Exponential Functions and Their Graphs

1
Exponential Functions and Their Graphs

• MATH 109 - Precalculus
• S. Rook

2
Overview

• Section 3.1 in the textbook
• Exponential functions
• Graphing exponential functions
• ex

3
Exponential Functions
4
Exponential Functions

• Thus far we have discussed linear and polynomial
  functions
• There exist applications which cannot be modeled
  by linear or polynomial functions
• e.g. bacteria reproduction, amount of a
  radioactive substance, continuous compounding of
  a bank account
• In all of these cases, a value in the model
  changes by a multiple of the previous value
• e.g. A population that starts with 2 members and
  doubles every hour

5
Exponential Functions (Continued)

• Exponential function f(x) ax where the base a
  gt 0, a ? 1 and x is a real number
• If the base were negative, some values of x would
  result in complex values
• To evaluate an exponential function
• Substitute the value for x and evaluate the
  expression

6
Evaluating an Exponential Function (Example)

• Ex 1 Use a calculator to estimate
• a) f(x) 3.4x when x 5.6 round to three
  decimal places
• b) g(x) 5x when x 2/3 round to three
  decimal places

7
Graphing Exponential Functions
8
Graphing Exponential Functions

• To graph an exponential function f(x) ax, make
  a table of values
• If a gt 1 and x gt 0, we
  will get a curve
  
  something like that on
  the right
• If 0 lt a lt 1 OR x lt 0,
  
  we will get a curve
  
  something like
  that on the right

9
Properties of Exponential Functions, a gt 0 and
x gt 0

• Does f(x) ax have an inverse?
• Yes, any horizontal line will
  
  cross f(x) only once
• What happens when x 0?
• a0 1 ? y-int (0, 1)
• What is the domain and
  
  range of f(x)?
• Domain (-oo, oo)
• Range (0, oo)
• What can be noticed about the end behavior of
  f(x)?
• As x ? -oo, f(x) ? 0 as x ? oo, f(x) ? oo

10
Properties of Exponential Functions, 0 lt a lt 1
or x lt 0

• Does f(x) a-x or f(x) ax (0 lt a lt 1) have an
  inverse?
• Yes, any horizontal line will
  
  cross f(x) only once
• What happens when x 0?
• a0 1 ? y-int (0, 1)
• What is the domain and
  range of f(x)?
• Domain (-oo, oo)
• Range (0, oo)
• What can be noticed about the end behavior of
  f(x)?
• As x ? -oo, f(x) ? oo as x ? oo, f(x) ? 0

11
Graphing Exponential Functions (Example)

• Ex 2 Use a calculator to obtain a table of
  values for the function and then sketch its
  graph
• a) f(x) 3x
• b) g(x) 6-x

12
Exponential Functions Transformations

• We can also apply transformations to graph
  exponential functions
• Recall the following types of transformations
• Horizontal and vertical shifts
• Horizontal and vertical stretches compressions
• Reflections over the x and y axis

13
Exponential Functions Transformations (Example)

• Ex 3 Use the graph of f to describe the
  transformation(s) that yield the graph of g
• a) f(x) 3x g(x) 3x 4 2
• b)

14
ex
15
ex

• e is a mathematical constant discovered by
  Leonhard Euler
• Used in many different applications
• Deriving the value of e is somewhat difficult and
  you will learn how to do so when you take
  Calculus
• Natural exponential function f(x) ex where
  e 2.718 (a constant)
• We can graph ex by creating a table of values and
  we can also apply translations
• f(x) ex

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ex ( Example)

• Ex 4 Use a calculator to estimate f(x) ex
  when x 10 and when x 7/4 round to three
  decimal places

17
One-to-One Property
18
One-to-One Property

• As previously discussed, exponential functions
  are one-to-one functions
• One value of y for every x and vice versa
• One-to-one Property If a gt 0, a ? 1, ax ay ?
  x y
• i.e. Obtain the same base and equate the exponents

19
One-to-One Property (Example)

• Ex 5 Use the One-to-One Property to solve the
  equation for x
• a) b)
• c)

20
Summary

• After studying these slides, you should be able
  to
• Understand the concept of an exponential function
  and the limitations on the base
• Describe the graph of an exponential function by
  looking at the base
• State the domain and range of an exponential
  function
• Graph an exponential function using a table of
  values and translations
• Understand the constant e and be able to graph
  the natural exponential function using a t-chart
  and translations
• Additional Practice
• See the list of suggested problems for 3.1
• Next lesson
• Logarithmic Functions and Their Graphs (Section
  3.2)