Quadratic Functions
as b varies
by Emily Kennedy

Again, consider the quadratic function

f(x) = ax² + bx + c
(Assume a ≠ 0.)

Suppose a and c are constant, and let b vary.

Click here for a Graphing Calculator file showing the graph of the parabola when a and c are constant. Click the Play button at the bottom of the screen to show an animation of the parabola as b varies and a and c are held constant.

What does the locus of the vertex look like as b varies?

You can also look at this GSP file, which will actually trace the locus of the vertex for you.

Make a conjecture about some properties of the equation describing the locus of the vertex as b varies. What shape is the locus? How do a and c figure into the equation?

Now that you've made a conjecture, let's rigorously determine an equation for the locus of the vertex as b varies.

We have shown that the vertex of the parabola is located at

Let . This is the x-coordinate of the vertex.

Since b is varying, we want to find the y-coordinate of the vertex in terms of only x, a, and c.

There is a b² in the numerator of , the first term of y, and there is a b in the numerator of .

So for some t that is in terms of a and c.

What is t?

So

And clearly c is already in terms of only x, a, and c.

Thus, the locus of the vertex as b varies is

This is a parabola (since we are assuming a ≠ 0).

Does this match your conjecture?

Click here to see a Graphing Calculator animation as b varies,
as well as the parabola we just found as the locus.
Note that, indeed, the vertex always lies on this parabola
(at least in this particular animation!).

Click here to continue and see what happens when c varies.

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