PAP Algebra 2

Unit 7A Exponentials

Name________________________

Period______

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Pre-AP Algebra Intro to Exponential Functions

After Test HW

Introduction to Exponential Growth & Decay Who gets paid more?

Median Income of Men and Women

Year Women Men

2000 21040 31650

2001 21356 32283

2002 21676 32929

2003 22001 33587

2004 22331 34259

2005 22666 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. 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?________

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Pre-AP Algebra Intro to Exponential Functions

After Test HW

Women Men 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.

W(x) = ______(_________)x

M(x) = ______(__________)x

Now, go to stat and enter

the years as 60 x 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?

W(x) = ______(_________)x

M(x) = ______(__________)x

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?

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?

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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?______________________________

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PAP Algebra II Notes 7.2 Graphing Exponential Functions

GRAPHING EXPONENTIAL FUNCTIONS: DAby CxB )(

1) xy 2 2) xy 5 3) xy

51 4) 423 xy

5) 35 xy 6) 12 3

1

x

y

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)()( xbAxf 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

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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) xy 3 2)

43 xy

3) 432 xy 4) )4(

21

3

x

y

5) xy 4 6) 54 3

1

x

y

9

7)

x

y

4

1 8) 3

4

12

x

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

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7.3 Activity – Writing Exponential Functions

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

-3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

4

5

6

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

9

10

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Writing Equations of Exponential Functions Name: ____________________

(Non-Calculator)

Write an exponential equation for these graphs.

1. 2.

____________________ ____________________

Write an exponential equation for these graphs. More than one transformation may be needed.

3. 4.

____________________ ____________________

5. Given x 2f(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.

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7.3 Activity – Writing Exponential Functions

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PAP Algebra II 7.3 Notes Growth & Decay & Half Life

EXPONENTIAL GROWTH EXPONENTIAL DECAY

(1 )ty A r (1 )ty A r

* 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?

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HALF LIFE is a specific type of exponential decay.

1

2

th

y A

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?

t is time, and h is half life. Make sure the units are the same on both!

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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?

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____________ 7) The table shows the population growth of one Knapwood grove over a 5-year period.

Time (years) Number of Plants

0 4

1 12

2 36

3 108

4 324

5 972

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●4x D) y = 3x + 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) 4096 ____________ 9) 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?

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PAP Algebra 2 Name ___________________

Notes 7.4 Compound Interest/Investments

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

( ) 1

nt

o

rA t A

n

where 0A 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: ( ) rt

oA 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:

_______________ a) annually?

_______________ b) quarterly?

_______________ c) monthly?

_______________ d) daily?

_______________e) compounded continuously? How much would you have

if your interest was compounded continuously?

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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 = 10ert where t represents

the year, with t = 0 corresponding to 1990. 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 2010.

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.

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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?

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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 30th

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?

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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?

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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?

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