U NIT 4: “P OWER T RIP ” Standard 4.1 : demonstrate understanding of the properties of exponents and to graph exponential functions (11-1, 11-2) Standard 4.2 Standard 4.2 : solve problems using exponential functions (11-2) Standard 4.3 Standard 4.3 : use the natural number e to solve problems (11-3) Standard 4.4 Standard 4.4 : demonstrate understanding of the properties of logarithms (11-4) Standard 4.5 Standard 4.5 : use common and natural logarithms to solve problems (11-5, 11-6)natural logarithms 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!

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?

E XPONENTIAL F UNCTIONS : Functions in which there is a variable acting as an exponent. The base will be some real number.

L ET ’ S CHECK THEM OUT ON THE CALCULATOR. How are these graphs alike? How are they different? What conclusions can I draw?

Review Warm- up…TRANSFORMATIONS!!!

L OOK AT THESE GUYS … What’s going on?

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?

W HAT EFFECT DOES A POWER HAVE ON A NUMBER ?

H OW DO EXPONENTS BEHAVE WHEN THEIR BASES ARE ADDED OR SUBTRACTED ?

H OW DO THEY BEHAVE WHEN THEIR BASES ARE MULTIPLIED ?

D IVIDED ?

R AISED TO A POWER ? Single base with an exponent? A product of bases? A quotient?

W HAT ABOUT NEGATIVE EXPONENTS ?

T RY SOME …

W HAT IF MY EXPONENT IS NOT AN INTEGER ?

S OLVING EQUATIONS INVOLVING RATIONAL EXPONENTS.

O KAY …W HAT IF MY EXPONENT IS NOT RATIONAL ?

S OLVING E QUATIONS

S AGE AND S CRIBE. 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.

S AGE AND S CRIBE. p A46 Lesson 11-1 #1 – 19 odd

H OMEWORK : p 700 #21 – 67 odd, 71, 73

W ARM -U P p 708 # 1 – 7 odd

H OMEWORK :

S TANDARD 4.2 S TANDARD 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?

U SING 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.

G ROWTH AND D ECAY General formula for exponential growth or decay is: N = Final amount, N o = 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?

G ROWTH AND D ECAY 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?

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

C OMPOUND I NTEREST 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.

C OMPOUND I NTEREST : 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.

P RESENT V ALUE OF AN A NNUITY Uses include mortgages and loans

P RESENT V ALUE OF AN A NNUITY You are buying your 1 st 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!

F UTURE V ALUE OF AN A NNUITY Uses include putting money into a retirement plan or other investment

F UTURE V ALUE OF AN A NNUITY 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?

H OMEWORK : p 708 #11 – 31 odd

W ARM -U P : 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 WITHOUT your calculator, sketch the graphs of the following:

H OMEWORK :

S TANDARD 4.3 S TANDARD 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?

R EMEMBER C OMPOUND I NTEREST FROM S ECTION 11-2? A = final amount P = principle or initial investment r = annual interest rate N = number of compoundings per year t = the number of years

F IGURE 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?

C ONTINUOUS C OMPOUNDING 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.

R EDO M ARIZA ’ S PROBLEM WITH CONTINUOUS COMPOUNDING

S O, 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).

L ET US CALCULATE WITH.

L ET ’ S GRAPH WITH !

M ORE U SES OF THE N UMBER Formulas in terms of e : Where: N is the final amount; N 0 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?

L ET ’ S DO SOME P HYSICS … According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume the initial temperature T 1 is 90º F and that k = 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.

L ET ’ S P ARTNER U P ! 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.

T HE N UMBER E P714 #1 – 9 all, 11(a and b), 13 and 17

W ARM - UP : Sketch the graph without your calculator: p 716 #18, 23 and 27

T RY 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?

O KAY … SO LET ’ S BRAINSTORM.

S TANDARD 4.4 S TANDARD 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?

D EFINING AN I NVERSE FOR E XPONENTIAL F UNCTIONS 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 = a x is…

H OW DO LOGS WORK ? First let’s look at its graph:

O KAY, MOVE IT AROUND …

N OW FOR SOMETHING A LITTLE CONFUSING …

O KAY, SIMPLER. It SHOULD equal the exponent. Remember, that’s what we were trying to solve for.

L ET ’ S PRACTICE MOVING FROM EXPONENTIAL TO LOGARITHMIC FORM Now from logarithmic to exponential. Okay, let’s solve some.

P ROPERTIES OF L OGARITHMS p 720 in your book. Let’s prove one…

P ROPERTIES OF L OGARITHMS p 720 in your book. Let’s prove one…

S OLVE SOME.

W ARM - UP

H OMEWORK : P A47 Lesson 11-4

S TANDARD 4.5 S TANDARD 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?

W HAT IS A COMMON LOG ?

H OW DO I EVALUATE COMMON LOGS ?

W HAT IS AN ANTILOG ? Just another name for an exponential function with a base of 10.

H OW DO I EVALUATE LOGS IN BASES OTHER THAN 10 AND E ? Change of Base Formula

T RY THIS …

S O WHAT ARE THESE LOGS GOOD FOR AND HOW DO I USE THEM ? Let’s try some from p 731

S OLVING BY GRAPHING

H OMEWORK : P 730 #19 – 51 odd, 53, 61

W ARM - UP : p 731 #58, 61 and 64

A NSWERS : 58. a) 1.7 miles b) 6 psi 61. Between 2 and 3 64.

H OMEWORK :

S TANDARD 4.5 S TANDARD 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?

W HAT IS A N ATURAL L OGARITHM ? It is a log whose base is e. It has its own abbreviation.

N ATURAL L OGS WORK JUST LIKE ANY OTHER LOG.

Y OU CAN USE THEM TO CHANGE BASES. You can solve equations. Let’s check that by graphing!

A ND 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?

C HAPTER 11 P RACTICE : S AGE AND S CRIBE 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.

C HAPTER 11 P RACTICE A NSWERS :

H OMEWORK : P 736 #19 – 51 odd, 55, 59 Unit 4 Test on Monday!!!

W ARM - UP : Pick up a couple pieces of graph paper p 71 #11, 13, 15 and 21

H OMEWORK :

S ECTION 11-7 : M ODELING R EAL -W ORLD D ATA WITH E XPONENTIAL AND L OGARITHMIC F UNCTIONS 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.

G ETTING READY … Go to STAT PLOT ( press 2 nd 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 1 st one the boxed point.

C HOOSING A M ATHEMATICAL M ODEL 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

L ET ’ S G IT ’ ER D ONE. 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.

T HE M & M P ROJECT : 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.

T HINGS TO R EMEMBER … You will need to adjust your window.

P ROJECT 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.