UNIT 4: “POWER TRIP”

Standard 4.1: demonstrate understanding of the properties of exponents and to graph exponential functions (11-1, 11-2)

Standard 4.2: solve problems using exponential functions (11-2)

Standard 4.3: use the natural number e to solve problems (11-3)

Standard 4.4: demonstrate understanding of the properties of logarithms (11-4)

Standard 4.5: use common and natural logarithms to solve problems (11-5, 11-6)

Modeling Data

Check it out!!!… Graph the following equations.

Compare and contrast the graphs. How are they alike? Different?

Use MATH terminology in your descriptions. You know…stuff like: translation, reflection,

continuity, domain, intercepts, asymptotes, behavior over intervals…yeah, stuff like that!

26 6 3 6 6x x x xy y y y

STANDARD 4.1: DEMONSTRATE UNDERSTANDING OF THE PROPERTIES OF EXPONENTS AND TO GRAPH EXPONENTIAL FUNCTIONS (11-2)

The essential questions are… What will happen if my variable is the

exponent instead the base? Can exponential behavior be predicted? What sorts of problems can be solved using

exponential functions in real life?

EXPONENTIAL FUNCTIONS:

Functions in which there is a variable acting as an exponent.

The base will be some real number.

2xy

LET’S CHECK THEM OUT ON THE CALCULATOR.

How are these graphs alike? How are they different? What conclusions can I draw?

Graph:

2 3 5 10x x x xy y y y

Review Warm-up…TRANSFORMATIONS!!!

LOOK AT THESE GUYS…

What’s going on?

14

4

xxy y

STANDARD 4.1: DEMONSTRATE UNDERSTANDING OF THE PROPERTIES OF EXPONENTS AND TO GRAPH EXPONENTIAL FUNCTIONS (11-1)

The essential questions are… What affect does a power have on a number? How do exponents behave when their bases

are added or subtracted? Multiplied? Divided? Raised to a power?

What does it mean when a base has a negative exponent? A fractional exponent? An irrational exponent?

WHAT EFFECT DOES A POWER HAVE ON A NUMBER?

where is an integer.nb b

HOW DO EXPONENTS BEHAVE WHEN THEIR BASES ARE ADDED OR SUBTRACTED?

2 2 2

2 3 2 2

4 3 9

5 3 6 7x 2 4 3

x x x y

x x x x x

HOW DO THEY BEHAVE WHEN THEIR BASES ARE MULTIPLIED?

5 3 2 41. ( ) 2. (5 )( 7 )x x z xz

DIVIDED?

2 54 7

6 2 4

3 (3 )1. 2.

3 ( )x yx

RAISED TO A POWER?

Single base with an exponent?

A product of bases?

A quotient?

WHAT ABOUT NEGATIVE EXPONENTS?

22 1

33 22

1. 4 4

2. What is the diff erence between

3 and 3 ?ff

TRY SOME…

14 2

5

2 534 7

42 3

2 2 31. 2.

2 4

3. 4. x y

s tx y

WHAT IF MY EXPONENT IS NOT AN INTEGER?

1

n

m

n

b

b

SOLVING EQUATIONS INVOLVING RATIONAL EXPONENTS.

OKAY…WHAT IF MY EXPONENT IS NOT RATIONAL?

26

SOLVING EQUATIONS

5

2724 15 12a

SAGE AND SCRIBE. One piece of paper per partnership. One person does the thinking - Sage, the

other writes - Scribe. Sage must tell the scribe exactly what to do

but may not write anything. The sage must describe with words only.

Scribe writes exactly what the Sage tells them to write.

For next problem switch roles.

SAGE AND SCRIBE.

p A46 Lesson 11-1 #1 – 19 odd

HOMEWORK:

p 700 #21 – 67 odd, 71, 73

WARM-UP

p 708 # 1 – 7 odd

HOMEWORK:

STANDARD 4.2: SOLVE PROBLEMS USING EXPONENTIAL FUNCTIONS (11-2)

The essential questions are… What sorts of problems can be solved using

exponential functions in real life?

USING EXPONENTIAL FUNCTIONS IN THE REAL WORLD…Natural Phenomena

Things like: bacterial growth, radio-active decay and human or animal populations.

Investing and Finance. Mortgages, depreciation and retirement

funds all use exponential models.

GROWTH AND DECAY

0(1 )tN N r

General formula for exponential growth or decay is:

N = Final amount, No = initial amount, r = rate of growth or decay per time period, and t = number of time periods.

When will this represent growth, and when will it represent decay?

GROWTH AND DECAY

A car depreciates (or loses value) at a rate of 20% per year.

If the car originally cost $21,000, write the a function in order to find the depreciation.

Graph the depreciation function.

What will it be worth after 4 years?

COMPOUND INTEREST

1nt

rA P

n

A = final amount P = principle or initial investment r = annual interest rate N = number of compoundings per year t = the number of years

COMPOUND INTEREST

Determine the amount of money in a savings account providing an annual rate of 5% compounded quarterly if $2000 is invested and is left in the account for 15 years.

COMPOUND INTEREST:

How much should Sabrina invest now in a money market account if she wishes to have $9000 in the account at the end of 10 years? The account provides an APR of 6% compounded daily.

PRESENT VALUE OF AN ANNUITY

Principle amount invested or borrowed Payment amount total number of payments = # payments per year x # years

APR Interest rate per payment period # paymen

1 (1 )

nPPn

i

nn

iP Pi

ts per year

Uses include mortgages and loans

PRESENT VALUE OF AN ANNUITY

You are buying your 1st home. You take out a $250,000, 30-year mortgage with an interest rate of 6.5%.A) What will the monthly payment for

principle and interest be?B) How much will they pay in interest

over the life of the loan?Wow! What a GREAT test question this would make!

FUTURE VALUE OF AN ANNUITY

Future Value Payment amount number of payments

APR Interest rate per payment period # payment periods per year

(1 ) 1

nF

Pn

i

nn

iF Pi

Uses include putting money into a retirement plan or other investment

FUTURE VALUE OF AN ANNUITY

If I open an individual retirement account (IRA) today that earns 8% interest and I contribute $2000 per year until I retire at age 62, how much money will I have when I retire?

HOMEWORK:

p 708 #11 – 31 odd

WARM-UP: In 1990, the population of Houston, TX

was 1,637,859. In 1998, the population was 1,786,691. Assuming the population increases by a certain percent each year, predict the population of Houston in 2014.

WITHOUT your calculator, sketch the graphs of the following:

2

3

4 4 2

4 4

x x

x x

y y

y y

HOMEWORK:

STANDARD 4.3: USE THE NATURAL NUMBER E TO SOLVE PROBLEMS (11-3)

We will answer these burning questions…

Since when is e a number? Can I perform mathematical operations on

an e? How on earth am I supposed to use the

number e in real life?

REMEMBER COMPOUND INTEREST FROM SECTION 11-2?

1nt

rA P

n

A = final amount P = principle or initial investment r = annual interest rate N = number of compoundings per year t = the number of years

FIGURE THIS OUT WITH YOUR GROUP.

Mariza has $1500 she wants to invest for 5 years.

She can get 6% interest and is offered 4 different plans: Compounded yearly, monthly, weekly or daily.

How much will each plan yield at the end of 5 years?

Which plan should she take?

CONTINUOUS COMPOUNDING

Where: P is the initial amount or

investment, A is the final amount, r is the interest rate and t is the time in years.

P rtA e

REDO MARIZA’S PROBLEM WITH CONTINUOUS COMPOUNDING

SO, JUST WHAT IS ?

e is called the “Natural Number”.

e is an irrational number which was found to occur regularly in nature, much as pi does.

It is found using the sum of an infinite series (more on those later).1 1 1 1 1

1 ...1 12 12 3 12 3 4 12 3 ...

2.718

en

e

LET US CALCULATE WITH .

5 33

2 1

2

e e e

e e e

e

LET’S GRAPH WITH !e

MORE USES OF THE NUMBER

0ktN N e

Formulas in terms of e:

Where: N is the final amount;

N0 is the initial amount, k is a constant and t is time.Does this formula sound familiar?

When is it growth and when is it decay?

e

LET’S DO SOME PHYSICS…

According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume the initial temperature T1 is 90º F and that k = 0.275.a. Write a function to model the rate at which

the liquid cools.b. Find the temperature T of the liquid after 4

minutes (t ).c. Graph the function on your calculator and

use the graph to verify your answer in part b.

LET’S PARTNER UP! You will work with a partner to complete

Lesson 11-3 p A46

One piece of paper. Take turns. One partner solves and the

other praises/coaches.

THE NUMBER E

P714 #1 – 9 all, 11(a and b), 13 and 17

WARM-UP:

Sketch the graph without your calculator:

p 716 #18, 23 and 27

13 4x x xy e y e y e

TRY THIS…

Sammy has $1000 she would like to have $4000 dollars to put down on a new car (Sammy says “Who cares about depreciation? I LOVE the smell of a new car!) If she can get continuously compounding interest at 6.5%, how long will it take her to reach her goal of $4000?

OKAY…SO LET’S BRAINSTORM.

STANDARD 4.4: DEMONSTRATE UNDERSTANDING OF THE PROPERTIES OF LOGARITHMS (11-4)

Our scintillating questions today are… How the heck do I solve when my

variable is acting as an exponent? Aren’t logs just hunks of wood? How do logs behave when you add or

subtract them? Multiply or divide? Raise to Powers?

How can I use logs to make my work simpler?

How do you graph a log?

DEFINING AN INVERSE FOR EXPONENTIAL FUNCTIONS We will call the inverse a logarithm. Well, why not? We called the inverse for

powers radicals and no one complained. We abbreviate it as log. So the inverse of y = ax is…

logay x

HOW DO LOGS WORK?

First let’s look at its graph:

OKAY, MOVE IT AROUND…

log( 3) log

log 4 log( )

y x y x

y x y x

NOW FOR SOMETHING A LITTLE CONFUSING…

I f you have

then log is its inverse

iff log is equivalent to .

x

a

ya

y a

y x

y x x a

OKAY, SIMPLER.

exponent log (answer)base

It SHOULD equal the exponent. Remember, that’s what we were trying to

solve for.

LET’S PRACTICE MOVING FROM EXPONENTIAL TO LOGARITHMIC FORM Now from logarithmic to exponential.

Okay, let’s solve some.

PROPERTIES OF LOGARITHMS

p 720 in your book.

Let’s prove one…

PROPERTIES OF LOGARITHMS

p 720 in your book.

Let’s prove one…

Let log and log

theref ore and .b b

yx

x m y n

m b n b

SOLVE SOME.

WARM-UP

HOMEWORK:

P A47 Lesson 11-4

STANDARD 4.5: USE COMMON LOGARITHMS TO SOLVE PROBLEMS (11-5)

The questions are… What is a common log? How do I evaluate common logs? What is an antilog? How do I evaluate logs in bases other than

10 and e? So what are these logs good for and how do I

use them?

WHAT IS A COMMON LOG?

HOW DO I EVALUATE COMMON LOGS?

3log5 2

219log

6

WHAT IS AN ANTILOG?

Just another name for an exponential function with a base of 10.

0.0600

2.3

antilog 0.0600 10

antilog 2.3 10

HOW DO I EVALUATE LOGS IN BASES OTHER THAN 10 AND E?

loglog

loga

nn

a

Change of Base Formula

3log 27

3log 13

TRY THIS…

0.5log 0.0675

SO WHAT ARE THESE LOGS GOOD FOR AND HOW DO I USE THEM?

45 73x

Let’s try some from p 731

SOLVING BY GRAPHING

29 2 8x x

HOMEWORK:

P 730 #19 – 51 odd, 53, 61

WARM-UP:

p 731 #58, 61 and 64

ANSWERS:

58. a) 1.7 miles b) 6 psi

61. Between 2 and 3

64. pry

q

HOMEWORK:

STANDARD 4.5: USE NATURAL LOGARITHMS TO SOLVE PROBLEMS (11-6)

In this section we will answer… What’s natural about a logarithm? Why would I ever need a natural log when

I’ve got common logs? When and how do I use natural logs? I wonder if Ms B. would really put a word

problem on a test?

WHAT IS A NATURAL LOGARITHM?

It is a log whose base is e. It has its own abbreviation.

log lne x x

NATURAL LOGS WORK JUST LIKE ANY OTHER LOG.

1. ln0.0089

2. antiln2.94y

YOU CAN USE THEM TO CHANGE BASES.

6log 256

You can solve equations.

6.5 16.25lnx

Let’s check that by graphing!

0.0451600 4 te

AND DON’T FORGET…WORD PROBLEMS! Remember this? Now you can solve it!

Sammy has $1000 she would like to have $4000 dollars to put down on a new car (Sammy says “Who cares about depreciation? I LOVE the smell of a new car!) If she can get continuously compounding interest at 6.5%, how long will it take her to reach her goal of $4000?

CHAPTER 11 PRACTICE: SAGE AND SCRIBE

You and your partner will take turns doing the problems.

Person 1 is in charge of recording the solving to the odd problems and Person 2 is responsible for the evens.

The person NOT writing leads the solving. The recorder MUST solve the problem according to the solver’s directions.

The recorder may offer help and suggestions but the solver has the final say.

CHAPTER 11 PRACTICE ANSWERS:

HOMEWORK:

P 736 #19 – 51 odd, 55, 59

Unit 4 Test on Monday!!!

WARM-UP:

Pick up a couple pieces of graph paper

p 71 #11, 13, 15 and 21

HOMEWORK:

SECTION 11-7: MODELING REAL-WORLD DATA WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS

In this section we will…In this section we will… Write exponential and logarithmic functions

to model real-world data

Use exponential and logarithmic functions to interpret real-world data.

GETTING READY…

Go to STAT PLOT ( press 2nd then Y = )

Choose 1:Plot 1

Turn to ON, make sure the TYPE shows scatter plot. Xlist should show L1and Ylist should show L2. Mark should be on 1st one the boxed point.

CHOOSING A MATHEMATICAL MODEL

p 745 #14 First graph your data and sketch a best fit

function. Decide which of our function parent graphs

best fits the data. Use the calculator to develop the regression

equation. p 746 #17

LET’S GIT’ER DONE. You need to set the window to fit your data.

Go to WINDOW change settings: Xmin=-1, Xmax=5, Xscl=1; Ymin= 0, Ymax=70, Yscl=10.

Press STAT, choose 1:Edit… In the L1 column enter the x-values; in the L2

column enter the f(x)-values. Press GRAPH. Press STAT, move to the CALC column,

choose 0:ExpReg. Press ENTER twice.

THE M&M PROJECT:

Graph your data by hand on graph paper.

Sketch the best fit function onto the graph.

Then using your calculator, find the equation for your best fit function.

Write the equation in proper form, under your graph.

Use the equation to complete the questions.

THINGS TO REMEMBER…

You will need to adjust your window.

PROJECT

Separate paper with cover page

All parts answered with real, honest-to-goodness sentences!

Neat, professional, very nice-nice!

Attach graph with data and best fit function drawn.

Due Monday.