The units on the average rate of change in this situation are.

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1 1 Objectives: I can analze and interpret the behavior of eponential functions. I can solve eponential equations analticall and graphicall. I can determine the domain and range of eponential functions. I can classif a function as eponential when represented numericall, analticall, or graphicall. I can determine regression models from data using appropriate technolog and interpret the results. I can justif and interpret solutions to application problems. Definitions / Vocabular / Graphical Interpretation: Doubling Pennies Problem: Da Pennies Average Rate of Change The units on the average rate of change in this situation are. Since our rate of change is doubling for each unit increase, we have an increasing rate of change, and we would epect the graph to be. Eponential Functions have the form f ( ) ab where a 0 and b 0. The input variable is located in the eponent. The output corresponding to an input of zero is called the and is represented b a. The base b represents repeated multiplication of the.

2 With linear functions we have a constant change in amount, but in eponential functions we have a constant. In the doubling pennies problem from page 1, since we were repeatedl multipling b, our base would be, and our initial value was. Therefore, the eponential model of this function f ( ) ab would be. Graphs of Eponential Functions: 1) For a positive initial value a, if the base b is >1, then the graph of the function looks like: It is alwas increasing and concave up. The domain is and the range is. The eponential function has a horizontal asmptote at. ) For a positive initial value a, if the base b is 0 < b <1, then the graph of the function looks like: It is alwas decreasing and concave up. Its domain is and its range is. It also has a horizontal asmptote at.

3 3 Finding an eponential function given points: We have two methods: elimination and substitution. Elimination Method: Given the points (, 6) and (,48) and recalling that between the two points. ab find the eponential function Step 1: Plug in the (,) coordinates of the given points into the eponential equation. 6 ab and 48 ab Step : Divide the equations to eliminate a and solve for b. 6 ab 48 ab Step 3: Plug our answer for b into either of our equations from Step 1 and solve for a. Substitution Method: Given the points (, 1) and (-1, 3/) and recalling that function between the two points. ab find the eponential Step 1: Plug in the (,) coordinates of the given points into the eponential equation: 1 ab and 3 ab 1 Step : Solve for a or b in one equation and substitute into the other equation to solve for the other variable. Step 3: Rewrite the equation using the factors a and b in ab form.

4 4 End Behavior: The parent graph of an eponential has the following properties: If b>1, then lim = and lim = If 0 < b < 1 then lim = and lim = Thus, graphs of eponential functions have horizontal asmptotes. Note: if the function has been verticall shifted, these limits will change! Ke idea: Growth factor (b) = (1 + growth rate) Growth factor (b) = (1+ constant percent change) Note: If the function is decaing, the growth rate is negative, thus the growth factor b is between zero and one. 0 < b < 1. Eample of eponential growth: Given an initial salar a =$0,000 and a guaranteed raise (constant percent change) of % per ear, determine a function S (t) that models our annual income each ear. New amount = old amount + % of old amount New amount = (100% + %) of old amount New amount = ( ) of old amount New amount (S) = 1.0 of old amount Let a = the initial salar and t = time in ears St () Eample of eponential deca: Given an initial population a of bacteria and a deca rate of 4% per hour, give a model that tells the population P(h) of bacteria after h hours. New amount = old amount 4% of old amount New amount = 100% 4% of old amount New amount = (1.4) of old amount New amount (P) = 0.76 of old amount Let a = the initial population and h = time in hours P (h)

5 Financial applications: Annual Compounding We can write a generalized formula for an interest rate r compounded annuall: B P(1 r) t Multiple Compoundings We can also write a generalized formula for interest compounded multiple times during the ear: r B P 1 n Where B = final balance in the account P = Initial amount deposited in the account r = annual % rate n = number of compounding periods per ear t = number of ears of compounding nt Continuous Compounding Finall, as the number of compounding times increases per ear, we can represent continuous compounding using the constant e (also called the natural base) as the growth factor, giving the formula: B Pe Where B = final balance in the account P = Initial amount deposited in the account r = annual % rate t = number of ears of compounding e =growth factor for continuous compounding where rt e 1 lim (1 ) n n n Changing between forms: Notice that there are two formulas to represent eponential growth, one for periodic compounding and one for continuous compounding. t ab for periodic compounding kt ae for continuous compounding. The difference between the two forms is the base growth factor where this fact to convert between the two forms of eponential equations. k b e. We use

6 6 Effective Yield Effective ield is the annual rate of return on an investment, and converts a nominal interest rate to an annual effective ield rate, based on the interest rate and the number of times of compounding. Thus, ou ma have different accounts with the same nominal rate, but the ma have different effective ields if the have different compounding periods. Effective ield allows ou to compare the nominal interest rates. The ke to finding the effective ield is to first calculate the growth factor based on the nominal rate and number of compounding periods, then subtract 1 and turn it into a percent. Eample 1: An account pas interest at the rate of % per ear compounded monthl. Nominal Rate = % (the advertised rate; does not account for compounding) Effective Rate =.1% (what ou actuall earn because of compounding): = Eample : Sa ou have $00 to invest at 6%. Set up a formula to calculate the amount in the account after t ears if interest is compounded: Annuall Quarterl Continuousl

7 7 Determining if data are eponential: Given a table of inputs and outputs, how does one determine what algebraic model best fits the data? We begin b looking at the first differences between the first outputs to determine whether a constant rate of change is shown impling a linear model. Second, we look at the second differences to see if a quadratic model might be a good fit. Third, we can look at the common ratio of successive terms to determine if an eponential model might be appropriate. Note: Inputs in this table are equall spaced. Does that matter? Wh or wh not? Stock Price Eample: f() 1st Diffs nd Diffs Ratio