Chapter 3 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Exponential Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Evaluate exponential functions. Graph exponential functions. Evaluate functions with base e. Use compound interest formulas. Objectives:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 What is an exponential function? What does an exponential function look like? Base Exponent and Independent Variable Just some number that’s not 0 Why not 0? Dependent Variable Obviously, it must have something to do with an exponent!

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 The Basis of Bases The base carries the meaning of the function. 1) determines exponential growth or decay. 2)base is a positive number; however, it cannot be 1.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent. Let’s look at the graph of this function by plotting some points. x 2 x /2 -2 1/4 -3 1/ Recall what a negative exponent means: BASE

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Using graphing calculator plot (graph) : Compare graphs in your groups: 1)Similarities 2)Differences

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Similarity of Graphs of Exponential Function where a > 1 What is the domain of an exponential function? 1. Domain is all real numbers What is the range of an exponential function? 2. Range is positive real numbers What is the x intercept of these exponential functions? 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 What is the y intercept of these exponential functions? 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing Are these exponential functions increasing or decreasing? 6. A horizontal asymptote y = 0 (x-axis) Can you see the horizontal asymptote for these functions? What is different? Steepness

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Graphing Exponential functions by Transformations What do you remember about Transformations of any functions: ---- shifts, reflections, stretching, compressing Coefficients/constants “a”, “h” and “k” On your graphing calculators :

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Transformations Shifts the graph up if k > 0 Shifts the graph down if k < 0 Vertical translation f(x) = b x + k

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Shifts to the left if h > 0 Shifts to the right if h < 0 On your graphing calculators : Horizontal translation: f(x)=b x-h

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Reflecting reflects over the x-axis. reflects over the y-axis. On your graphing calculators :

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Stretches the graph if a > 1 Shrinks the graph if 0 < a < 1 Vertical stretching or shrinking, f(x)=ab x : On your graphing calculators :

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Foldable: Graphing Exponential Functions 13

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 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?

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Evaluating an Exponential Function The exponential function models the average amount spent, f(x), in dollars, at a shopping mall after x hours. What is the average amount spent, to the nearest dollar, after three hours at a shopping mall? We substitute 3 for x and evaluate the function. After 3 hours at a shopping mall, the average amount spent is $160.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing an Exponential Function Graph: One of the ways to graph it: “T” chart - plot points x –2–2 –1–1 0 1 We set up a table of coordinates, then plot these points, connecting them with a smooth, continuous curve.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Characteristics of Exponential Functions of the Form

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Transformations Involving Exponential Functions Use the graph of to obtain the graph of Begin with We’ve identified three points and the asymptote. Horizontal asymptote y = 0

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Transformations Involving Exponential Functions (continued) Use the graph of to obtain the graph of The graph will shift 1 unit to the right. Add 1 to each x-coordinate. Horizontal asymptote y = 0

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 The Natural Base e The number e is defined as the value that approaches as n gets larger and larger. As the approximate value of e to nine decimal places is The irrational number, e, approximately 2.72, is called the natural base. The function is called the natural exponential function.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Evaluating Functions with Base e The exponential function models the gray wolf population of the Western Great Lakes, f(x), x years after Project the gray wolf’s population in the recovery area in Because 2012 is 34 years after 1978, we substitute 34 for x in the given function. This indicates that the gray wolf population in the Western Great Lakes in the year 2012 is projected to be approximately 4446.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Formulas for Compound Interest After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1. For n compounding periods per year: 2. For continuous compounding:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Using Compound Interest Formulas A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to quarterly compounding. We will use the formula for n compounding periods per year, with n = 4. The balance of the account after 5 years subject to quarterly compounding will be $14,

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Using Compound Interest Formulas A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to continuous compounding. We will use the formula for continuous compounding. The balance in the account after 5 years subject to continuous compounding will be $14,