Substitution has many uses. It can be used to simplify
differential equations and make them easier to solve. It can by use
when integrating to transform integrands into forms that can be
integrated, it can be used to transform graphs, and used correctly it
can make many intractable problems solvable.
Example:
Find
Write down the equation of the graph that results when y=sinx is
translated by the vector
The x value is increased bybut
counter intuitively this means that sin x becomes
-
the y coordinate behaves as you expect, so the new equation is,
but it is also sufficient to say that ifis
translated by the vector
then
the new function isThis
is the same as making the substitutionsand
ExampleBy
making the substitutionsolve
the equationand
find
On substitutingthe
equation becomesThis
expression factorises into
hence
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