**Now consider the graph below
of for b = -3, -2,
-1, 0, 1, 2, 3, note the "movement" of a parabola as
the value of b is changed. Also, all parabolas of this type passes
through the same point on the y-axis i.e(0,1). For more illustration
click Graphing Calc
#3. For b = -2, the
parabola is tangent to the X- axis hence, one real root at the
point of tangency. For -2<b<2, the parabola does not intersect
the X= axis, therefore, 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 intersects the x-axis twice hence two
real roots.**

**Examining the locus of vertices of the parabolas
as b varies is also instructive in this situation. This time the
locus of vertices forms a parabola, whose equation is,
as you can see by viewing the grey graph below**.

**To prove that is
the locus of the vertices of the parabola, consider the set of
vertices of all
quadratics of the form
By rewriting in vertex form**

,

the general form of the vertices can be

seen to be

.

Substituting 3 values for b will

identify 3 particular points. The

unique quadratic through those three

points can identified with linear

algebra.

**Hence, passes
through all vertices.**

Back to Samuel's page