In many mathematical investigations we encounter the need
to compare two functions. Such comparisons can be of two types:
Equations - when we want to know for which values
of variable x the two functions are equal.
Inequalities - when we want to know for which values
of varible x one function is greater than the other.
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When the two functions involved are linear, we call
the comparison a linear comparison.
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When at least one of the functions is not linear,
we refer to a non-linear comparison.
Non-linear comparisons form a wide and rich field of study.
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This activity focuses mainly on one type of non-linear
comparison: the quadratic comparison
between a quadratic and a linear function or between
two quadratic functions.
As you investigate quadratic comparisons,
it will be useful to think about the similarities and differences
between them and linear comparisons, and about the
new phenomena that arise.
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The tools used in this activity help you investigate
comparisons in symbolic and graphic forms, and
make changes to comparisons (either to one side or
to both sides simultaneously) using various methods. These changes
will enable you to:
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Construct various examples of comparisons and investigate
different possibilities for their sets of solutions.
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1.
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Examine various operations for changing comparisons and
study changes that affect (or do not affect) the set
of solutions of comparisons.
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2.
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In addition to the tools specific to this activity, you may also find the
Representation of comparisons
tool from the general tools list to be useful.
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Prepare a "comparisons gallery"
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Examine different common positions between
a quadratic function and a linear function, and between two
quadratic functions. How would you present a "gallery"
of different comparisons to give a "visitor"
a general picture of various possible types of comparisons?
Consider the following properties of comparisons:
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The type of functions compared
(quadratic, linear, constant, increasing...).
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The type of comparison (equation, inequality...).
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The nature of the set of solutions (no solutions,
a single solution, several solutions, an interval of solutions,
several intervals of solutions).
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