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.