pre-ap pre-calculus chapter 3, section 1 exponential and logistic functions 2013 - 2014
TRANSCRIPT
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Pre-AP Pre-CalculusChapter 3, Section 1
Exponential and Logistic Functions2013 - 2014
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Function vs. Function
• What do you notice about the two functions?
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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 _________________________________.
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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 _________.
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Are they exponential functions?
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Compute the exponential function value
• Use
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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
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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
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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.
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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.
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Exponential Growth & Decay
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The Natural Base e
• Domain:• Range:• Continuity:• Increasing:• Symmetry:• Extrema:• Asymptotes:• End Behavior:
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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 _____________________________________.
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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.
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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.
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Basic Function: The Logistic Function
• [-4.7, 4.7] by [-0.5, 1.5]
• Domain:• Range:• Continuity:• Extrema:• Asymptotes: • End Behavior:
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Graphing Logistic Growth Functions
• Graph the function. Find the y-intercept and the horizontal asymptotes.
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Graphing Logistic Growth Functions
• Graph the function. Find the y-intercept and the horizontal asymptotes.
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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
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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?
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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