Locus of Parabolas

By Joshua Singer

In this investigation, we are going to examine the graphs of when b is changed while a and c are held constant. Then we are going to further examine the vertices of the set of parabolas with different b's and show that the graph of these vertices, called the locus, is also a parabola with an equation that has some interesting characteristics.

First we will look at the graph of the parent equation . This is the case where a = 1, b = 0, and c = 0.

What observations of the graph do we notice? First, the vertex of the graph lies on the y-axis. The graph opens upward since a is positive. The vertex also lies on the x-axis since c = 0. So what happens when we fix a and c and change b? Let's look at the graphs of when a = 1, c = 1, and b = -3, -2, -1, 0, 1, 2, and 3.

As we can see from this set of parabolas, the vertex is displaced from the y-axis as b is changed. When b = 0 (orange), the vertex is on the y-axis. When b > 0, the vertex is to the left of the y-axis and when b < 0, the vertex is to the right of the y-axis. All of the parabolas pass through the same point (in this case it's (0,1) since c = 1). We also notice that when b < -2, the parabola intersects the x-axis at two points with positive x values. This means that the equation has to real roots, both positive. When b = -2, the parabola is tangent to the x-axis (intersects at one point with positive x value) which means that the equation will have one real root that is positive. When -2 < b < 2, the parabola does not intersect the x-axis and therefore, the equation has no real roots (it does have complex roots). When b = 2, the parabola is tangent to the x-axis (intersects at one point with negative x value) which means that the equation will have one real root this is negative. When b > 2, the parabola intersects the x-axis at two points with negative x values. This means that the equation has two real roots, both negative.

Now that we see how changing b affects the graph of , let's focus our attention to the vertices of the parabolas. It seems that if we graphed just the vertices, we would get another parabola. We call this the locus of the vertices. But what would the equation of the locus be? Let's see if we can come up with one. We know that the vertex of the locus will be at (0,1) because one of our original parabolas has a vertex there. Since the vertex is on the y-axis, the equation of the locus must have b = 0. Also, since the vertex is at (0,1), the equation of the locus must have c = 1. Since the parabola opens downward, we know that a < 0. Using another point on the locus, say (1,0), we can solve for a and get a = -1. So, the equation for the locus of the vertices is . Let's graph this along with the above graph and see what it looks like.

We see that the locus does indeed pass through each of the vertices of the original set of parabolas. What observations can be made about the equation of the locus and the equations of the original set of parabolas? First, we see that the equation of the locus has the same c value. Second, we see that the equation of the locus has the opposite a value. And last, we see that the equation of the locus has a b value of 0. Will this always work? Let's check out the graphs of where a = 5, c = 3, and b = -3, -2, -1, 0, 1, 2, and 3.

From our previous observations about the equation of the locus of these vertices, let's guess that the equation of the locus is . Are we correct? Let's see.

Looks like we're correct! Now we can generalize the equation of the locus for a set of parabolas where a and c are fixed and b changes. The equation of the locus will have -a, b = 0, and c. So, for a set of parabolas with equations , a and c fixed and b variable, the equation of the locus will be .

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