by Margaret Trandel
The general form of a quadratic equation is: y = a x2+ bx +c. This investigation will discuss the movement of the quadratic graph as b changes value. In the equations below, b is an integer such that -4 < b < 4, and a and c each remain constant at a = 1 and c = 1 in all of the equations. Looking at a graph of the quadratic equations below, we can see some similarities.
Graph of Quadratic Functions with varying values of b
Let's now observe the graphs above:
The parabolas all pass through (0, 1). For b = -2, the parabola is tangent to the x-axis, so there is only one positive root. For b < -2, the parabola will intersect the x axis twice and both of the roots are positive real numbers. When b = 2, the parabola is tangent to the x axis, so again there is only one root, but this time negative. Finally, when b>2, there are two intersection points which are negative real numbers.
Looking closely at all of these quadratic functions, and marking the vertices it looks like there is a locus of points at y = -x2 + 1
As we know, a proof is not in a graph. We know that the vertices of the quadratic function y = x2 + bx +1 are the points where dy/dx = 0. The derivative is 2x+b so we can set that = 0, or:
2x + b = 0
b = -2x
Now we can substitute -2x into our equation for b:
y = x2 + (-2x)x + 1 = -x2 + 1
The vertices of the parabola indeed form the locus of points on y = - x2 + 1.
Note this applies to a general quadratic equation, but the vertices are the points where:
dy/dx = 0 = 2ax + b.
thus b = -2ax
As before we can substitute -2ax into our equation
y = a x2 + (-2ax)x + c = -a x2 + c
resulting in our general locus of vertices in a parabola.