Pre-AP Pre-CalculusChapter 3, Section 1

Exponential and Logistic Functions2013 - 2014

Function vs. Function

• What do you notice about the two functions?

Exponential Functions & Their Graphs• The functions and each involve a base raised

to a power, but the roles are reversed. • For the base is the __________, and the

exponent is a ___________. This is a monomial and a power function.

• For , the base is a _______, and the exponent is a _________. This is called an _________________________________.

Definition: Exponential Function

• An exponential function can be written in the form:

where a & b are real number __________. a is ____________ & the ________________. b is ____________ and does not __________. b is also called the _________.

Are they exponential functions?

Compute the exponential function value

• Use

Finding an Exponential Function from its Table of Values

x g(x) h(x)

-2 4/9 128

-1 4/3 32

0 4 8

1 12 2

2 36 1/2

Finding an Exponential Function from its Table of Values

x f(x) t(x)

-2 6 108

-1 3 36

0 3/2 12

1 3/4 4

2 3/8 4/3

Graph of Exponential Functions• Graph each function in the viewing window [-2, 2]

by [-1, 6]

• Which point is common to all four graphs?• Analyze each graph for domain, range, extrema,

continuity, increasing or decreasing, symmetry, asymptotes, and end behavior.

Graph of Exponential Functions• Graph each function in the viewing window [-2, 2]

by [-1, 6]

• Which point is common to all four graphs?• Analyze each graph for domain, range, extrema,

continuity, increasing or decreasing, symmetry, asymptotes, and end behavior.

Exponential Growth & Decay

The Natural Base e

• Domain:• Range:• Continuity:• Increasing:• Symmetry:• Extrema:• Asymptotes:• End Behavior:

Facts about e

• The number e is _______________.• Since the graph is increasing, e is considered

_____________________________. • Because of specific calculus properties, the

function e is considered the _____________ _________ of exponential functions.

• is considered the _____________________________________.

Logistic Functions & Their Graphs• Exponential growth is unrestricted. Meaning,

_____________________________________.• In many growth situations, there is a limit to possible

growth. A plant can only grow so tall. The number of goldfish in an aquarium is limited by the size of the aquarium.

• In such growth situations, the beginning is exponential in manner, but slows down and eventually levels out.

• These types of growth situations have horizontal asymptotes.

Logistic Growth Functions

• Let a, b, c, and k be positive constants, with b < 1. A logistic growth function in x in a function that can be written in the form

where the constant c is the ______________.

Unless otherwise noted, all logistic functions in the book are logistic growth functions.

Basic Function: The Logistic Function

• [-4.7, 4.7] by [-0.5, 1.5]

• Domain:• Range:• Continuity:• Extrema:• Asymptotes: • End Behavior:

Graphing Logistic Growth Functions

• Graph the function. Find the y-intercept and the horizontal asymptotes.

Graphing Logistic Growth Functions

• Graph the function. Find the y-intercept and the horizontal asymptotes.

Population Growth• Use the data in the table. Assuming the growth is

exponential, when will the population of San Jose surpass 1 million persons? The Population of San

Jose, California

Year Population

1990 782,248

2000 895,193

Modeling Dallas’s Population• Based on recent census data, a logistic model for the

population of Dallas, t years after 1900, is as follows:

According to this model, when was the population 1 million?

Ch 3.1 Homework

• Page 286 – 288, #’s: 3, 4, 5, 8, 10, 13, 28, 42, 56, 61

• 10 total problems

• Gray book: pg 261 - 263