Warm up Express the following in simplest form: 7 −1 9 1 2 8 2 3 6 −3 9 1 2 8 2 3 6 −3 16 −3 4

3.1 Exponential Functions

Sketch & Analyze Graphs: Exponential Functions 𝑓 𝑥 = 3 𝑥 x -4 -2 -1 2 4 f(x) Domain: Range: y-intercept: Asymptote: End behavior: Increasing: Decreasing:

Sketch & Analyze Graphs: Exponential Functions 𝑓 𝑥 = 2 −𝑥 x -4 -2 -1 2 4 f(x) Domain: Range: y-intercept: Asymptote: End behavior: Increasing: Decreasing:

What is an exponential function? Domain: Range: y-intercept: x-intercept: Extrema: Asymptote: End behavior: Continuity: Domain: Range: y-intercept: x-intercept: Extrema: Asymptote: End behavior: Continuity:

Transformations of exponential functions Using the rules we have already learned, describe the following transformations given that 𝑓 𝑥 = 2 𝑥 is the parent function. 𝑔 𝑥 = 2 𝑥+1 ℎ 𝑥 = 2 −𝑥 𝑗 𝑥 =−3( 2 𝑥 ) 𝑘 𝑥 = 2 𝑥 −2

Natural Base exponential function Most real world applications don’t use base 2 or base 10, but instead use an irrational number called e. e is an irrational number called the natural base. 𝑒= lim 𝑥→∞ 1+ 1 𝑥 𝑥 𝑒=2.7182818281828 𝑓 𝑥 = 𝑒 𝑥 is called the natural base exponential function.

Graphing the natural base exponential Using a graphing calculator, graph: 𝑓 𝑥 = 𝑒 𝑥 a 𝑥 = 𝑒 4𝑥 b 𝑥 = 𝑒 −𝑥 +3 𝑐 𝑥 = 1 2 𝑒 𝑥

Real world applications: compound interest Suppose an initial principle P is invested into an account with an annual interest rate r, and the interest rate is compounded or reinvested annually. At the end of each year, the interest earned is added to the account balance. The sum is the new principle for the next year. Let’s derive an equation for compound interest! 𝐴 𝑡 =𝑃 (1+𝑟) 𝑡

To allow for quarterly, monthly, or even daily compoundings, let n be the number of times the interest is compounded. The rate per compounding is: The number of compoundings after t years is: So the equation becomes: 𝐴 𝑡 =𝑃 1+ 𝑟 𝑛 𝑛𝑡

Krysti invests $300 in an account with a 6% interest rate, making no other deposits or withdrawals. What will Krysti’s account balance be after 20 years if the interest is compounded: Semi-annually: Monthly: Daily:

Financial literacy: your turn If $10,000 is invested in an online savings account earning 8% per year, how much will be in the account at the end of10 years if there are no other deposits or withdrawals and interest is compounded: Semiannually? Quarterly? Daily?

How does n affect account balance? Compounding n 𝑨=𝟑𝟎𝟎 𝟏+ 𝟎.𝟎𝟔 𝒏 𝟐𝟎𝒏 Annually 1 Semiannually 2 Quarterly 4 Monthly 12 Daily 365 Hourly 8760

Continuous compound interest Suppose to compound continuously so that there is no waiting period between interest payments. Let’s derive! Replace 𝑟 𝑛 with 1 𝑛 𝑟 Recall that 𝑒= lim 𝑥→∞ 1+ 1 𝑥 𝑥 , so 𝐴=𝑃 𝑒 𝑟𝑡

Suppose Krysti finds an account that will allow her to invest her $300 at a 6% rate compounded continuously. If there are no other deposits or withdrawals, what will Krysti’s account balance be after 20 years?

If $10,000 is invested in an online savings account earning 8% per year compounded continuously, how much will be in the account at the end of 10 years if there are no other deposits or withdrawals?

Exponential growth or decay Exponential growth and decay models apply to any situation where growth is proportional to the initial size of the quantity being considered. Rate r or k must be expressed as a decimal.

Mexico has a population of approximately 110 million Mexico has a population of approximately 110 million. If Mexico’s population continues to grow at the described rate, predict the population of Mexico in 10 and 20 years. 1.42% annually 1.42% continuously 𝑁= 𝑁 0 (1+𝑟) 𝑡 𝑁= 𝑁 0 𝑒 𝑘𝑡

The population of a town is declining at a rate of 6% The population of a town is declining at a rate of 6%. If the current population is 12,426 people, predict the population in 5 and 1o years using each model. Annually Continuously

Exponential Modeling The table shows the number of reported cases of chicken pox in the US in 1980 and 2005. If the number of reported cases of chicken pox is decreasing at an exponential rate, identify the rate of decline and write an exponential equation to model this situation Use your model to predict when the number of cases will drop below 20,000.

Use the data in the table and assume that the population of Miami-Dade County is growing exponentially. Identify the growth and write an exponential equation to model this growth. Use your model to predict in which year the population of Miami-Dade County will surpass 2.7 million.