Quadratic Functions
as a varies
by Emily Kennedy

Again, consider the quadratic function

f(x) = ax2 + bx + c
(Assume a ≠ 0.)

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


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

What does the locus of the vertex look like as a 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 a varies. What shape is the locus? How do b and c figure into the equation? Be sure to look at the case when b = 0.

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

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

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

Since a is varying, we want to find the y-coordinate of the vertex in terms of only x, b, and c. In order to "swallow up" all of the a's in the equation, we have to use x's (since x depends on a, but b and c don't).

There is an a in the denominator of , the first term of y, and there is an a in the denominator of .

So for some s that is in terms of b and c.

What is s?

So

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

Thus, the locus of the vertex as a varies is

This is a straight line.

(Be careful! Note that when b = 0,
we have that x and y are both constant, since
and
so the vertex does not move as a varies

Does this match your conjecture?

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

Check the interesting b = 0 case as well. What do you see?

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

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