**Exploration of Quadratic Equations**

**By**

**Moni Olubuyide**

Let us investigate the equation:

for various values of a, b, and c...

Let's start with keeping a and c constant at 1 with various values for b.

+

Notice that all the parabola have a y-intercept at (0, 1). This is because when x=0, y = c...therefore the y-intercept will always be (0, c)

Now when b < 0, we observe that the vertex of the parabola lie on the right side of the y-axis and when b=0 (the green graph) the vertex is on the y-axis. When b = -1 we have no real roots because it does not cross the x-axis. When b= -2, we have one root, and the graph intercepts the x-axis at one point. When b < -2 the graph intercepts the x-axis twice which corresponds to the two roots.

+

Now when b > 0, all the vertex of the parabolas lie on the left side of the y-axis. When b= 1 then there are no real roots, when b= 2 there is one root, and when b >2 there are two roots. Therefore we can generalize that when b = 1, the graph will intercept the x-axis once. When -2 > b or b> 2 the parabola will intercept the x-axis twice. Lastly when -1 < b < 1, the parabola will not intersect the x-axis....

Observe the generalization...

Here is the locus of the vertices of the parabolas (the black graph)...notice that the locus is also a parabola

Given a general equation you can observe how parabola moves along the equation of the locus...

The equation of the locus is represented by

Click here to animate the locus of the vertices..