PAP Algebra 2. Unit 7A. Exponentials Name Period

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1 PAP Algebra 2 Unit 7A Exponentials Name Period 1

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34944 2006 23006 35643 Data estimated from US Census data 1. Make a scatterplot of the data on the 2 graphs below. 2. What trend do you notice?

3 Pre-AP Algebra After Test HW Intro to Exponential Functions Introduction to Exponential Growth & Decay Who gets paid more? Median Income of Men and Women Year Women Men Data estimated from US Census data 1. Make a scatterplot of the data on the 2 graphs below. 2. What trend do you notice? 3. Taking the women s incomes, what common number are they multiplying by each time? 4. Taking the men s incomes, what common number are they multiplying by each time? 3

4 Pre-AP Algebra After Test HW What would you predict the earnings to be in 2010? F(x) = a(b) x is the standard form for this data. Initial amount is a amount. The rate by which the table is growing is the b amount. Now, go to stat and enter the years as 0 x 6 in L1 and the women income in L2. Go to your regression menu and select exponent regression (expreg). Fill in the equation it gives you. How does that compare to your women function? a) Set your window to: X min: 0 X max: 10 Y min: 0 Y max: max value in table b) Make a scatterplot of your data on the calculator. c) Type W(x) or M(x) into y= and graph. See if the curve fits the data. d) What was the median income in 1990? Intro to Exponential Functions Women Men W(x) = ( ) x M(x) = ( ) x W(x) = ( ) x M(x) = ( ) x Median income in 1990: Median income in 1990: 1. Will the graph ever reach the negative numbers? How low do you think it will get? When a graph looks like it approaches a number but doesn t touch it, it s called an asymptote. This graph has an imaginary horizontal asymptote at y =. 2. Will women s earnings ever catch up to men s? Why or why not? 4

5 Pre-AP Algebra Intro to Exponential Functions After Test HW 3. Refer back to ACTIVITY 1. Is it increasing or decreasing? What is the domain and range? What is the initial value? What is the percent of increase or decrease? 4. Refer back to ACTIVITY 2. Is it increasing or decreasing? What is the domain and range? What is the initial value? What is the percent of increase or decrease? 5

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7 PAP Algebra II Notes 7.2 Graphing Exponential Functions GRAPHING EXPONENTIAL FUNCTIONS: y Ab B( x C) D 1) x y 2 2) x y 5 y 5 1 3) x 4) y 3 2 x 4 5) y 5 x 3 6) y 1 2 1x 3 7

8 x f ( x) A( b ) is the basic parent exponential function. In this equation b is the base and A is the coefficient. Only b is being raised to the power of x! The base is always positive!!! Given any two points, there is one exponential function that would pass through those points. Find the following exponential functions that pass through 1) (0,4) and (2, 36) 2) f(0)=9 and f(3)=1/3 3) f(1)=10 and f(3)=40 8

9 PAP Algebra II WS 7.2 Graphing Exponential Fns NONCALCULATOR Name: Graph the following exponential functions. Label at least 2 points and the horizontal asymptote. 1) x y 3 2) y 3 x 4 x 3) y ) y ( x 4) 5) x y 4 1 6) y 4 3 x 5 9

10 7) y 1 4 x x 1 8) y Find the exponential function passing through the given points. NO CALCULATOR!!! 9) (0,3) and (1,15) 10) f(0)=5 and f(4)=80 11) f(0)=64 and f(2)=4 12) f(0)=80 and f(4)=5 13) f(1)=12 and f(2)=48 14) f(1)=6 and f(3)=54 10

11 7.3 Activity Writing Exponential Functions Writing Equations of Exponential Functions (Non-Calculator) Write an exponential equation for these graphs Name: Write an exponential equation for these graphs. More than one transformation may be needed Given x 2 f(x) 2 3 and g(x) described by the graph below, a. describe the transformations from f(x) to g(x). b. write the equation of h(x) whose graph is the same as g(x) reflected in the x-axis and shifted two units left. 11

12 7.3 Activity Writing Exponential Functions 12

13 PAP Algebra II 7.3 Notes Growth & Decay & Half Life EXPONENTIAL GROWTH EXPONENTIAL DECAY y A(1 r ) t y A(1 r ) t * r is the rate written as a decimal! * t is the amount of time that has passed! EX1: A species of rabbits in increasing at a rate of 7% per year. If there are currently 10,000 rabbits, how many will there be in 15 years? EX2: A home in Frisco is currently worth $150,000. If homes in Frisco are appreciating at a rate of 2.4% per year, when will the home be worth $200,000? EX3: If you buy a new car for $25,000 and cars depreciate at a rate of about 8% per year, how much could you sell it for in 5 years? EX4: If you bought your car 7 years ago for $10,000 and today you can sell it for $3500, what was it s rate of depreciation? 13

14 HALF LIFE is a specific type of exponential decay. 1 y A 2 t h t is time, and h is half life. Make sure the units are the same on both! EX5: The half life of carbon 10 is 20 seconds. If you start out with 100 grams, how much will be left in 2 minutes? EX6: Carbon-14 decays with a half life of about 5730 years by the emission of an electron of energy. If archeologists find a rock that had 5 grams of carbon-14, and it now has 3.7 grams of carbon-14, how old is the rock? 14

15 PAP Algebra II WS 7.3 Name: 1) A species of dolphins is decreasing at a rate of 3.1% per year. If there are currently 20,000 dolphins, how many will there be in 30 years? 2) A home in Frisco is currently worth $300,000. If homes in Frisco are appreciating at a rate of 4.7% % per year, when will the home be worth $350,000? 3) If you buy a new car for $18,000 and cars depreciate at a rate of about 7% per year, how much could you sell it for in 3 years? 4) If you bought your car 5 years ago for $15,000 and today you can sell it for $7000, what was it s rate of depreciation? 5) The half life of beryllium 11 is 13.8 seconds. If you start out with 50 grams, how much will be left in 3 minutes? 6) Carbon-14 decays with a halflife of about 5730 years. If archeologists find a rock that had 10 grams of carbon-14, and it now has 7.2 grams of carbon-14, how old is the rock? 15

16 7) The table shows the population growth of one Knapwood grove over a 5-year period. Time (years) Number of Plants Which function rule best models the relationship between the number of years, x, and the number of Knapwood plants produced, y? A) y = 3x B) y = 4(3) x C) y = 3 4 x D) y = 3 x + 4 8) Cell growth can be modeled using the exponential functions. What will be the number of cells present after a single cell doubles 12 times? A) 24 B) 144 C) 2048 D) ) Suppose a culture of 1 bacterium is put into a Petri dish and the culture triples every hour. Identify the number of bacteria in the culture after 12 hours. 10) A rubber ball rebounds ¼ of the height from which it is dropped. How high is the seventh bounce if the ball is dropped from a height of 200 cm? Be sure to show your thinking. 11) Salem s little brother is learning to talk. He has a vocabulary of 30 words. Each week the number of words in his vocabulary doubles. At this rate, how many words will he know 5 weeks from now? 16

17 PAP Algebra 2 Notes 7.4 Compound Interest/Investments Name Compound Interest Formula The growth in the value of investments earning compound interest is modeled by an exponential function. The total amount of an investment, A, earning compound interest is r A( t) Ao 1 n where A 0 is the principal, r is the annual interest rate (expressed as a decimal), n is the number of times interest is compounded per year, and t is the time in years. If the compounding is done continuously, the formula becomes: A() t A e Ex1: If you won $1,000,000 in the lottery today and put it into savings for 10 years at an interest rate of 2.9%, how much would you have if your interest was compounded: nt o rt a) annually? b) quarterly? c) monthly? d) daily? e) compounded continuously? How much would you have if your interest was compounded continuously? 17

18 EX 2: Determine the starting amount that must be invested at 7.5%, compounded quarterly, so that $500,000 will be available for retirement in 20 years. EX3: If you invest $5,000 at a 3.9% interest rate compounded weekly, how long would it take for it to grow to $7000? EX4: If interest is compounded continuously at a rate of 5.12%, how long will it take for your money to double? Ex 5: The value V (in millions of dollars) of a famous painting can be modeled by V = 10e rt where t represents the year, with t = 0 corresponding to In 2004, the same painting was sold for $65 million. Find the value of r, and use this result to predict the value of the painting in Ex6: If the half life (in years) is 1599 for the radioactive isotope 226RA, how much would remain after 1000 years if the initial quantity is 10g. 18

19 PAP Algebra II WS 7.4 Name: Per 1) If you get a total of $3000 as gifts when you graduate from high school and you put it in a savings account that earns interest at a rate of 3.7% per year, how much would you have in 4 years when you graduate from college if your interest is compounded a) annually? b) quarterly? c) weekly? d) continuously? 2) If you invest $10,000 at a 2.6% interest rate compounded monthly, how long would it take for your investment to grow to $15,000? 3) If you invested $100 in an account that compounded a 1.9% interest rate continuously, how long would it take for you to have $1000? 4) If you find an account that has an interest rate of 2.8% compounded quarterly, how long would it take for your money to triple? 19 1 P a g e

20 5. A population of 150 deer in a state park quadruples every 10 years. a. Find the model that represents this situation. b. Find the population of deer in 55 years. 6. Suppose a retired professor who lives next door offers to pay you to work for him for the next 30 days. He says he will pay you $500 per day, or if you would like he will pay you $0.01 for the first day and double your pay each day after that. How much money would you make on the 30 th day with each option? Which option would you choose? 7. Suppose your parents invested $2000 in a money market account on the day you were born. The money market averages 6% interest compounded monthly. How much will you have in your college account after 18 years? 8. Find the accumulated amount when $8000 is invested for 25 years in an account that pays 4.6% annual interest compounded continuously. 9. A certain radioactive isotope has a half-life of 16 days. If one starts with 15 grams of the isotope, how much is left after 4 days? 20 2 P a g e

21 10. Suppose your current annual salary is $75,000 per year. Suppose inflation remains at a flat rate of 3.5% per year for the next 15 years. What will your salary need to be in order to have the same purchasing power you have now? Assume that inflation compounds continuously. 11. A certain drug has a half-life of 5 hours. Suppose you take a dose of 750 milligrams of the drug. How much of the drug is left in your bloodstream 24 hours later? 12. A certain radioactive isotope has a half-life of 27 years. How many years will it take 500 grams to decay to 25 grams? 13. How much money should be invested at 4% compounded quarterly for 15 years so that you have $25,000 at the end of 15 years? 14. A certain radioactive isotope decays from 70 to 36 grams in 15 years. What is the half-life of the isotope? 15. In the spring squirrels reproduce at an astounding rate. The number of squirrels quadruples every week. If there are currently 259 squirrels, how many will there be in 10 weeks? 21 3 P a g e

22 16. In Vegas there is a game that has a small chance of doubling your money each hand you win. If you have $500 and put it all in, how many hands will it take winning to give you $10,000? 17) You have a thick piece of cardstock that measures.2cm in thickness. Each time you fold it in half it doubles in thickness. How thick will the cardstock be if you fold it 10 times? 18) The value of a dollar is three times less today than it was 15 years ago due to inflation. What will be the value of a dollar 50 years from now? 22 4 P a g e

Objectives: Students will be able to model word problems with exponential functions and use logs to solve exponential models.

Pre-AP Algebra 2 Unit 9 - Lesson 6 Exponential Modeling Objectives: Students will be able to model word problems with exponential functions and use logs to solve exponential models. Materials: Hw #9-5