One way to try to understand complex phenomena is to
decomopose them into simpler components. If you can understand each
simple component seperately, you may understand the
phenomenon as a whole more easily.
Here is an example of such a process in the field of geometry. Consider a polygon. You can decompose it into
simpler components: triangles.
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If you know, for example, what the sum of angles is in a triangle,
or how to find the area of a triangle, you can use this knowledge
to find the sum of angles (or the area) of the polygon.
Similarly, when investigating functions, you can try to decompose a
complicated function into simpler ones. For example, to understand
the properties of a function with the following correspondence rule:
f(x) = 3x + 7
you can decompose it into a sum of the following two functions:
g(x) = 3x and h(x) = 7.
This activity examines quadratic functions with correspondence rules
of the type:
f(x) = 2x² - 3x + 1.
Consider these functions as a sum of three basic components:
g(x) = 2x², h(x) = -3x, and k(x) = 1.
Does familiarity with each of the basic components
help you understand the sum function?
More generally, you will also examine the addition of functions
and the uses to which you can put this operation in constructing new functions.
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The following tools are available for this activity to help you
investigate the addition of functions:
The Addition of functions tool
helps you present any pair of functions and their sum. The
Three components
tool presents the three basic components ax², bx, and c,
and the quadratic function that is their sum.
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Prepare a report on the addition of functions
and on the construction of quadratic functions as a
sum of the three basic components
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Show different examples of sums of linear and quadratic
functions and describe the relations between the summed functions and
the sum function.
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Describe how each basic component
affects the features of a quadratic function constructed
as a sum of basic components.
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Describe the course of your investigation,
your intermediate conclusions, your decisions, and their mathematical
justifications. In each case state what is your level of confidence
in your conclusions and why.
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Use the available tools to record important points for discussion and
interesting or problematic cases.
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