name: period: estimated test date: unit 3a modeling with … · 2019-11-28 · common core math 2...

Common Core Math 2 Unit 3A Modeling with Exponential Functions

1

Name:____________________________ Period: _____ Estimated Test Date: ___________

Unit 3A

Modeling with Exponential

Functions


2


3

Main Concepts Page # Study Guide 4-6 Vocabulary 7 Equivalent Forms of exponential Expressions 7-10 Simplifying Radicals 11-13 Operations with Radicals 14-17 Rational Exponents and Radicals 18-21 Exponential Growth and Decay 22-31 Test Review 38-39 Homework Answers 40-


4

Common Core Standards

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

c. Use the properties of exponents to transform expressions for exponential functions.

A-CED.1 Create equations and inequalities in one variable and use them to solve problems.

A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

F-BF.1 Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-IF-7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior.

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Unit Description In this unit, students will continue their exploration of exponential functions begun in Math I. Students will extend the properties of exponents to rational exponents. Students will continue to explore transformations of exponential functions, adding vertical stretches and compressions. Students will also use exponential functions and equations to solve more complex real world problems (that may involve rational exponents), interpreting values in context. Essential Questions By the end of this unit, I will be able to answer the following questions: Exponential functions model real world problems, of growth and decay, such as monetary growth,

population increases or decreases, car values, half-life, etc. One type of function does not fit all situations in life.

Exponential functions can be written as explicit expressions or using a recursive process.


5

Unit Skills I can: Solving exponential equations The method of graphing or making a table to solve an exponential equation is appropriate when an exact

answer is not required.

Graphs and tables of exponential functions The general form of an exponential function is 𝑓(𝑥) = 𝑎 · 𝑏𝑥, where a is the initial value and b is the

base/common ratio. (F-BF.1)

Every exponential function has a horizontal asymptote. (F-IF.4)

The graph of 𝑓(𝑥) + 𝑘 is shifted k units vertically as compared to the parent graph of 𝑓(𝑥). If k is positive, the

graph shifts up. If k is negative, the graph shifts down.

The graph of 𝑓(𝑥 + 𝑘) is shifted k units horizontally as compared to the parent graph of 𝑓(𝑥). If k is positive,

the graph shifts left. If k is negative, the graph shifts right.

The graph of 𝑘 ∙ 𝑓(𝑥) is scaled vertically by a factor of k as compared to the parent graph of 𝑓(𝑥) (i.e. the point

(𝑥, 𝑦) maps to the point (𝑥, 𝑘 ∙ 𝑦). If the absolute value of k is greater than 1, then the graph is stretched

vertically. If the absolute value of k is less than 1, then the graph is compressed vertically.

Applications of exponential functions Exponential functions model real world problems of growth and decay including, but not limited to, monetary

growth, population increases or decreases, car values, and half-life.

The equation for exponential growth is given by 𝑦 = 𝑎(1 + 𝑟)𝑥 and the equation for exponential decay is given by 𝑦 = 𝑎(1 − 𝑟)𝑥, where a is the initial amount, r is the growth or decay rate in decimal form, and x is the number of time intervals that have passed.


6

Vocabulary: Define each word and give examples and notes that will help you remember the word/phrase.

Common Difference

Example and Notes to help YOU remember:

Common Ratio


Domain


Explicit function


Practical Domain


Theoretical Domain


Recursive function



7

Equivalent Forms of Exponential Expressions

Before we begin today’s lesson, how much do you remember about exponents? Use expanded form to write the rules for the exponents.

OBJECTIVE 1 Multiplying Exponential Expressions

32 ∙ 34 𝑦4 ∙ 𝑦10 123 ∙ 125

SUMMARY: 𝑎𝑚 ∙ 𝑎𝑛 = ___________

OBJECTIVE 2 Dividing Exponential Expressions (Remember: 𝑥

𝑥= 1)

36

32

𝑦10

𝑦4

125

123

SUMMARY: 𝑎𝑚

𝑎𝑛 = ___________

OBJECTIVE 3 Negative Exponential Expressions: Simplify 2 WAYS using expanded form AND the rule from OBJECTIVE 2

32

36

𝑦4

𝑦10

123

125

SUMMARY: 1

𝑎𝑛 = ___________

OBJECTIVE 4 Exponential Expressions Raised to a Power

(36)2 (𝑦3)4 (12𝑚)5

SUMMARY: (𝑎𝑚)𝑛 = ___________ SUMMARY: (𝑎 ∙ 𝑏)𝑛 = ____________ We’ve learned how to simplify exponential expressions in the past and reviewed those just now. Next we need to use those properties to find some missing values.


8

Find the value of x in each of the following expressions.

5𝑥 ∙ 52 = 57

3−2 ∙ 3𝑥 = 32 42 3⁄ ∙ 4𝑥 = 4

(53)𝑥 = 56

(3−2)𝑥 =1

32

(4𝑥)1 2⁄ = 4

56

5𝑥= 54

3𝑥

312=

1

32

43 2⁄

4𝑥= 4

52

5𝑥= 1

(3𝑥)2 = 1 464𝑥 = 1

Find the values of x and y in each of the following expressions.

5𝑥

3𝑦= (

5

3)

2

(23

3𝑥)

−2

=36

2𝑦 (

𝑥2

5)

3

=26

5𝑦

(5 ∙ 6)2 = 5𝑥6𝑦

(2𝑥 ∙ 63)4 = 28 ∙ 6𝑦 (3𝑥42)3 = 4𝑦


9

These problems will have more than one correct solution pair for x and y. Find at least 3 solution pairs.

(5𝑥)(5𝑦) = 512 Possible Solutions

Option 1

Option 2

Option 3

(3𝑥)𝑦 = 33 Possible Solutions

Option 1

Option 2

Option 3

4𝑥

4𝑦= 1

Possible Solutions

Option 1

Option 2

Option 3

When there are so many rules to keep track of, it’s very easy to make careless mistakes. To help you guard against that, it helps to become a critical thinker. Take a look at the expanded and simplified examples below. One of them has been simplified correctly and there’s an error in the other two. Identify the correctly simplified example with a . For the incorrectly simplified examples, write the correct answer and provide suggestions so that the same mistake is not made again.

𝑥2

𝑥3= 𝑥

(4𝑥)(𝑥) = 4𝑥2

50𝑐2𝑑2

5𝑐𝑑5= 45𝑐2𝑑3

You’ve seen some of the more common mistakes that can happen when simplifying exponential expressions, and you may have made similar mistakes in the past. For each of the next rows of problems, complete one of the problems correctly and two of the problems incorrectly. For the incorrect problems, try to use errors that you think might go unnoticed if someone wasn’t paying close attention. When you finish, you’ll switch papers with two different neighbors (one for each row) so that they can check your work, find, fix, and write suggestions for how those mistakes can be avoided.

(2𝑥2𝑦3)5 (3𝑥)−2(𝑥2) −3𝑥2

𝑦6

2𝑥𝑦2

8𝑥2𝑦

3−224𝑥3𝑥 (−2𝑥𝑦)4


10

Properties of Exponents (©Kuta Software – Infinite Algebra I)


11

Simplifying Radicals & Basic Operations

We are familiar with taking square roots (√ ) or with taking cubed roots (√3

), but you may not be as familiar with the elements of a radical.

√𝑥𝑛

= 𝑟

An index in a radical tells you how many times you have to multiply the root times itself to get the radicand.

For example, in √81 = 9, 81 is the radicand, 9 is the root, and the index is 2 because you have to multiply the root by itself twice to get the radicand (9 ∙ 9 = 92 = 81). When a radical is written without an index, there is an understood index of 2.

√643

=? Radicand Index Root is ______ because ___ ∙ ___ ∙ ___ = ___3 = 64

√32𝑥55=?

Radicand Index Root is ______ because ___ ∙ ___ ∙ ___ ∙ ___ ∙ ___ = ___5 = 32𝑥5

Yes…you can use your calculator to do this, but for some of the more simple problems, you should be able to figure them out in your head. To use your calculator Step 1: Type in the index. Step 2: Press MATH

Step 3: Choose 5:√𝑥

Step 4: Type in the radicand.

√243𝑦55

√1296𝑚4𝑛84 √144𝑣8

BE CAREFUL THAT YOUR VARIABLE ONLY STANDS FOR A POSITIVE NUMBER.

For instance, √𝑎2 = 𝑎, 𝑖𝑓 𝑎 ≥ 0 , but if a < 0, then √𝑎2 = −𝑎 (the opposite of a), since the square root sign

always indicates the positive square root. Since there is no way for us to know if a is positive or negative, we

use absolute value. So, √𝑎2 = |𝑎|

Example: √𝑎3𝑏2 = √𝑎 ∙ 𝑎2 ∙ 𝑏2 = 𝑎|𝑏|√𝑎

a cannot be negative because we would not have been able to take the square root of a3. However, b could

be negative, so use absolute value signs.


12

BUT not every problem will work out that nicely!

Try using your calculator to find an exact answer for √243

= ________________ The calculator will give us an estimation, but we can’t write down an irrational number like this exactly – it can’t be written as a fraction and the decimal never repeats or terminates. The best we can do for an exact answer is use simplest radical form. Here are some examples of how to write these in simplest radical form. See if you can come up with a method for doing this. Compare your method with your neighbor’s and be prepared to share it with the class. (Hint: do you remember how to make a factor tree?)

√𝟏𝟐 = 𝟐√𝟑

√𝟐𝟒𝟑

= 𝟐√𝟑𝟑

√𝟒𝟖𝟒

= 𝟐√𝟑𝟒

Simplifying Radicals: 1) ________________________________________________________

2) ________________________________________________________

3) ________________________________________________________ Examples:

216x x8 315x

3 8 4 96

3 404 364x

5 6332 yx

3 42381 zyx 3 275192 zyx 4 241875 zx

3 580n


13


14

Multiplying Radicals –

When written in radical form, it’s only possible to write two multiplied radicals as one if the index is the same. As long as this requirement is met,

1) multiply the _____________

2) multiply the _____________ 3) Simplify!

2532 283 10354

xyyx 53 2 323 1086 xyyx 5234 155 yxyx

33 2 854 xyx 4 334 5 402 yxx 5 535 3 9274 yxx

33 3 50253 yx 33 249 44 328


15

Adding & Subtracting Radicals

You’ve been combining like terms in algebraic expressions for a long time! Show your skills by simplifying the following expressions. 2𝑥 − 𝑥 + 4𝑥 = ______________________ 3𝑦 − 2𝑥 + 𝑦 − 6𝑦 = ____________________ Usually we say that like terms are those that contain the same variable expression, but they can also contain the same radical expression. When you add or subtract radicals, you can only do so if they contain the same index and radicand. Just like we don’t change the variable expression when we add or subtract, we’re not going to change the radical expression either. All we are going to do is add or subtract the coefficients. Always simplify the radical before you decide that you can’t add or subtract.

3 + 4 + 2 + 3 4 -

-

3 3 5 5 5 12 75

3x45 3x2033 1082325 33 54163

33 4 851252 aa 33 1357409 aa 33 4 27165 yy

503186 33 1654 44 4832


16

Simplify each expression.

9√3 + 2√3

5√2 + 2√3 3√7 − 7√73

3√32 + 2√50

√200 − √72 14√𝑥𝑦3 − 3√𝑥𝑦3

4√813

− 3√723

− 3√243

3√12 + 7√75 − √54

Simplify 2 – 7

A. -15√𝑥 B. -15x C. -5x

D. -5√𝑥

5√32 - 7√8

-7√11 + 3 √11

Multiply • . Simplify. A. 25

B. 5√53

C. 5√253

D. 5

A garden has width √13 and length 7√13. What is the perimeter of the garden in simplest radical form?


17

HW Operations with Radicals


18

Rational Exponents & Radicals Raising a number to the power of ½ is the same as performing a familiar operation. Let’s take a

look at the graph of 𝑦 = 𝑥1

2⁄ to discover that operation.

Step 1: Type 𝑥1

2⁄ into the y= screen on your graphing calculator. Step 2: Look at the table of values generated by this function. Verify that you have the same values as the rest of your class. (It is very easy to make a mistake when you type in the exponents here!) Step 3: Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Where have you seen this relationship before? Summarize your findings in a sentence.

Step 4: Type 𝑥1

3⁄ into the y= screen on your graphing calculator. Step 5: Look at the table of values generated by this function. Verify that you have the same values as the rest of your class. (It is very easy to make a mistake when you type in the exponents here!) Step 6: Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Have you seen this relationship before? Summarize your findings in a sentence. Step 7: Type 25𝑥 into the y= screen on your graphing calculator. Step 8: Adjust your table so that the values go up by ½ and begin at 0. Verify that your table contains the same values as the rest of your class. Step 9: Discuss with your classmates the pattern you see. Use the table below to help you see the pattern. (One row has been completed for you). Summarize your findings in the space beside the table.

X (exponent)

X (exponent) as a fraction with a

denominator of 2 Y1 (25x)

Rewrite Y1 as a power of 25 with fraction

exponents

Rewrite Y1 as a power of

√25

0

.5

1

1.5 3

2 125 253 2⁄ (√25)3

2

2.5

3

3.5

How could you use this pattern to find the value of 363/2? Check your answer in the calculator.


19

How could you use this pattern to find the value of 272/3? Check your answer in the calculator. How could you use this pattern to find the value of 815/4? Check your answer in the calculator. Step 10: Generally speaking, how can you find the value of an expression containing a rational exponent. Use the expression 𝑎𝑚/𝑛 to help you in your explanation. You try: Rewrite each of the following expressions in radical form.

y-9/8

(𝟑𝟐𝟓)𝟓

x 1.2

Now, reverse the rule you developed to change radical expressions into rational expressions.

2

3

x 3

2

)27( 4

5

)16( x

4

1

2a 2

7

4

5 2 53 )6( 75

7 4 39 27 3x


20

Earlier in this unit, you learned that when written in radical form, it’s only possible to write two multiplied radicals as one if the index is the same. However, if you convert the radical expressions into expressions with rational exponents, you CAN multiply or divide them (as you saw in your warm-up)! Give it a try Write your final answer as a simplified radical.

12√𝑦3

4√𝑦

(√𝑎23

√𝑏)

−6

(2√𝑎

4)

3∙ √𝑎3

√𝑥124∙ √𝑦−2 √64𝑥3

√512𝑥93 √625𝑥84

√𝑥27∙ √𝑥314

1

√−27𝑥93 (√𝑥 ∙ √𝑦23)

−6

How does the idea of simplifying radicals relate to the idea of rational exponents? There are several ways to approach this. Develop your own method for calculating simplest radical form of an expression without converting to radical form until the very last step!

𝑎32

𝑏64

𝑐105

𝑑253

Describe your method for simplifying radicals from rational exponents. Share your method with the class.


21

Radical and Rational Exponents (Kuta Software – Infinite Algebra 2)

©P F2

Q011

P21 rK

DuUt

oaN rS

Qoqfft8w

GarrQ

es 3L

ILsC

6.E j

dAkldl

0 XrCilg

FhEtT

sS Kr

seks

WeJrh

vFe

xdS.u Y

BMja

jdYe

7 Rw

ZigtUh

B RIU

nJfSiNn

5iatze

O mA

il2gVe

wb2rN

aM 22

U.gW

ork

shee

t by K

uta

So

ftw

are

LL

C

17)

( 6

v)1

.5

18)

m

-1 2

Writ

e ea

ch e

xp

ress

ion

in

exp

on

enti

al

form

.

19)

(4

m)3

20)

(3

6

x)4

21)

4

v22)

6

p

23)

(3

3

a)4

24)

1

(3

k)5

Sim

pli

fy.

25)

9

1 226)

34

3

-4 3

27)

10

00

00

0

1 628)

36

3 2

29)

(

x6)

1 230)

( 9

n4)

1 2

31)

( 64

n1

2)

-1 6

32)

( 81

m6)

1 2

-2-

©a X

2T0

I1q2

a pKhu

Rtaa

0 lSAo

jf2tjw

6a2rk

eE rL

xLZC

g.W A

4Akl2l

l 0rwiV

gCh

Ptlso hr

Sems

TeurO

vZe

qdp.7 o

oMia

2dKe

K 7w

Lijtuh

F AIU

nNf

4iBn

yi0t2e

U GA

HlGg

Be4b

lrGaj n2

y.iW

ork

shee

t by K

uta

So

ftw

are

LL

C

Ku

ta S

oft

war

e -

Infi

nit

e A

lgeb

ra 2

Nam

e__

___

__

__

___

__

__

__

__

__

__

__

__

__

__

__

_

P

erio

d_

__

_D

ate_

__

__

__

__

__

__

__

_R

adic

als

and

Rat

ion

al E

xp

on

ents

Wri

te e

ach

ex

pre

ssio

n i

n r

ad

ica

l fo

rm.

1)

7

1 22

)

4

4 3

3)

2

5 34

)

7

4 3

5)

6

3 26

)

2

1 6

Wri

te e

ach

ex

pre

ssio

n i

n e

xp

on

enti

al

form

.

7)

(1

0)3

8)

6

2

9)

(4

2)5

10

)

(4

5)5

11

)

32

12

)

61

0

Wri

te e

ach

ex

pre

ssio

n i

n r

ad

ica

l fo

rm.

13

)

( 5

x)

-5 4

14

)

( 5

x)

-1 2

15

)

( 10

n)

3 21

6)

a

6 5

-1-


22

Investigation : Exponential Growth & Decay Materials Needed:

Graphing Calculator (to serve as a random number generator) To use the calculator’s random integer generator feature:

1. Type any number besides zero into your

calculator, press STO , MATH, , ENTER,

ENTER

2. Press MATH 5 1 , 6 )

You can use numbers other than 1 and 6 here. The calculator will choose numbers between and including these numbers when you press enter. Continue pressing enter for more numbers.

Investigation: 1. Choose a recorder to collect the class’s data on the board. You’ll copy the data down in your table later. 2. Everyone in the class should stand so that the recorder can count everyone and record the number of people

standing in a table for “Stage 0”. 3. Use your calculator to find a random integer between 1 and 6. If you roll a 1, sit down. Before proceeding,

allow time for the recorder to count the number of people still standing. When the recorder is finished counting, (s)he will let you know.

4. Repeat step 3 until fewer than 3 people are standing (or you run out of room on the table). 5. Record the data in your table.

Stage 0 1 2 3 4 5 6 7 8 9 10

People Standing

Questions: 1. What is your initial value for this set of data? What does it represent in the investigation? 2. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 3. Based on your answer to Question 2, find the r OR d for the data you collected. Show the process you used to do

so. 4. Could you estimate your answer to Question 3 without conducting the exploration? If so, how? 5. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 6. Write an explicit function using function notation that would help you make predictions for this data. In your

function let x be the stage of the investigation and let f(x) equal the number of people standing in that stage.


23

Investigation 2: Half of a radioactive substance decays every 53 years. How much will remain of a 12 milligram sample after 530 years?

Complete the table. Years 0 53 106 159 212 265 318 371 424 477 530

Remaining radioactive substance

Questions: 7. What is your initial value for this set of data? What does it represent in the investigation? 8. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 9. Based on your answer to Question 8, find the r OR d for the data you collected. Show the process you used to do so. 10. Could you estimate your answer to Question 9 without filling in the table? If so, how? 11. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 12. Write an explicit function using function notation that would help you make predictions for this data. In your

function let x be the number of 53 year increments in the investigation and let f(x) equal the amount of radioactive substance remaining.

13. Write an explicit function using function notation that would help you make predictions for this data. In your

function let x be the number of years in the investigation and let f(x) equal the amount of radioactive substance remaining. Use your equation to determine how much radioactive substance will remain after 500 years.

14. When will there be exactly 5 milligrams of the radioactive substance? Determine your answer to the nearest

month. 15. Compare Investigation 1 and Investigation 2. What are the similarities and differences?


24

Investigation 3: You invest money in a savings account that earns 2.5% interest annually. How much money will you have at the end of 10 years if you begin with $1000?

Complete the table. Years 0 1 2 3 4 5 6 7 8 9 10

Money in your account

Questions: 16. What is your initial value for this set of data? What does it represent in the investigation? 17. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 18. Based on your answer to Question 17, find the r OR d for the data you collected. Show the process you used to

do so. 19. Could you estimate your answer to Question 18 without filling in the table? If so, how? 20. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 21. Write an explicit function using function notation that would help you make predictions for this data. In your

function let x be the number of years in the investigation and let f(x) equal the amount of money in the account. 22. You find a bank that will pay you 3% interest annually, so you consider moving your account. Your current

bank decides you’re a good customer and offers you a special opportunity to compound your interest semiannually‼! (They make it sound like it’s a really good deal, so you’re curious). You don’t play around with your money, so you ask what exactly that means. They explain that you’ll still get 2.5% interest, but they’ll give you 1.25% interest at the end of June and 1% interest at the end of December. You want to see if you make more money than you would if you just switched banks, so you do the calculations. Which bank is giving you a better deal? Explain your answer.


25

When you write an equation for a situation and use it to make predictions, you assume that other people who use it will understand the situation as well as you do. That’s not always the case when you take away the context, so we sometimes need to provide some additional information to accompany the equation.

23. The domain of a function is the set of all the possible input values that can be used when evaluating it. If you remove your functions in Questions 6 and 13 and 21 from the context of this situation and simply look at the

table and/or graph of the function, what numbers are part of the theoretical domain of the function? Will this be the case with all exponential functions? Why or why not?

24. When you consider the context, however, not all of the numbers in the theoretical domain really make sense.

We call the numbers in the theoretical domain that make sense in our situation the practical domain. For instance, in the first investigation, our input values are “Stages”. If you look at the tables you created, what numbers would be a part of the practical domain for these investigations?

Investigation 1 Investigation 2 Investigation 3 Make note of any similarities and differences and explain why they exist.


26

Summary of Findings

The results of Investigation 1 & 2 simulate exponential decay because the values are being multiplied by the same value between 0 and 1 each time. The results of Investigation 3 simulate exponential growth because the values are being multiplied by the same value greater than 1 each time. As you may have noticed in Questions 4, the common ratio is related to the probability that you will be randomly

assigned a 1 out of 6 options. Since this probability is 1:6, or 1

6, or 0.16̅, or 16. 6̅% that fraction of the class will sit

down. These numbers do not represent our common ratio, however. That’s because when 1 of 6 people sit down,

that leaves 5:6, or 5

6 or 0.83 or 83% of the class still standing. To find the common ratio (b) using this probability

(p) or any percentage for that matter, you can use the following equations. The probability or percent MUST be written as a decimal to use this formula.

Exponential Growth 𝑏 = |1 + 𝑝| This is because you begin with 100% (1 when written as a decimal) and add the same percentage each time.

Exponential Decay 𝑏 = |1 − 𝑝| This is because you begin with 100% (1 when written as a decimal) and subtract the same percentage each time.

In problems 5, 11, and 20, you wrote recursive functions for the exponential data. Make sure you include the starting value of your data (a) – this will always be where the independent variable is 0. 𝑁𝐸𝑋𝑇 = 𝑁𝑂𝑊 ∙ 𝑏, Starting at 𝑎 𝑦 = 𝑎𝑏𝑥 For each of these Investigations, we came up with functions with theoretical domains of all real numbers. In fact, all exponential functions will have domains of all real numbers. However, when we considered the context, we realized that the practical domain for these the first investigation only involves the whole numbers because all of the other real numbers would not work as “stages”. We also saw that in most scenarios where the domain is related to time, the domain is only positive values.


27

HW: ZOMBIES!

SCENARIO 1

A rabid pack of zombies is growing exponentially! After an hour, the original zombie infected 5 people total, and those 5 zombies went on to infect 5 people, etc. After a zombie bite, it takes an hour to infect others. Develop a plan to determine how many newly infected zombies will be created after 4 hours.

If possible, draw a diagram, create a table, a graph, and an equation.

SCENARIO 2

During this attack, a pack of 4 zombies walked into town last night around midnight. Each zombie infected 3 people total, and those 3 zombies went on to infect 3 people, etc. After a zombie bite, it takes an hour to infect others. Develop a plan to determine the number of newly infected zombies at 6 am this morning. If possible, draw a diagram, create a table, a graph, and an equation.

SCENARIO 3

At 9:00 am, the official count of the zombie infestation was 16384. Every hour the number of zombies quadruples. Around what time did the first zombie roll into town?


28

Guided Practice : Exponential Growth & Decay

1. You decide to conduct an investigation in a room of 100 standing students who were to randomly choose a number between and including 1 through 20. If they chose a multiple of 4, they were to sit down. You record the number of people still standing after each turn.

a. What is the probability of choosing a multiple of 4?

b. What is the common ratio for this investigation?

c. What is the initial value of this investigation?

d. Write a recursive equation for the investigation.

e. Write an explicit equation for the investigation.

f. Using your equations, how many stages of the investigation will occur before there are fewer than 3 people standing?

2. The amount of radioactive ore in a sample can be modeled by the equation 𝑦 = 20(0.997)𝑥, where x represents

years and y is the amount of ore remaining in milligrams. a. What was the initial amount of radioactive ore?

b. Is this an example of exponential growth or decay?

c. What percentage of the ore is being lost or gained according to this model?

d. When will there be half of the initial amount of the radioactive ore?

3. Complete the following table.

Explicit Function Recursive Function Initial Value Common Ratio Growth or Decay?

𝑦 = 2(3)𝑥

𝑁𝐸𝑋𝑇 = 𝑁𝑂𝑊 ∙ 0.5,𝑆𝑡𝑎𝑟𝑡 𝑎𝑡 − 3

0.125 4

𝑦 = 𝑎𝑏𝑥

growth


29

4. The height of a plant can be modeled by the table below. Day 0 1 2 3 4 Height (in) 2.56 6.4 16 40 100

a) What is the initial height of the plant? Explain how you found your answer?

b) What is the common ratio? Explain how you found your answer.

c) Is this an example of exponential growth or decay? How could you find the answer if you only had the common ratio?

d) Write the recursive function for this situation.

e) Write the explicit function for this situation.

f) How tall will the plant be on the 12th day of data collection? How tall is this in inches, feet, yards, and miles? Does this seem realistic to you?

g) On what day will the plant first be over 100 yards tall? 5. When you opened a savings account on your 15th birthday, you deposited most of the money from your

summer job ($2000). The banker who helped you informed you that you would received 1.5% interest each year.

a. How much money will you have in the account when you turn 21?

b. Use the properties of exponents to determine the monthly percentage interest rate that you could have gotten from the bank that would have given you the same amount of money when you turned 21.

6. For each scenario, write an explicit equation, define your variables, and determine the practical and theoretical

domain. a) The town of Braeford was first established in 1854 when it had a population of 24. Since then it has grown

by a percentage of 1.25% each year.

b) A species of bacteria reproduces exactly once each hour on the hour. At this exact time, each organism present divides into two organisms. One of these bacteria is placed into a petri-dish at 8:00 this morning.


30

HW : Exponential Growth & Decay

1. In 1990, Florida’s population was about 13 million. Since1990, the state’s population has grown about 1.7%

each year. This means that Florida’s population is growing exponentially.

Year Population 1990 1991 1992 1993 1994

a) Write an explicit function in the form 𝑦 = 𝑎𝑏𝑥 that models the values in the table.

b) What does x represent in your function?

c) What is the “a” value in the equation and what does it represent in this context?

d) What is the “b” value in the equation and what does it represent in this context? 2. Since 1985, the daily cost of patient care in community hospitals in the United States has increased about 8.1%

per year. In 1985, such hospital costs were an average of $460 per day. a) Write an equation to model the cost of hospital care. Let x = the number of years after 1985.

b) Find the approximate cost per day in 2012.

c) When was the cost per day $1000

d) When was the cost per day $2000?

3. To treat some forms of cancer, doctors use Iodine-131 which has a half-life of 8 days. If a patient received 12

millicuries of Iodine-131, how much of the substance will remain in the patient 2 weeks later?


31

4. Suppose your parents deposited $1500 in an account paying 6.5% interest compounded annually when you were born. a) Find the account balance after 18 years.

b) What would be the difference in the balance after 18 years if the interest rate in the original problem was 8% instead of 6.5%?

c) What would be the difference in the balance if the interest was 6.5% and was compounded monthly instead of annually?

5. Since 1980, the number of gallons of whole milk each person in the US drinks in a year has

decreased 4.1% each year. In1980, each person drank an average of 16.5 gallons of whole milk per year.

a) Write a recursive function for the data in the table.

b) Write an explicit function in the form y = abx that models the values in the table. Define your variables.

c) According to this same trend, how many gallons of milk did a person drink in a year in 1970?

6. The model 𝑦 = 604000(0.982)𝑥 represents the population in Washington, D.C. 𝑥 years after 1990.

a) How many people were there in 1990?

b) What percentage growth or decay does this model imply?

c) Write a recursive function to represent the same model as the provided explicit function.

d) Suppose the current trend continues, predict the number of people in DC in 2013.

Suppose the current trend continues, when will the population of DC be approximately half what it was in 1990?

Year Population 1980 1981 1982 1983 1984


32

Unit 3A Test Review

Simplify each expression, and write your final answer with rational exponents.

1. √36𝑠2 ∙ (𝑠6)1/3 2. 2𝑘2/3 ∙1

4𝑘5/6 3. 𝑥 √16

4∙ 24𝑥

Simplify each expression, and write your final answer in simplest radical form. 4. m1/2 ∙m4/3 5. (12n2∙24n1/4)3 6. √256𝑥84

∙ √8𝑥3

7. Explain why 1613/4=163∙√164

is a true statement.

Fill in the blank to make each statement true. 8. 2x6∙____= 8x 9. (10x-3)/____ = 5x 10. (____)2 = 32x 11. (____)-3 = 8m9

Write each expression in simplified radical form.

12. √8𝑚2𝑛45∙ √20𝑚4𝑛

5 13. √𝑘

3∙ 𝑘6/4 14. −3√16𝑦94

15. √72 − √75 + √98 16. √81𝑥3𝑦6 17. √(𝑏 − 5)2(𝑏 − 5)43

18. Explain how to calculate the value of 813/4 without using a calculator.


33

19. The function y = 290,000 (0.92)x represents the value of an old home that has been abandoned by its

owners x years ago. Find the decay rate of the old home.

20. If the population in 1995 for a Town A is 1,500, and is increasing at a rate of 2.3% every 5 years, what is the projected population of the Town A in 2025?

21. 2. If the population in 2001 for a Town B is 100,000, and is increasing at a rate of 1.5% every year, what is the projected population of the Town A in 2012?

22. If the population in 1999 for a Town C is 95,000, and is decreasing at a rate of 2.5% every 2 years, what is the projected population of the Town C in 2003?

23. If the population in 2005 for a Town D is 5,500, and is decreasing at a rate of 1.9% every year, what is the projected population of the Town C in 2009?


34

Homework Answers Properties of Exponents; Page 10 1) 4m5

2) 2m

3) 8

𝑟

4) 8n

5) 8k5

6) 4x2

7) 6y2x

8) 4v4u2

9) 12

𝑎𝑏

10) 𝑥5

𝑦2

11) 1

12) 1

16𝑥8

13) 256

14) 16a6

15) 81k16

16) 1

4𝑥𝑦

17) 1

2𝑏4

18) 𝑥4

𝑦2

19) 𝑦3

2𝑥4

20) 1

9𝑚2

21) 1

2𝑟

22) 1

4𝑥5

23) n

24) 1

2

25) 3

𝑚7

26) 2𝑥2

3𝑦𝑧7

27) 𝑧3

𝑥𝑦2

28) 2ℎ3𝑘3

3𝑗4

29) 4𝑚2𝑛

3𝑝

30) 3𝑥7

𝑦𝑧

Simplifying Square Roots; pages 13

THEY DON’T HAVE THE VEGAS IDEA.

Operations with Radicals; pages 17

I DON’T KNOW AND I DON’T CARE

1) 3√5

2) 5𝑥2√2

3) 11√3

4) 4√6

5) −60√2

6) 2𝑥𝑦√5y

7) 7√𝑥 + 3√𝑦

8) 2√30

5

9) √2

4

10) 6𝑛𝑡 √𝑛3

11) −30√6

12) 4√15

3

13) 30𝑛4𝑡2√7𝑡

14) 59√3

15) 3√11 + 2√22

16) √6

10

Radicals and Rational Exponents; pages 21

1) √7

2) (√43

)4

3) (√23

)5

4) (√73

)4

5) (√6)3

6) √26

7) 103

2

8) 21

6

9) 25

4

10) 55

4

11) 21

3

12) 101

6

13) 1

( √5𝑋4

)5

14) 1

√5𝑋

15) (√10𝑛)3

16) (√𝑎5

)6

17) (√6𝑣)3

18) 1

√𝑚

19) 𝑚3

4

20) (6𝑥)4

3

21) 𝑣1

4

22) (6𝑝)1

2

23) (3𝑎)4

3

24) (3𝑘)−5

2 25) 3

26) 1

2401

27) 10

28) 216

29) 𝑥3

30) 3𝑛2

31) 1

2𝑛2

32) 9𝑚3


35

Zombies; Page 27

1.

y = 1(5)x

Hours New zombies

0 1 1 5 2 125 3 625 4 3125

2.

y = 4(3)x

Hours New zombies

0 4 1 12 2 36 3 108 4 324

3.

y = 16384(4)-7 = 1

Hours New zombies

9 am 16384 8 am 4096 7 am 1024 6 am 256 5 am 64 4 am 16 3 am 4 2 am 1

Exponential Growth and Decay; pages 30-31 1. a. y = 13(1.017)x b. years since 1990 c. 13 mil in FL in 1990 d. 1.017 growth of 100% + 1.7% 2. a. y = 460(1.081)x

b. x = 27 y = $3767.49 c. x = 9.97 yrs y = $1000 at end of 1994 d. x = 18.869 yrs y = $2000 at end of 2003

3. y = 12(.5)14/8 = 3.57 millicuries

4. a. $4659.98 b. $1334.05 c. $157.77

5. a. NEXT = NOW * 1.041 start at 16.5 b. y = 16.5(1.041)x c. 11.04 gal

6. a. 604000 b. 18% decay c. NEXT = NOW * 0.982 start at 604000 d. 397741 e. 2028

Test Review; pages 32-33

1. 6𝑠3

2. 1

2𝑘

3

2

3. 32𝑥2

4. 𝑚√𝑚56

5. 23,887,872𝑛6 √𝑛34

6. 8𝑥3√2𝑥

7. 163

4 8. 4𝑥−5

9. 2𝑥−4

10. √32𝑥2 or 25

2𝑥2

10. √32𝑥2 or 25

2𝑥2

11. 1

2𝑚−3 or 2−1𝑚3

12. 2𝑚𝑛 √5𝑚5

13. 𝑘 √𝑘56

14. −6𝑦2 √𝑦4

15. 13√2 − 5√3

16. 27 17. (𝑏 − 5)2 18. Answers vary 19. 8% 20. 1,719 21. 117,795 22. 90,309 23. 5,094


36

name: period: estimated test date: unit 3a modeling with … · 2019-11-28 · common core math 2...

Documents