Parabolic Transformations Part B (aka part 3)


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

Without calculus, show that the locus is a parabola.

This exploration is a continuation of the previous exploration about quadratic equations. Although it is the second part (part b, named for the variable we will be exploring), it is in some ways the third part, since we discussed how two constants influence equations of the form in the previous exploration. In this exploration we are going to look at b.

Lets go ahead and see varying values of b, ranging from 4 to -3, with an a and c of 1.

Here we can definitely see that something is going on. As we can see, b has a powerful influence on the vertex of the parabola as a and c remain constant. Let us see an animation:

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By viewing this animation we can observe that the "shape" of the parabola is preserved throughout the varying b. In addition, the path that the vertex takes (the locus) is also more apparent. It appears that the path follows the shape of a parabola, one that is concave down and has a vertex at (0,1). Is this true? Can we prove this from another angle?

Let us go to the quadratic formula for answers. Recall the quadratic formula:

The quadratic formula is used to determine the roots of a quadratic equation. Or, in other words, it provides the (usually 2, but sometimes 1 or 0) values for which x is zero given a particular a, b, and c.

Another interesting feature of the quadratic formula, however, is that when the discriminant (b^2 - 4ac, or the value underneath the radical) is equal to zero, that presents the x-coordinate of the vertex. This is why we have a double root (when the vertex is on the x-axis) when x and the discriminant are both zero.

Therefore, we can say that the vertex is located at an x-coordinate of , when a is one, as is seen in this example. So, what about the y-coordinate (a = 1)?

What does that look like? Without the "+c" it looks like the y-coordinate is equal to the x-coordinate squared, times negative one! This relationship would indicate a parabola, as each succeeding x-coordinate (for the varying values of b) would produce a y-coordinate that is simply the x-coordinate squared (and reflected across the y-axis) and translated by c.

Therefore, in our example, with a c of 1 (as given), we are left with this function:

Lets trace the path of that vertex for varying b. This is the locus of that vertex.

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What is the equation of the locus?

Look familiar?