## Substitution

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 vectorthen the new function isThis is the same as making the substitutionsand

ExampleBy making the substitutionsolve the equationand find

On substitutingthe equation becomesThis expression factorises intohence