Now consider the locus of the vertices of the set of parabolas graphed from

Show that the locus is the parabola

First of all, you may be asking yourself, "What is a locus, exactly?" Let's just be clear on the definition.

Now, what do we know about vertices of a parabola?

1. They are either a relative maximum or a relative minimum value.

2. The slope of the graph is positve on one side of a vertex and negative on the other side.

3. The f ' (x) = 0 at the vertex.

So, a locus of the vertices of the set of parabolas graphed from our given equation would somehow be connected to the derivative of the equation.

Now, remember, we have to find each vertex of the graph. So we have to set the derivative equal to 0 and solve for b.

So this is the relationship with each b value and the corresponding x value at the vertex. Now just plug in b into the original equation.

So this is the equation for the locus of the vertices of our given equation. Check it out!