Kathy Radford

EMAT 6680

Assignment 3, Investigation #1

For this assignment we are going to look at the graphs of
equations of the form:

In particular we are going to graph the equation , for values of b from -4 to 4, as shown below:

It can be seen from the graph that these equations are all
parabolas of the same size, but shifted horizontally to the left of the origin for positive
b, and right of the origin for negative b, and also vertically. The graph of the equation where b=0, is
the parabola centered on the y-axis with a vertex at (0,1). The graphs for
values of b other than zero are increasingly lower.

To find the equation of the locus of the vertices of this
set of parabolas, we will take the equation

and put it in vertex
form by completing the square

From this form it can be seen that the vertex of each
parabola is

Setting , we see that each vertex is located on the graph of

or

which is the equation
for a downward-opening parabola translated upward or vertically by 1.

In the graph shown below it can be seen that the graph of
this equation, shown in red, contains, or intersects the vertices of each
parabola.