An important way of characterizing functions is
by describing their rate of change.
Linear functions are characterized by
a constant rate of change -- so they are
useful for describing such phenomena as motion
at a constant speed, where the distance travelled changes
at a constant rate. However, many phenomena do not
happen at a constant rate, and to describe them it is necessary to
use functions that are not linear.
This activity focuses on one of the most useful
types of functions used to describe change at a
non-constant rate: quadratic functions.
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Use the tool below to control the change of a function
(represented in the lower window). Click the graph button to construct
the graph of the function.
You can now construct different
functions describing quadratic growth,
where the rate of change changes at a constant rate.
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Prepare an essay on quadratic growth
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Explore how the various quantities under
your control (difference between x-values, first change,
difference between changes) affect the change. Explore the properties
of the resulting graph. Categorize the different types of graphs you can
obtain in a way that you think is useful.
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Present examples of tables of values and correspondence rules
that describe quadratic growth, and others that do
not describe such growth.
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Present stories with mathematical descriptions that
lead to quadratic growth. Construct various
representations (graphs, correspondence rules, value tables)
for functions describing the stories. Propose interesting
questions and try to answer them using the mathematical
representations.
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Use the tools available for this activity to study the concepts
of change and quadratic growth.
The From function to change
tool from the general tools list can also help.
After an initial investigation using the tools,
work on some of the tasks; they provide examples of
stories with mathematical descriptions that lead to
quadratic growth. Examine and
apply what you have learned and raise question for further
investigation.
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