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
Hence, passes through all vertices.
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