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Quadratic Functions
Definition: A quadratic function is a function
which can be put in the form
(General Form) or
(Standard or Vertex Form )
For a quadratic functions or :
0) The domain of a quadratic function is
1) Graph : a (vertical) parabola
2) Opens up if
3) Opens down if
4) Vertex: with
and
5) The x intercept (s) can be found by setting
and solving for x .
6) The y intercept an be found by setting
and solving for y .
Remark: The above items can be used to sketch the graph of a quadratic
function
Examples:
Remark: The conversion from vertex form to general
from is very easy. For conversion from general form to vertex form
we use the method of Completing Square .
Example:
The Maximum and the Minimum of a Quadratic Function (Parabola):
As was discussed above, the vertex of a parabola is
with
and
a) If the parabola opens up, the vertex will be the lowest point and we say that
the parabola has the minimum value
of at .
b) If the parabola opes down, the vertex will be the highest point and we say
that the parabola has the maximum
value of at .
Remark: The coordinates
of the vertex can be
found by a graphing calculator .
Examples:
Application:
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