Danie Brink

THE PARABOLA AND ITS ROOTS

Investigation 1: Let us investigate the effect in a change in b, the coefficient of x in the standard parabola equation .

click on the graph above to see an animation

From the graphs above we learn that:

The coeficient of the x in has an effect on the axis of symmetry of a parabola and on the roots (x-intercepts) of the parabola.

As b varies, the locus of the vertices of the parabolas form another parabola with equation . Click here to see an animation of this.

The vertex of a parabola is given by the point .

Investigation 2: We will now keep a = 1 and c = 1 in the equation . We essentially have the equation and can now sketch the equation (We sketch this equation and not the equation just because Graphing Calculator draws graphs in the xy plane)

From the graph above we learn that:

The graph drawn in the xb-plane has a discontinuity at x = 0.

The two parts of the graph have vertices at x = 1 and x = -1. The significance of this will be explained in investigation 3.

Investigation 3: Now we will place two specific parabolas and the hyperbola on the same set of axes and see what we learn.

From the graphs above we learn that:

The vertices of the hyperbolas correspond with the places where and touch the x-axis.

For b values inbetween -2 and 2 the parabola has no roots. For b values 2 and -2 the parabola has only one root. For b values greater than 2 or smaller than -2, the parabola has two roots. To see this vividly we can look at the following graphs.

click on the graph to see an animation

From the graph above we learn that:

The intersection between the graphs of the line y = b and imitates the roots (x-intercepts) of the parabola .

For values of b > 2, the parabola will have two, negative real roots.

For b = 2, the parabola will have one negative real root.

For values -2 < b < 2, the parabola will have no real roots.

For b = -2, the parabola will have one positive real root.

For values of b < -2, the parabola will have two, positive, real roots.

Investigation 4: Let us look at a parabola with a c-value of -1.

From the graph above we learn that:

The parabola will always have real roots. Click on the graph above to confirm this fact.

The graph of has no discontinuity because of the fact that the parabola always has real roots. The blue line y = b in the animation above confirms this fact.

Investigation 5: Let us investigate a cubic function and its roots.

From the graph above we learn that:

The same relationship between the three graphs exists as in the case of the parabola.

When b = -2.5, the cubic graph has exactly two roots.

When b < -2.5, the cubic graph has three roots.

When b > -2.5, the cubic graph has one root . Click on the graph above to see an animation of this fact.

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