MAN: Determine the equation of the vertical trace of the function g of y equals negative x squared minus y squared plus 2x plus 4y minus 1 corresponding to y equals 3 and describe its graph. I'm going to actually complete the square so that g of xy, I'm going to write this as negative parentheses x squared minus 2x and leave some blank some space and close parentheses minus parentheses y squared minus 4y, leave some space close parentheses, minus 1. All my terms are there, but I'm grouping my x and my y terms together.

Now for x squared minus 2x to be a perfect square I really need a plus 1 here. Now because I'm adding 1 inside the parentheses-- that's actually I'm putting in a negative 1. I'm going to add another minus 1 to the right end of my equation to keep this balanced. Still the same function. Now also y squared minus 4y plus something. For it to be a perfect square, I need to have a plus 4, which means what I really just did was subtract 4.

I said that wrong earlier. Let me go back. I'm completing my completing the square but with the x squared minus 2x plus 1 I just subtracted ones to balance it I'm going to add 1 to the equation. And y squared minus 4y plus 4 makes it a perfect square, but I really just subtracted 4 from the equation. So I need to add 4 to the equation as well.

All right so this is equal to negative parentheses x minus 1 squared minus parenthesis y minus 2 close parentheses squared plus 4. Again, I designed those, I added the 1 and the 4, so that those would factor as perfect squares. So g of xy is equal to negative parentheses x minus 1 squared minus parenthesis y minus 2 close parentheses squared plus 4.

Now if y equals 3 this equation becomes g of x, which is now y is a constant, equals negative parentheses x minus 1 close parentheses squared plus 3. So that is the vertical trace of this. And then to describe this is a parabola. It's a parabola opening downward in the plane the plane y equals 3.