14 16 17 21 28 36 40 52 63 Warm-Up: Exponential Functions lesson By hand, graph the following data, draw a line of best fit, and determine a linear model for the data (an equation). X 14 16 17 21 28 Y 36 40 52 63 use y = mx + b OR y – y1 = m(x – x1)

Exploring Exponential Growth and Decay Models

What am I going to learn? Day 1: Concept of an exponential function Models for exponential growth Models for exponential decay Meaning of an asymptote Day 2 Finding the equation of an exponential function Applying exponential functions to models in finance

Recall Independent variable is another name for domain or input, which is typically but not always represented using the variable, x. Dependent variable is another name for range or output, which is typically but not always represented using the variable, y.

What is an exponential function? Obviously, it must have something to do with an exponent! An exponential function is a function whose independent variable is an exponent.

What does an exponential function look like? y=ab Exponent and Independent Variable Dependent Variable Just some number that’s not 0 Why not 0? Base

The Basis of Bases The base of an exponential function carries much of the meaning of the function. The base determines exponential growth or decay. The base is a positive number; however, it cannot be 1. We will return later to the reason behind this part of the definition .

Exponential Growth An exponential function models growth whenever its base > 1. (Why?) If the base b is larger than 1, then b is referred to as the growth factor.

What does Exponential Growth look like? Consider y = 2x Table of Values: Cool Fact: All exponential growth functions look like this! x 2x y -3 2-3 -2 2-2 ¼ -1 2-1 ½ 20 1 21 2 22 4 3 23 8 Graph:

Exponential Decay An exponential function models decay whenever its 0 < base < 1. (Why?) If the base b is between 0 and 1, then b is referred to as the decay factor.

What does Exponential Decay look like? Consider y = (½)x Graph: Cool Fact: All exponential decay functions look like this! Table of Values: x (½)x y -2 ½-2 4 -1 ½-1 2 ½0 1 ½1 ½ ½2 ¼ 3 ½3 1/8

End Behavior Notice the end behavior of the first graph-exponential growth. Go back and look at your graph. as you move to the right, the graph goes up without bound. as you move to the left, the graph levels off-getting close to but not touching the x-axis (y = 0).

End Behavior Notice the end behavior of the second graph-exponential decay. Go back and look at your graph. as you move to the right, the graph levels off-getting close to but not touching the x-axis (y = 0). as you move to the left, the graph goes up without bound.

Asymptotes One side of each of the graphs appears to flatten out into a horizontal line. An asymptote is a line that a graph approaches but never touches or intersects.

Asymptotes Notice that the left side of the graph gets really close to y = 0 as . We call the line y = 0 an asymptote of the graph. Think about why the curve will never take on a value of zero and will never be negative.

Asymptotes Notice the right side of the graph gets really close to y = 0 as . We call the line y = 0 an asymptote of the graph. Think about why the graph will never take on a value of zero and will never be negative.

Let’s take a second look at the base of an exponential function Let’s take a second look at the base of an exponential function. (It can be helpful to think about the base as the object that is being multiplied by itself repeatedly.) Why can’t the base be negative? Why can’t the base be zero? Why can’t the base be one?

Closure Determine if the function represents exponential growth or decay. 1. 2. 3. Exponential Growth Exponential Decay Exponential Decay

Warm-Up The population of Florida is given for various years from 1970-1994 in the table below. Years After 1970 10 15 20 22 24 Population (in thousands) 6791 9746 11351 12938 12510 13953 a) Use your calculator to find an exponential model for the data. (We use the same steps as linear regression, but we now do STAT, CALC, 0 (ExpReg) b) Use the model to find the population in 1987. c) Is this interpolation or extrapolation ? Why?

Exponential Modeling, Day 2

Example 4 Writing an Exponential Function Write an exponential function for a graph that includes (2, 2) and (3, 4). Use the general form. Substitute using (2, 2). Solve for a. Substitute using (3, 4). Substitute in for a.

Example 4 Writing an Exponential Function Write an exponential function for a graph that includes (2, 2) and (3, 4). Simplify. Backsubstitute to get a. Plug in a and b into the general formula to get equation.

Modeling Data At your tables, give 3 reasons that would prompt you to use exponential modeling instead of linear modeling.

Exponential Models Summary An exponential model is in the form of y= abx The horizontal asymptote is y = 0 When a > 0, the graph is concave up; when a < 0, the graph is concave down. When 0 < b < 1, lim x→∞ a b x =0 When b > 1, if a>0, lim x→∞ a b x =∞ if a<0, lim x→∞ a b x =−∞

Modeling in Finance Continuously Compounding Interest Compound Interest Formula 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 Continuously Compounding Interest A = Pert P = original amount invested r = interest rate (as a decimal) n = compounding periods t = time (in years)

Modeling in Finance, Examples Determine the amount of money in an account after 2 years, 10 months, when $575 was deposited initially, and the interest compounded is 4.7% APR? Semi-Annually $655.89 Quarterly $656.39 Monthly $656.73 Continuously $656.90

Exponential Modeling: You Practice In 1993, the population of the US was about 258 million. During the early 1990’s, the population of the US was growing at a rate of about 1% annually. Suppose this growth rate continues. Let p(x) = the population x years after 1993. a) Find an exponential formula for p(x). b) What would be the population of the US in the year 2013? c) In about what year will the population of the US reach 400 million?

Answers P(x) = 258 (1.01)x 314.81 million people Between year 44 and 45, so 2037

Closure Enter the data in your calculator to determine an exponential regression. Use your equation to predict the following: The debt (in trillions of dollars) for 2014 When will the debt be more than 10 trillion dollars? How did you determine your answer?

Warm-Up: Quiz Day Linear and Exponential Models Using your calculator, determine the equations for the following models: Linear Exponential Which model is better suited to the data? Justify your answer.