Being very proud of myself for my current discoveries, I continued to focus on reconstructing my Grandma's design. I realized after the fact that all the coeffceint c that I had been selecting were positive. I thought for a moment well if they are negative. This opened up the other half of my Grandma's design as shown below in the chart so we can see the how c affects the graph.
| -1 | -2 | -3 | -4 |
|---|---|---|---|
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One can see that this graph is completely different than its positive c counterparts. I remembered what Grandma had told me before I left and if that were true then that would mean that these graphs tell us that no matter what b is the original equation will have 2 real roots one positive and one negative. I wondered why this was true, and I remembered that in the quadratic equation if one finds what the value would be under the radical (b2 - 4ac) then one can determine how many solutions one has. Since we know that a = 1, we have the equation be b2 - 4c. So if c is greater than 0 then it would depend on whether 4c > b2 or if 4c < b2. If 4c > b2 then we will have no real solutions (complex solutions). If 4c < b2 then we will have 2 real solutions. However, if c < 0 then their is only the possiblity of 2 real solutions since a negative number (c) and -4 will change into a positive so we will be adding to b2 a multiple of 4, so their is no possibility of a negative number.
One thing to note that almost went unnoticed is that I was unable to pinpoint the coordinate of the minimum if it had not been a whole number. If we look at the graphs above we can see that when c = -1 it crosses the x-axis at 1 and -1. If we look at the graph for c = 1, we see that the minimum is at (-1, 2) and the maximum is at (1, -2). If we look at the graph when c = -4, it crosses the x-axis at 2 and -2. If we look at the graph for c = -4, we see that the minimum is at (-2, 4) and the maximum is at (2, -4). This leads to the fact that if we find were the opposite of c crosses the x-axis, we can create a perpendicular line to see where it crosses our locus, and that will be x coordinate of that c. From their a little algebra and we can find the y-coordinate.
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After all of this work I was able to reconstruct my Grandma's design and once I had completed this I decided not to show her all my work. It would make her way to happy to see this and after the con she pulled on my parents, I think it would be best to take this as personal victory.
Part II: The Manipulation of C