Exponential equations will be written as, where a =. Example 1: Determine a formula for the exponential function whose graph is shown below.

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1 .1 Eponential and Logistic Functions PreCalculus.1 EXPONENTIAL AND LOGISTIC FUNCTIONS 1. Recognize eponential growth and deca functions 2. Write an eponential function given the -intercept and another point (from a tale or a graph).. Be ale to define the numer e 4. Use transformations to graph eponential functions without a calculator. 5. Recognize a logistic growth function and when it is appropriate to use. 6. Use a logistic growth model to answer questions in contet. So far we have discussed polnomial functions (linear, quadratic, cuic, etc.) and rational functions. In all of these functions, the variale was the ase. In eponential functions, the variale is the eponent. Eponential equations will e written as, where a =. Eponential functions are classified as growth (if > 1) or deca (if 0 < < 1). Eample 1: Determine a formula for the eponential function whose graph is shown elow. (2, 6) (0, 2) Eample 2: Descrie how to transform the graph of f 2 in to graph of g 1. Eample : Sketch the graph of the eponential function. a) g ) g 2 The Numer e Man eponential functions in the real world (ones that grow/deca on a continuous asis) are modeled using the ase of e. 1 Just like p».14, we sa as follows: e». We can also define e using the function ( ) 1 As, ( 1+ ) e Tr convincing ourself that this function approaches e using the TABLE function of our calculator. - 1

2 .1 Eponential and Logistic Functions PreCalculus Do ou think it is reasonale for a population to grow eponentiall indefinitel? Logistic Growth Functions functions that model situations where eponential growth is limited. An equation of the form or. where c =. The graph of a logistic function looks like an eponential function at first, ut then levels off at = c. Rememer from our Parent Functions in chapter 1, that the logistic function has two HA: = 0 and = c. Eample 4: The numer of students infected with flu after t das at Springfield High School is modeled the following function: 1600 P t 0.4t 1 99e a) What was the initial numer of infected students t = 0? ) After 5 das, how man students will e infected? c) What is the maimum numer of students that will e infected? d) According to this model, when will the numer of students infected e 800? Eample 5: Find the intercept and the horizontal asmptotes. Sketch the graph. 9 a) f ) f e - 2

3 .2 Eponential and Logistic Modeling PreCalculus.2 EXPONENTIAL AND LOGISTIC MODELING 1. Write an eponential growth or deca model f ( ) a and use it to answer questions in contet. Eponential Growth Factor = 1 + % rate (as a decimal) Eponential Deca Factor = 1 % rate (as a decimal) 2. Write a logistic growth function given the -intercept, oth horizontal asmptotes, and another point. Eample 1: The population of Glenrook in the ear 1910 was Assume the population increased at the rate of 2.25% per ear. a) Write an eponential model for the population of Glenrook. Define our variales. ) Determine the population in 190 and c) Determine when the population is doule the original amount. Eample 2: The half-life of a certain radioactive sustance is 14 das. There are 10 grams present initiall. a) Epress the amount of sustance remaining as an eponential function of time. Define our variales. ) When will there e less than 1 gram remaining? Eample : Find a logistic equation of the form (24, 15) is on the curve. c = that fits the graph elow, if the -intercept is 5 and the point 1 + ae - -

4 . Logarithmic Functions and Their Graphs PreCalculus. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS 1. Rewrite logarithmic epressions as eponential epressions (and vice-versa). 2. Evaluate logarithmic epressions with and without a calculator.. Graph and translate a logarithmic function. Rewriting Logarithmic Epressions A logarithmic function is simpl an inverse of an eponential function. The following definition relates the two functions: log = if and onl if = If ou think of the graph of the eponential function =, could e an real numer, ut > 0. These same limitations on the variales are true for the logarithm function as well. Eample 1: Evaluate each epression without a calculator. a) log5 125 ) log7 1 c) log 1 81 d) log8 2 Two Special Logarithms A logarithm with ase 10 is called a logarithm and is written. A logarithm with ase e is called a logarithm and is written. Eample 2: Evaluate each epression without a calculator. a) 4 1 log 10 ) ln e A Consequence of the Inverse Properties of Logarithms: log M M = M and log ( ) = M Eample : Evaluate each logarithmic epression without a calculator. a) log6 6 ) 5 log10 c) ln 5 log8 7 e d) 8-4

5 . Logarithmic Functions and Their Graphs PreCalculus Eample 4: Graph the following functions without a calculator. a) = log ) = ln Eample 5: Graph the following transformations of the two functions aove without a calculator. a) = log ( - ) + 1 ) = ln( - ) Eample 6: Graph = log ( 2- ) without a calculator. - 5

6 .4 Properties of Logarithms PreCalculus.4 PROPERTIES OF LOGARITHMS 1. Use properties of logarithms to epand a logarithmic epression. 2. Use properties of logarithms to write a logarithmic epression as a single logarithm.. Evaluate logarithms with ases other than e or 10 using a calculator. Properties of Logarithms these properties follow from the properties of eponents Rules of Logarithms: 1. log( MN) = log( M) + log( N) æm ö 2. log ç = log( M )-log( N) çèn ø k log M = k log M. ( ) ( ) Eample 1: Epand each logarithmic epression. a) log( ) ) æ 2 q ö ln ç t çè ø Eample 2: Condense each logarithmic epression into a single logarithm. 1 a) ln ln 5ln z - + ) ln ( 1 ln 5ln z) c) 2 4 log 27 2 log 9 Your calculator (unless ou have the new TI-84+ operating sstem) will onl evaluate a logarithm with ase 10 or e. If ou need to evaluate a logarithm with a different ase, ou must use the change of ase formula. Change of Base Formula: log a log a = log or ln a ln Eample : Evaluate each logarithmic epression using our calculator. a) log7 5 ) log

7 .5 Equation Solving and Modeling PreCalculus.5 EQUATION SOLVING AND MODELING 1. Solve eponential and logarithmic equations. When ou solve an equation, ou undo what has een done addition to undo sutraction, multiplication to undo division. Since eponents and logarithms are inverses of each other, it follows that in order to solve a logarithmic equation, ou can write it as an eponent to undo the logarithm, and if ou are solving for an eponent, ou write the equation as a logarithm. NOTE: You can onl switch etween eponential and logarithmic forms when ou have log = or = Eample 1: Solve the following equations: a) 2 e - = 19 ) ( ) =- 14 c) log ( ) = 4 d) 7- log( ) =- 5 2 e) log + log( + 21) = 2 f) log ( + 4) -log ( - 5) = 2-7

8 .6 Mathematics of Finanace PreCalculus.6 MATHEMATICS OF FINANCE 1. Appl correct compounding interest formulas (either continuousl or n times per ear) 2. Use the FINANCE application on our TI-8 or 84 calculator to solve for unknown quantities. Compounding Interest n Times Per Year When the interest earned is added to the original amount and then interest is calculated on that interest (in other words, ou are earning interest on our interest), we sa the interest is compounded. The formula for compounding interest n times each ear is given Where r A P 1 n A = new amount P = original amount or Principle r = the interest rate as a decimal n = the numer of times a ear interest is calculated t = the numer of ears nt Eample 1: Consider the situation where n = 1. a) Eplain what this means in the contet of compounding interest. ) What does the compounding interest rate look like when n = 1? Eample 2: Victor deposits $500 into a savings account earning 6% interest compounded monthl. How much mone is in his account in 12 ears? Compounding Interest Continuousl The formula for compounding interest continuousl is. Notice in this formula, ou need the interest rate r and NOT the growth factor. Eample 5: Bo invests $500 into a savings account earning 6% interest compounded continuousl. How much mone does Bo have in 12 ears? - 8

9 .6 Mathematics of Finanace PreCalculus Using the FINANCE App On Your TI-8 or TI-84 Calculator Your graphingg calculator has a ver powerful financial calculator application uilt into its programming called the TMV Solver. TMV stands for Time Value of Mone. The application solves for one variale when given values for an four of the five remaining variales. It is especiall useful when ou are making the same periodic pament (or investment) more than once a ear i.e. car pament. First we need to understand the 5 variales involved Press APPS, Finance, TMV Solver to get started 1. n = the numer of compounding periods 2. I% = the annual percentage rate. PV = the Present Value of the account orr investment 4. PMT = the pament amountt 5. FV = the Future Value of the account or investment 6. P/ /Y = paments per ear 7. C/Y = compounding periods per ear for our use C/ /Y = P/Y 8. PMT: END BEGIN means pament at the end or eginning of the pament period. NOTE: Enter an mone received in as positive numers and an mone paid out as negative numers. Eample 6: Sal opened a retirement account and pas $50 at the end off each quarter (4 times/ear) into it. The account earns 6.12% annual interest. Sal plans to retire in 15 ears. How muchh mone will ee in his account at that time? Eample 7: Kim wants to u a used car for $9000. The ank offers her a 4-ear loan with an APR of 7.95%. What will Kim s monthl car pament e? Assume she makes her pament on thee 1 st of each month. - 9