 grams? Use proper units. e) Use the graph to read the coordinates of 4 points related to the half-life information starting with the Y-intercept. Record the coordinates on the table. X Y f) Use algebra and the first two points from the table in part (e) to find the eponential function y = a b that fits the data; round to three decimal places. g) Use the points from part (e) and the eponential regression feature of the calculator to find the eponential function y = a b that fits the data; round to 3 decimal places. h) Complete the following: Initially there were of radioactive substance in the tank. Every year, the amount remaining is decreasing by subtracting/multiplying (circle one) by. Every % of the substance remains and % decays 14

15 30) A storage tank contains a radioactive element. Let p = f(t) be the amount (in grams) of the element that remains at t years after now. The graph for f is shown below. (1) Use -scale = y-scale = 10 and label the tic-marks along the aes. (2) Use -scale = 20, y-scale = 30 and label the tic-marks along the aes a) How many grams of the radioactive substance does the tank contain today? Use proper units. b) Use the graph to determine the half life of the element? Use proper units. c) Use the graph to read the coordinates of 4 points related to the half-life information starting with the Y-intercept. Record the coordinates on the table. y y d) Use the graph to estimate the amount remaining in 25 years. Use proper units. e) Use the graph to estimate when will there be left 40 grams of radioactive substance left. Use proper units 15

16 Problem continued: f) Use algebra and the first two points from the table in part (c) to find the eponential function that fits the data; round to three decimal places. g) Use the points from part (c) and the eponential regression feature of the calculator to find the eponential function y = a b that fits the data; round to 3 decimal places. a) Interpret the coefficient and the base of the eponential model 16

17 Problem continued i) Use the model equation to find the amount remaining 25 years from today. Use proper units. d) Use the model equation and the graphical approach to determine when there will be 40 grams in the tank. Graphical Approach i. Write the equation ii. Enter the left hand side of the equation in Y1 of the calculator iii. Enter the right hand side of the equation in the Y2 of the calculator iv. Enter appropriate window values. v. Press 2 nd TRACE [CALC] vi. Select 5:intersect vii. Press ENTER three times until you find the point of intersection viii. Answer the problem Graphical Approach i. Write the equation ii. Enter the left hand side of the equation in Y1 of the calculator iii. Enter the right hand side of the equation in the Y2 of the calculator iv. Enter appropriate window values. v. Press 2 nd TRACE [CALC] vi. Select 5:intersect vii. Press ENTER three times until you find the point of intersection viii. Answer the problem 17

18 Math 117A and Math 093 Eponential Decay - Half Life In each of the following two situations do the following: a. Identify the variables and corresponding units b. Write the coordinates of 4 points satisfying the pattern if originally we have 100% of the radioactive material. c. Use algebra to construct the function. d. Use the regression feature of your calculator to construct the function. e. Use your knowledge of eponential functions to sketch the graph and label in contet. f. Use the graphical approach to find how long will it take for 20% of the substance to remain 31) Strontium 90 is a radioactive material with half-life 28.4 years. 32) The half-life of radioactive potassium is 1.3 billion years. 18

19 Math 117A and Math 093 Eponential Decay - Half Life In each of the following two situations do the following: a. Identify the variables and corresponding units b. Write the coordinates of 4 points satisfying the pattern if originally we have 400 grams of the radioactive material. c. Use algebra to construct the function. d. Use the regression feature of your calculator to construct the function. e. Use your knowledge of eponential functions to sketch the graph and label in contet. f. Use the graphical approach to find how long will it take for 100 grams of the substance to remain 33) The half-life of radium is 1690 years. 34) The half-life of iodine 131 is 8 days 19

20 Math 117A and Math 093 Mied Practice 35) The number of web pages contained in Google s inde, in millions, at t years since 2012 can be modeled with the function Gt ( ) = 30(1.95) t. a) Find G(10) and interpret in contet. b) Find the G-intercept and interpret in contet. 36) The average number of cable channels per household N, at t years since 2010 can be modeled with the function Nt ( ) = 135(1.08) t a) Interpret in contet the meaning of the mathematical statement N(8) = 250 b) Interpret in contet the meaning of the base b = 1.08; as a multiplier and as a rate. 37) We deposit $30,000 in an account that pays 12% compounded annually. a) Construct the corresponding function. b) When will the amount in the account double? Use the graphical approach to answer. 38) Sketch the scatter-diagram for the data and use the regression feature of the calculator to construct an equation for the best model that fits the data. Flouride Water Content 39) There are 20 grams of bacteria in a dish. Assuming that the bacteria triples every 5 hours, construct an eponential model to fit the data. 20