Chapter 3 Exponential and Logarithmic Functions 3 1

Objectives: • • Evaluate exponential functions. Graph exponential functions. Evaluate functions with base e.

What is an exponential function? What does an exponential function look like? Obviously, it

The Basis of Bases The base carries the meaning of the function. 1) determines

Let’s examine exponential functions. They are different than any of the other types of

Using graphing calculator plot (graph) : Compare graphs in your groups: 1) Similarities 2)

Similarity of Graphs of Exponential Function where a > 1 1. Domain is all

Graphing Exponential functions by Transformations What do you remember about Transformations of any functions:

Transformations Shifts the graph up if k>0 Shifts the graph down if k <

On your graphing calculators : Shifts to the left if h > 0 Shifts

On your graphing calculators : Reflecting reflects over the x-axis. reflects over the y-axis.

On your graphing calculators : Stretches the graph if a > 1 Shrinks the

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

Let’s take a second look at the base of an exponential function. (It can

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

Example: Evaluating an Exponential Function The exponential function models the average amount spent, f(x),

Example: Graphing an Exponential Function Graph: x – 2 – 1 One of the

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

Example: Transformations Involving Exponential Functions Use the graph of to obtain the graph of

Example: Transformations Involving Exponential Functions (continued) Use the graph of to obtain the graph

The Natural Base e The number e is defined as the value that approaches

Example: Evaluating Functions with Base e The exponential function models the gray wolf population

Formulas for Compound Interest After t years, the balance, A, in an account with

Example: Using Compound Interest Formulas A sum of $10, 000 is invested at an

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

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. 4

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. x 3 2 1 0 -1 -2 -3 2 x 8 4 2 1 1/2 1/4 1/8 Let’s look at the graph of this function by plotting some points. 8 7 6 5 4 3 2 1 BASE Recall what a negative exponent means: -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

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

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. 8

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. 14

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. 16

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

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. 19

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. 20

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. 21

Example: Evaluating Functions with Base e The exponential function models the gray wolf population of the Western Great Lakes, f(x), x years after 1978. Project the gray wolf’s population in the recovery area in 2012. 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. 22

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. 23

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, 859. 47. 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 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, 918. 25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25