Exponential Functions Objectives: To define and graph exponential growth and decay functions To model real-life exponential situations To calculate compound interest To define the number e and use it as the base of exponential functions

7.1-7.3: Exponential Functions Objective 1 You will be able to define and graph exponential growth and decay functions http://25.media.tumblr.com/tumblr_lz23ylenZl1ro09hco1_r1_1280.jpg

Exponential Functions An exponential function with base 𝑏 is denoted as 𝑓(𝑥)=𝑏𝑥, where 𝑏>0 and 𝑏≠1. If 𝑏=1, then we’d have 𝑦=1𝑥. It’s constant. Which is just 𝑦=1. And that’s not exponential.

7.1-7.3: Exponential Functions Exercise 1 What is the difference between a power function and an exponential function? Which one gets bigger faster? 𝒙 1 2 3 4 5 6 7 8 9 𝒙 𝟐 𝟐 𝒙 1 4 9 16 25 36 49 64 81 2 4 8 16 32 64 128 256 512

Exercise 2 Without a calculator, graph each of the following: 𝑓(𝑥)=2𝑥 𝑔(𝑥)=3𝑥 ℎ(𝑥)=4𝑥

Exercise 3 Without a calculator, graph each of the following: 𝑓(𝑥)=(1/2)𝑥 𝑔(𝑥)=(1/3)𝑥 ℎ(𝑥)=(1/4)𝑥

An exponential function is of the form 𝑦=𝑎∙ 𝑏 𝑥 0<𝑏<1 𝑏>1 Exponential Decay Exponential Growth 𝑏 is the Decay Factor 𝑏 is the Growth Factor

Exponential: bx Graph of 𝑦=𝑏𝑥, 𝑏>1 Domain: (−∞, ∞) Range: (0, ∞) Horizontal asymptote: 𝑦=0 𝑦-intercept: 1 Increasing (growth) Continuous One-to-one

Graph of 𝑦=𝑏𝑥, 𝟎<𝑏<1 Exponential: bx Graph of 𝑦=𝑏𝑥, 𝟎<𝑏<1 Domain: (−∞, ∞) Range: (0, ∞) Horizontal asymptote: 𝑦=0 𝑦-intercept: 1 Decreasing (decay) Continuous One-to-one

Exponential Transformations The role of a, h, and k in the exponential function: a Scaling 0 < |a| < 1: Shrink vertically |a| > 1: Stretch vertically a < 0: Flips across 𝑥-axis h Horizontal translation k Vertical translation 𝑦=𝑎∙ 𝑏 𝑥−ℎ +𝑘

Exercise 4 Graph the following. State the domain and range. 𝑦= 4 𝑥 𝑦=2∙ 4 𝑥+2 +1

Exercise 5 Graph the following. State the domain and range. 𝑦= 1 3 𝑥 𝑦= 1 3 𝑥 𝑦=2 1 3 𝑥−2 −1

7.1-7.3: Exponential Functions Objective 2 You will be able to model real-life exponential situations http://podcasts.shelbyed.k12.al.us/p2morris/page/7/

A Helpful Tip Often in a problem, you have to add a percentage of a price back to the price, like taxes and tips. To do this, add one to the percent, and then multiply. Original amount $5.50+8% $5.50 = $5.50+.08 $5.50 = 𝑎+𝑟𝑎= $5.50 1+.08 = 𝑎 1+𝑟 $5.50 1.08 One + the rate

A Taxing Result In many real-life situations, a quantity will grow at a constant rate. This is like the previous tip/tax problem, except it is done over and over again. $5.50 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.0 Original amount $5.50 1.08 𝑡 One + the rate

A Generous Discount Often in a problem, you have to subtract a percentage of a price from the price, like discounts. To do this, subtract the percent from one, and then multiply. Original amount $5.50−10% $5.50 = $5.50−.1 $5.50 = 𝑎−𝑟𝑎= $5.50 1−.1 = 𝑎 1−𝑟 $5.50 .9 One − the rate

A Shrinking Return In many real-life situations, a quantity will shrink at a constant rate. This is like the previous discount problem, except it is done over and over again. $5.50 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 Original amount $5.50 .9 𝑡 One − the rate

Exponential Growth/Decay Model 7.1-7.3: Exponential Functions Exponential Growth/Decay Model In a real-life situation, when a quantity a continues to increased or decrease by a fixed percent r each year (or some other time frame), the amount y of the quantity after time t can be modeled by: Exponential Growth Exponential Decay 𝑦=𝑎 1+𝑟 𝑡 𝑦=𝑎 1−𝑟 𝑡 Growth Factor Decay Factor

Exercise 6 According to some website, the population of Denton, TX was 119,454 in 2008. This was up from 3.01% from the previous year. If the population continues to grow at this rate, what do you expect the population of Denton to be in 2020?

Exercise 7 Estimate the year when the population of Denton, TX will be 1 million, assuming that its population continues to grow at a yearly rate of 3.01%.

Exercise 8 Let’s say you bought a new car for $20,000 with a 5-year loan. If the car depreciates in value at a continuous rate of 15% per year, how much will your “new” car be worth when you finish paying for it?

Exercise 9 Assume the car from the previous example continues to decrease in value by 15% every year. In how many years with the value of the car be ½ the original value?

7.1-7.3: Exponential Functions You will be able to calculate compound interest Objective 3 http://pencilskirtbound.blogspot.com/

Very Interesting When you stash your money in a financial institution, depending on the type of account you choose, the bank will pay you for using your money while it is stashed in their FDIC-protected vaults. This is called interest, and it is added to your bank account.

Very Interesting Also, when you borrow money for a car loan, a mortgage, or a credit card, you have to pay a percentage back to the bank for borrowing that money. (This is similar to the bank borrowing your money while you stash it with them.) That is also called interest.

Very Interesting Finally, interest comes in two basic flavors: Simple: Interest is only ever calculated on the initial amount (called the principal) Compound: Interest is calculated on the principal plus the previously earned interest

Compound Interest 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 Consider an initial principal 𝑃 deposited in an account that pays interest at an annual rate 𝑟 (expressed as a decimal), compounded 𝑛 times per year. The amount 𝐴 in the account after 𝑡 years is given by the: Compounding refers to adding the interest back to the principal. This can be done yearly, monthly, quarterly, daily, or even continuously. 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

Exercise 10 You deposit $5,500 in an account that pays 3.6% annual interest. Find the balance after 10 years if the interest is compounded the given frequency: Semiannually Quarterly Monthly Daily

Exercise 11 You deposit $1,000 in an account that pays 5% annual interest. Find the balance in the account after 10 years if the interest is compounded a) daily, b) hourly, and c) by the minute. As you increase the number of compoundings, does your account increase without bound?

Exercise 11 𝐴=1,000 1+ .05 𝑛 10𝑛 Daily Hourly Minutely 𝑛=365 𝑛=365∙24 𝐴=1,000 1+ .05 𝑛 10𝑛 Daily Hourly Minutely 𝑛=365 𝑛=365∙24 𝑛=8,760 𝑛=8,760∙60 𝑛=525,600 𝐴=$1,648.66 𝐴=$1,648.72 𝐴=$1,648.72

Exercise 11 𝐴=1,000 1+ .05 𝑛 10𝑛

You will be able to define the number 𝑒 and use it as the base of exponential functions Objective 4

Exercise 12 Let’s say you had $1 to put into an account for a year, and the interest rate was 100%. How much money would you have in the account if the interest was compounded once (n = 1)? Twice (n = 2)? Will this amount continue to grow as n increases? What’s the best possible value of n? How much money would you have in the bank at the end of the year with this value of n?

Exercise 12 𝐴=1 1+ 1 𝑛 𝑛 $2.00 $2.25 $2.44 $2.61 $2.71 $2.72 Limit! 𝐴=1 1+ 1 𝑛 𝑛 𝑛=1 𝑛=2 𝑛=4 𝑛=12 𝑛=365 𝑛=8,760 $2.00 $2.25 $2.44 $2.61 $2.71 $2.72 +.25 +.19 +.17 +.10 +.01 Limit!

Exercise 12 𝐴=1 1+ 1 𝑛 𝑛

Natural Base e The natural base 𝑒 is an irrational number such that lim 𝑛→∞ 1+ 1 𝑛 𝑛 =𝑒≈2.718281828459045… Given a letter by Leonhard Euler (1707-1783) Sometimes called the Euler number

Natural Base e The natural base 𝑒 is an irrational number such that lim 𝑛→∞ 1+ 1 𝑛 𝑛 =𝑒≈2.718281828459045… Like 𝜋, 𝑒 is a transcendental number since it is not the root of any polynomial equation (Euler’s Formula, not Euler’s Arm)

Exercise 13e Simplify the expression. 𝑒 9 ∙ 𝑒 6 60 𝑒 8 12 𝑒 3

Exercise 13a Use a calculator to approximate. 𝑒 −1 𝑒 𝑒 2

Exponential Functions with e The natural base exponential function is of the form 𝑦=𝒂𝑒𝒓𝑥 𝒂>0 𝒓<0 𝒂>0 𝒓>0 Exponential Decay Exponential Growth 𝑒 𝒓 is the Decay Factor 𝑒 𝒓 is the Growth Factor

Exponential Functions with e The natural base exponential function is of the form 𝑦=𝒂𝑒𝒓𝑥

Continuously Compounded Interest Recall that the formula below is used to calculate the amount of money in an account after 𝑡 years with interest compounded 𝑛 times per year. 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

Continuously Compounded Interest When interest is compounded continuously, the amount 𝐴 in an account after 𝑡 years is given by the formula: Where 𝑃 is the principal and 𝑟 is the interest rate expressed as a decimal. 𝐴=𝑃 𝑒 𝑟𝑡

Exercise 14 You deposit $ 5,500 in an account that pays 3.6% annual interest compounded continuously. What is the balance after 10 years?

Exercise 15 What formula could you use to calculate the amount of interest earned on an account in which the interest is compounded continously?

Continuous Growth/Decay The formula 𝑦=𝑃𝑒𝑟𝑡 is often an accurate model for growth or decay problems because things usually grow or decay continuously, not at specific time intervals. Exponential Decay Exponential Growth 𝑟<0 𝑟>0

7.1-7.3: Exponential Functions Exercise 16 According to some website, the population of Denton, TX was 119,454 in 2008. This was up from 3.01% from the previous year. If the population continues to grow continuously at this rate, what do you expect the population of Denton to be in 2020?

Exponential Functions Objectives: To define and graph exponential growth and decay functions To model real-life exponential situations To calculate compound interest To define the number e and use it as the base of exponential functions http://amb63.deviantart.com/art/Multiply-like-Rabbits-195975203