Quadratic Equations

by Laura Trkovsky

In this investigation we looked at the quadratic equation:

And how when we fix c and a, then we can change b, but it will always have the point of (0,c) in common in every graph no matter what the value of b. The example shown is when

The graph of this equation with different b values looks similar to this:

The b values in the about picture go from -2 to 2. It demonstrates how the parabolas all have the point (0,1) in common in each.

Now we want to look at the locus of this equation is the parabola with the equation:


Let's investigate this: The point where this curve passes the y-axis is at the point (0,1). It is the same as where all the equations with different b values from our given equation hit. Now the vertex of each of the equations with b=1 and b=-1 also intersect this parabola at either the point (-1,0) or (1,0). From the previous assignment we saw how the negative in front of the x squared turns the parabola upside down. So this parabola is upside down with the vertex at (0,1). In this locus parabola it has a vertex at (0,1) and passes through the same points (1,0) and (-1,0). We see this in the graph below.

Now let's look at this with the original parabolas.

We can see from the graph that the purple graph which corresponds to the locus parabola does appear to be just that.

In general we can say that a parabola with the form has a locus parabola in the form of . Let's explore this with a couple examples. Let's see what happens when a=5 and c=3.

It is obvious to see that the locus parabola is just that when a=5 and c=3. Now let's explore this with more numbers like a=-2 and c=6.

In this example I used different values again for b ranging from -8 to 8. This graph is better to see how the locus parabola (the one in purple) is hitting through the loci of the original parabola with different b values. Therefore, the generalization is correct.