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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.