Page Bird

Investigation 1.

Consider the locus of the vertices of the set of parabolas graphed from

Show that the locus is the parabola

In this graph, the values of b range from –3 to 3.

The parabolas all intercept at the point (0,1).

Using the graphing calculator, it is clear that the vertices of the parabolas graphed above are

(–1.5, -1.25), ( -1, 0), ( -.5, .75), (0, 1), ( .5, .75), ( 1, 0),

(1.5, -1.25).

Using slope and y-intercept form, the equations of the lines that go through points (1, 0), ( -1, 0), and (0, 1) are

and

Interestingly,

The graph below shows that the product of the equations of the two lines is a parabola. The vertices of each parabola with the equation

maps to the graph below.

To view a GSP file of this parabola click here.