3.1 – Exponential Functions and Their Graphs

Pre-Calculus

Mr. Niedert

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 1 / 20

3.1 – Exponential Functions and Their Graphs

1 Exponential Functions

2 Graphs of Exponential Functions

3 The Natural Base e

4 Solving Simple Logarithmic Equations

5 Compound Interest

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 2 / 20

3.1 – Exponential Functions and Their Graphs

1 Exponential Functions

2 Graphs of Exponential Functions

3 The Natural Base e

4 Solving Simple Logarithmic Equations

5 Compound Interest

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 2 / 20

3.1 – Exponential Functions and Their Graphs

1 Exponential Functions

2 Graphs of Exponential Functions

3 The Natural Base e

4 Solving Simple Logarithmic Equations

5 Compound Interest

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 2 / 20

3.1 – Exponential Functions and Their Graphs

1 Exponential Functions

2 Graphs of Exponential Functions

3 The Natural Base e

4 Solving Simple Logarithmic Equations

5 Compound Interest

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 2 / 20

3.1 – Exponential Functions and Their Graphs

1 Exponential Functions

2 Graphs of Exponential Functions

3 The Natural Base e

4 Solving Simple Logarithmic Equations

5 Compound Interest

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 2 / 20

Definition of Exponential Function

Definition of Exponential Function

The exponential function f with base a is denoted by f (x) = ax wherea > 0, a 6= 1, and x is any real number.

The base a = 1 is excluded because it yields f (x) = 1x = 1. This isactually a constant function, since it is always equal to 1, so it is notan exponential function.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 3 / 20

Definition of Exponential Function

Definition of Exponential Function

The exponential function f with base a is denoted by f (x) = ax wherea > 0, a 6= 1, and x is any real number.

The base a = 1 is excluded because it yields f (x) = 1x = 1. This isactually a constant function, since it is always equal to 1, so it is notan exponential function.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 3 / 20

Graphs of y = ax and y = a−x

Practice

Plug in points (of your choosing) and sketch the graph of each function onthe same coordinate plane. Use different colors if possible.

a f (x) = 2x

b g(x) = 4x

c h(x) = 2−x

d j(x) = 4−x

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 4 / 20

Characteristics of Exponential Functions

Investigation

Consider the graphs of f (x) = 2x and h(x) = 2−x from earlier.

a For each function, determine the domain, range, intercepts, andasymptotes.

b For each function, determine when the function is increasing andwhen the function is decreasing.

c For each function, determine if the function is continuous.

d Using parts a-c, make some general conclusions about the graphs ofexponential functions.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 5 / 20

Characteristics of Exponential Functions

Investigation

Consider the graphs of f (x) = 2x and h(x) = 2−x from earlier.

a For each function, determine the domain, range, intercepts, andasymptotes.

b For each function, determine when the function is increasing andwhen the function is decreasing.

c For each function, determine if the function is continuous.

d Using parts a-c, make some general conclusions about the graphs ofexponential functions.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 5 / 20

Characteristics of Exponential Functions

Investigation

Consider the graphs of f (x) = 2x and h(x) = 2−x from earlier.

a For each function, determine the domain, range, intercepts, andasymptotes.

b For each function, determine when the function is increasing andwhen the function is decreasing.

c For each function, determine if the function is continuous.

d Using parts a-c, make some general conclusions about the graphs ofexponential functions.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 5 / 20

Characteristics of Exponential Functions

Investigation

Consider the graphs of f (x) = 2x and h(x) = 2−x from earlier.

a For each function, determine the domain, range, intercepts, andasymptotes.

b For each function, determine when the function is increasing andwhen the function is decreasing.

c For each function, determine if the function is continuous.

d Using parts a-c, make some general conclusions about the graphs ofexponential functions.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 5 / 20

Characteristics of Exponential Functions

Investigation

Consider the graphs of f (x) = 2x and h(x) = 2−x from earlier.

a For each function, determine the domain, range, intercepts, andasymptotes.

b For each function, determine when the function is increasing andwhen the function is decreasing.

c For each function, determine if the function is continuous.

d Using parts a-c, make some general conclusions about the graphs ofexponential functions.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 5 / 20

Transformations of Graphs of Exponential Functions

Practice

Graph f (x) = 3x . Then graph each of the following and describe thetranslations necessary to graph each from the parent function f (x) = 3x .

a g(x) = 3x+1

b h(x) = 3x − 2

c j(x) = −3x

d k(x) = 3−x

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 6 / 20

3.1 – Exponential Functions and Their Graphs (Part 1 of4) Assignment

Part 1: pg. 226 #7-22

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 7 / 20

The Natural Base e

The number e ≈ 2.718281828.

It is referred to as the natural base and is the most convenient choicefor a base in many applications that incorporate exponential functions.

f (x) = ex is called the natural exponential function.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 8 / 20

The Natural Base e

The number e ≈ 2.718281828.

It is referred to as the natural base and is the most convenient choicefor a base in many applications that incorporate exponential functions.

f (x) = ex is called the natural exponential function.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 8 / 20

The Natural Base e

The number e ≈ 2.718281828.

It is referred to as the natural base and is the most convenient choicefor a base in many applications that incorporate exponential functions.

f (x) = ex is called the natural exponential function.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 8 / 20

Evaluating the Natural Exponential Function

Example

Use a calculator to evaluate the function given by f (x) = ex when x = −2.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 9 / 20

Evaluating the Natural Exponential Function

Practice

Use a calculator to evaluate the function given by f (x) = ex at eachindicated value of x .

a x = −1

b x = 0.25

c x = −0.3

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 10 / 20

Graphing Natural Exponential Functions

Practice

Use a graphing calculator to construct a table of values for the function.Then sketch the graph of the function.

a f (x) = 2e0.24x

b f (x) = 12e

−0.58x

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 11 / 20

3.1 – Exponential Functions and Their Graphs (Part 2 of4) Assignment

Part 1: pg. 226 #7-22Part 2: pg. 226-227 #27-38

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 12 / 20

One-to-One Property

We have already seen that an exponential function is either alwaysincreasing or always decreasing.

As a result, the graph passes the Horizontal Line Test, meaning thatthe functions are always one-to-one.

One-to-One Property

For a > 0 and a 6= 1, ax = ay if and only if x = y .

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 13 / 20

One-to-One Property

We have already seen that an exponential function is either alwaysincreasing or always decreasing.

As a result, the graph passes the Horizontal Line Test, meaning thatthe functions are always one-to-one.

One-to-One Property

For a > 0 and a 6= 1, ax = ay if and only if x = y .

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 13 / 20

One-to-One Property

We have already seen that an exponential function is either alwaysincreasing or always decreasing.

As a result, the graph passes the Horizontal Line Test, meaning thatthe functions are always one-to-one.

One-to-One Property

For a > 0 and a 6= 1, ax = ay if and only if x = y .

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 13 / 20

Using the One-to-One Property

Example

Solve the equation 9 = 3x+1

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 14 / 20

Using the One-to-One Property

Practice

Solve the equation(12

)x= 8.

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 15 / 20

Compound Interest

Compound interest refers to interest being charged based on interestcharged in the past.

Interest can be charged yearly, monthly, weekly, daily, hourly, or evencontinuously.

Formulas for Compound Interest

After t years, the balance A in an account with principal P and annualinterest rate r (in decimal form) is given by the following formulas.

I For n compoundings per year: A = P(

1 +r

n

)nt

I For continuous compounding: A = Pert

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 16 / 20

Compound Interest

Compound interest refers to interest being charged based on interestcharged in the past.

Interest can be charged yearly, monthly, weekly, daily, hourly, or evencontinuously.

Formulas for Compound Interest

After t years, the balance A in an account with principal P and annualinterest rate r (in decimal form) is given by the following formulas.

I For n compoundings per year: A = P(

1 +r

n

)nt

I For continuous compounding: A = Pert

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 16 / 20

Compound Interest

Compound interest refers to interest being charged based on interestcharged in the past.

Interest can be charged yearly, monthly, weekly, daily, hourly, or evencontinuously.

Formulas for Compound Interest

After t years, the balance A in an account with principal P and annualinterest rate r (in decimal form) is given by the following formulas.

I For n compoundings per year: A = P(

1 +r

n

)nt

I For continuous compounding: A = Pert

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 16 / 20

Compound Interest

Example

A total of $10,000 is invested at an annual interest rate of 6%. Find thebalance after 8 years if it is compounded

a weekly

b semiannually

c continuously

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 17 / 20

Compound Interest

Practice

A total of $12,000 is invested at an annual interest rate of 9%. Find thebalance after 5 years if it is compounded

a quarterly

b monthly

c continuously

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 18 / 20

3.1 – Exponential Functions and Their Graphs (Part 3 of3) Assignment

Part 1: pg. 226 #7-22Part 2: pg. 226-227 #27-38Part 3: pg. 226-227 #45-52, 54-60 even, 61-62

3.1 – Exponential Functions and Their Graphs Assignmentpg. 226-227 #7-22, 27-38, 45-52, 54-60 even, 61-62

Pre-Calculus 3.1 – Exp. Func. and Their Graphs Mr. Niedert 19 / 20