Quadratic Functions
as c varies
by Emily Kennedy

Again, consider the quadratic function

f(x) = ax² + bx + c
(Assume a ≠ 0.)

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

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

What does the locus of the vertex look like as c 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 c varies. What shape is the locus? How do a and b figure into the equation?

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

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

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

Note that, since a and b are constant,
the x-coordinate of the locus is constant.

And y can be anything, since c is varying.

Thus, the locus of the vertex as c varies is

This is a vertical line.

Does this match your conjecture?

Click here to see a Graphing Calculator animation as c 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!).

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