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
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
a = 1, b = 1, c = 1, and k = 9
What do you think!!!!
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