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This course examines the families of polynomial,
exponential, inverse, logarithmic, rational, and periodic functions.
Emphasis is placed on studying functions as bridges between the
mathematics and the situations they model. Graphing calculator
skills are reinforced and expanded. Prerequisite: Math X464A (Perspectives on Functions 2)
Pascal’s Triangle
(PASCAL)
Participants investigate several number patterns stemming from
geometric patterns and from Pascal’s triangle. Participants
learn to identify polynomial patterns and to develop appropriate
polynomial models using the finite differences method. Participants
learn to identify exponential patterns and develop appropriate
exponential models using the “finite quotients” method.
Participants review the basic forms and growth behaviors of linear,
quadratic, cubic and exponential functions.
Exponential Functions
2 (EXPO2)
Participants investigate exponential functions in the context
of applications, such as rebound height of a bouncing ball,
population
growth, investing, and rumor propagation. Participants learn
the basic form of the exponential function and the meaning
of its parameters
in context. They learn to create exponential models for appropriate
applications. They investigate the concepts of domain, range,
limits, and end behavior in the context of exponential functions. Inverse
Functions (INV)
Participants are introduced to inverse functions through a hands-on
activity, where they graphically find the reflection of a function
along the y = x line. Participants explore inverse functions
through a bottle exploration activity and through conversion
formulas such
as Celsius and Fahrenheit temperatures. Participants investigate
methods of finding inverse functions, graphical relationships
between inverses, domain and range relationships between inverses,
and
invertibility.
Logarithms (LOGS)
Participants investigate inverses of exponentials by analyzing
a concrete data set. Participants investigate properties
of logarithm functions and their proofs. Participants practice
computing logarithms
and using properties of logarithms. Participants investigate
applications of logarithms in linearizing data, in the
context of earthquake
data. The Exponential Function (E)
Participants investigate the effects of compounding interest
over shorter and shorter time intervals. In a numerical
and graphical investigation, they are led to the base e for
natural
logarithms
as the limit of (1 + 1/n)n as n increases to infinity.
Participants investigate graphically the inverse function of
f(x) =
ex .
Rational Functions (RAT)
Participants practice computation skills with rational
expressions. Participants investigate intercepts, asymptotes,
and graphs
of rational functions. Participants investigate a fixed
area problem
that leads to a rational function. Participants investigate
the properties of rational functions and elementary limit
concepts. Participants investigate other modeling problems
that lead
to
rational functions.
The Kite Project (KITE)
Participants construct tetrahedral kites using drinking
straws and tissue paper. Participants then combine
their small kites
to form two different larger kites (Sierpinski and
solid) and investigate
the patterns among the tetrahedra. Finally, participants
use the fourfold way to write generalizations about
the relationships discovered.
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