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 by
but 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 if
is translated by the vector
then the new function is
This is the same as making the substitutions
and
Example
By making the substitution
solve the equation
and find
On substituting
the equation becomes
This expression factorises into
hence
