Completing the Square
Completing the square is when either of the following is performed:
is written in the form
is written in the form
For the simpler case where the coefficient of is 1:
- Take
to be half of
.
- Expand
.
- Choose
so as to adjust the constant so that the original quadratic expression is obtained.
See Example 1.
For the more complicated case where the coefficient of is not 1, remove a factor of
from the original quadratic and perform the above on the inside of the brackets before expanding again in the final step. See Example 2.
Sketching Quadratics
The graph of a quadratic can easily be sketched if you think about the transformations that have been applied to the graph of .
- Consider the graph of
- The graph of say
can be sketched by shifting the graph of
to the left by 1. See x-transformations on the Transformations page.
- The graph of
can be sketched by stretching the graph of
about the x-axis by a factor of 3. See y-transformations on the Transformations page.
- The graph of
can then be sketched by shifting the graph of
down by 4. See y-transformations again on the Transformations page.