Exponential Functions and Their Graphs

MATH 109 - Precalculus S. Rook

Overview

• Section 3.1 in the textbook: – Exponential functions – Graphing exponential functions –

e

x

2

Exponential Functions

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 4

Exponential Functions (Continued) • •

Exponential function: a

> 0,

a

f(x) =

a

x

where the base ≠ 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 5

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) = 5

x

when x = 2 ⁄ 3 decimal places – round to three 6

Graphing Exponential Functions

Graphing Exponential Functions

• To graph an exponential function f(x) =

a

x

, make a table of values: – If

a

> 1 and x > 0, we will get a curve something like that on the right – If 0 <

a

< 1 OR x < 0, we will get a curve something like that on the right 8

• • • • Properties of Exponential Functions, a > 0 and x > 0 Does f(x) =

a

x

– have an inverse?

Yes, any horizontal line will cross f(x) only once What happens when x = 0?

a

0 = 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 9

• • • • Properties of Exponential Functions, 0 < a < 1 or x < 0 Does f(x) =

a

-x – or f(x) =

a

x

(0 <

a

< 1) have an inverse?

Yes, any horizontal line will cross f(x) only once What happens when x = 0?

a

0 = 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 10

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) = 3

x

b) g(x) = 6 -x 11

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 12

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) = 3

x

b)

f

 7 2

x g

g(x) = 3 x – 4 – 2    7 2

x

 3  1 13

e

x

e

x

• • • 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:

e ≈ 2.718 (a constant) f(x) = e

x

where We can graph e

x

by creating a table of values and we can also apply translations f(x) = e

x

15

e

x

( Example)

Ex 4:

Use a calculator to estimate f(x) = e

x

x = 10 and when x = 7 ⁄ 4 decimal places – round to three when 16

One-to-One Property

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:

x = y If

a

> 0,

a

≠ 1, a – i.e. Obtain the same base and equate the exponents

x

= a

y

→ 18

One-to-One Property (Example)

Ex 5:

Use the One-to-One Property to solve the equation for x: a) 3

x

 1  27 c) 2

x

 2  1 32 b)

e x

2  3 

e

2

x

19

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) 20