Investigation of

With this investigation we will look at the graph of the above equation when a = 1, k = 1, b = 1 and c = 1

It appears to be a simple graph of a line...with the x-intercept = 1, the y-intercept = 1, and the slope equalling -1...what happens when we change the value of a to 4.

In this case it appears that the y-intercept is still equall to 1 but the x-intercept is now 1/4...does this mean that the y-intercept could be equal c/b or b/c and x-intercept equal to either b/a or c/a?  Let's explore this further by leaving k = 1 but varying the other variables...how about a = 10, b = 20, and c = 60

Here it appears that the x-intercept is equal to 6...which would be c/a; and the y-intercept is equal to 3...which would be c/b...Let's continue to explore this to see if our theory continues to stand true.  Let's look at when a = -5; b = 15, and c = - 60

Yes, it looks to be true...the x-intercept is equal to 12, the ratio c/a and the y-intercept is equal to -4, the ratio c/b.

Now that we know how the graph changes with the variations of the variables a, b, and c...let's see what happens when we change the value of k.

Let's let a = 1, b = 1, c = 1, and k = 2

Oh my...look what we have...it appears that we have four lines that form a rectangle (possible square) and then we have what appears to be 4 parabolas...it also looks like the lines that form the rectangle could also be asymptotes of the parabolas.

What happens when we change the value of k to 3?

This time the lines form a triangle whose sides could be asymptotes to the 3 parabolas that are formed.

Does this mean that the theory we saw in the very first investigation continues to work here where if k is even the lines of the graph will form a polygon with 2k number of sides and 2k parabolas and if k is even the lines form a polygon with k number of sides and k number of parabolas?

Let's look at a couple more graphs to see what happens.

a = 1, b = 1, c = 1, and k = 6

and

a = 1, b = 1, c = 1, and k = 9

What do you think!!!!