Showing the XB-plane, is no Plane Jane

An Investigation on :

By Ryan Shannon

An investigation on what will happen as we change the values of coefficient (b)

Let us take a look at the graph in the xb Plane, A great way to start is with:

clicking the graph above will show the animation as "b" changes between 0,10.

To better depict the zeros of the graph we place b=-4 to b=4 on the graph.

Here we can clearly see that the graph will only intersect once when b=2 or b=-2 and there will be no intersection in between or at -2<b<2. Also we will have two intersections when b<-2 and b>2.

Another way to find these roots is to try the quadratic formula. We know that there will be one root when the discriminate is equal to zero, there will be two roots when the discriminate is greater than 0, and there will be two non real roots when the discriminate is less than zero.

What if we are to change the coefficient of x to a negative number as in:

Not only has the graph changed from a parabolic like shape, to a hyperbolic like shape. We have changed in the where we find our roots of the graph. Here we notice that we will continue to have real roots. Which must mean that our discriminate will never be less than zero.

When adding two positive numbers we know that we will always be positive, thus we will two real roots for the whole line.

Now lets investigate when "c" is negative one.

Again by looking at the graph we see that there is going to be two real rots for all values. That is the discriminate will be greater than zero.

As we change c in the we notice that c is the y-intercept.

Any graph where c>1 we will have no real roots.

Here we notice that all values will have roots. The graph is being translated downward on the y-axis.

One observation that is obvious is that the y-intercepts are going to be -5,-4,-3,-2,-1,0,1,2,3,4,5 respectively.

Then changing to investigate what would happen in the plane when we have the coffin have a value we will see:

We can start to see how the "parabolas" have been distorted in a way.

We see that all the graphs have the same vertex value. Why is it that when we change a we only change how fast the parabola opens up? That is were are increasing y faster. (i.e.) let x=1 when a=1 and let x=1 when a=5. For values of y we get, 3,7 respectively.

Thus now we can see that changing the coefficient "a" will only change the stretch and shrink the graph. There will be no vertical/horizontal shift.

Noting that "a" is now negative we will have the same effect with the changing of the stretch and shrink.