by: Katie Gilbert

It has now become a rather standard exercise, with available technology, to construct graphs to investigate the equation:

and to overlay several graphs of:

for different values of a, b, or c as the other two are held constant. From these graphs, discussion of the patterns for the roots of:

can be followed.

To start our investigation of "how changing our b-value" effects our graph, let's first take a look at the equation, ......(where our b = 0)

Let's take a look at the graph by itself to get a good look at it and use it as our basis....

We see that our parabola intersects the y-axis at 1, which is our "c-value", and it's vertices is on the y-axis. Now let's investigate our other equations where we vary our b value and see what we find.....

For example, if we set:

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

Let's discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation). We can see that when b>0 the vertex of the graph is on the left-hand side of the y-axis, and when b<0 then the vertex of the parabola is on the right-hand side of the y-axis. We also see that for b < -2 the parabola will intersect the x-axis in two points with positive x values (i.e. the original equation will have two real roots, both positive). For b = -2, the parabola is tangent to the x-axis and so the original equation has one real and positive root at the point of tangency. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots. Similarly for b = 2 the parabola is tangent to the x-axis (one real negative root) and for b > 2, the parabola intersets the x-axis twice to show two negative real roots for each b.


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


When we look at our locus, the equation , is a parabola with y-intercept at 1, and it opens downward. Let's take a look at another set of graphs and see if we can come up with a general equation for our locus.

Let's look at the equation:

Where we vary our b value from -3 to 3 like we did above......

We see that our graph below has the locus of

We can generalize the equation of our locus, for any equation of form

as the equation of :


CLICK HERE if you wish to explore for other equations.