**The Task:**
Construct the graphs for a quadratic equation in standard form
where two of the values for **a**, **b**, and **c** are fixed. Interpret the graphs.

The standard from of a quadratic function is:

For this investigation focuses on fixing
the values of **a** and **c** while varying the value for **b**. As a beginning,
the value of **a** and **c** are assigned an arbitrary value of **1**. A set of
functions with values of **b** ranging from **-5** to **5** is presented below.

One interesting aspect of this graph
is the location of the vertex for each parabola and how this location
changes as **b**
changes. While it may be difficult to grasp this relationship
from the graph above, it is easily seen by viewing an animated
graph.

Click here to see the Animated Graph. (Your computer must have Graphing Calculator software to exercise this option.)

As **b** changes, the vertex of the parabola appears
to move along a parabolic path. This can be confirmed algebraically.

For the quadratic function,

the vertex has the coordinates

Substituting the value x for gives us the coordinates . Thus, the locus of the vertices for these quadratic functions is

Inserting the graph of **v** into the graph
above confirms that the vertex of each parabola lies on **v**.

Click here to see the Animated Graph
tracing the function **v**.

What happens when the values of **a** and **c** are changed?
Using the same algebraic derivation from above will result in
a more general form of the function v.

Theorem: Let a family of parabolas
be of the form , where **a** and **c** are fixed values and **b**
is any real number. The locus of points formed by the vertices
of the graphs in this family is a parabola defined by the function

Can you derive this function? Click here to see the Derivation of the General Form.