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Expert: - 9/21/2009


Question
I need help with the STEPS (preferrably very detailed steps) on graphing the following equation (and yes, this is all the information my Prof. gave us for the question: "Graph:" and then this equation):

x2 (x to the 2nd power)+20-y2 (y to the 2nd power)-7x-2y+21-8x+ 6y-2 = 2-4y-5x-2y2 (2y to the 2nd power)

So far I have the following, and I'd like to know if I'm correct:

x2-7x-8x -y2-2y+6y +20+21-2 = 2-4y-2y2-5x

This math class is called "Mathematics - A Modeling Approach" and it's about using math in the real world and is not about any particular kind of math - it jumps around from Algebra II to Calculus to Trigonometry, to I don't know what. And I'm having trouble trying to figure out (from teacher's notes handed out or from my own notes taken in class) what in the world formula I'm supposed to use, and what kind of graph I'm supposed to be making.  Thank you SO much for ANY kind of light you can shed, and any way you can help me with the formula so I can "graph" it.

Answer
The equation is x² - 7x - 8x - y² - 2y + 6y +20 + 21 - 2 = 2 - 4y - 2y² - 5x.

Combining like terms:
 there is a -7x and a -8x, which make -15x;
 there is a -2y + 6y, which makes 4y; and
 20 + 21 - 2 makes 39.

This gives x² - 15x - y² + 4y + 39 = 2 - 4y - 2y² - 5x.

Now add -2 + 4y + 2y² + 5x to both sides, giving x² - 10x + y² + 8y + 37 = 0.

We will split this equation into pieces.  They are x²-10x, y²+8y, and 37.

Note that for x²-10x, the -10 over 2 is -5, and -5² = 25,
so it needs to be changed to x²-10x+25 - 25 so it will turn into (x-5)² - 25.

In the same way, y²+8y, 8/2=4, and 4²=16,
so it needs to be changed to y²+8y+16 - 16 so it will turn into (y+5)² - 16.

The 37 changes since we have a -25 and a -16 in the equation now.
37 - 25 is 12, and 12 - 16 is -4.

All of these, when combined, give us (x-5)² + (y+4)² - 4 = 0.

Noting that 4 = 2² and adding 4 to both sides gives (x-5)² + (y+4)² = 2².

Now that it is in it's proper form, it can be seen to be a circle of radius 2.
The center is at (5,-4).  Adjusting with the radius will give us the four edges.
This means the top is at (5,-4+2) = (5,-2), the right side is at (5+2,-4) = (7,-4),
the bottom is at (5,-4-2) = (5,-6) and the left side is at (5-2,-4) = (3,-4).

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