Weird Math Theory Help (Parabolas) (This is not a normal math problem)?

So, I figured out a while ago that you could plot a parabola mentally when it is in vertex form (ex: 2(x-6)+3) more obviously by first finding the vertex and then plotting the vertex by taking the opposite of the number in front of x as the x coordinate and using the number attached to the end as the y coordinate. Everyone knows this much. I, however found a neat little trick to plug in the number outside of the parentheses and multiply it to x^2 to get the y value to pair to each x value (ex: 3(x-5)+3: 1 unit over to either side pairs with 3 units above vertex, 2 units over to the side pairs with 12 units above the vertex, etc...). Using this, you can plot any parabola in vertex form. However, I want to find the way to do this in standard form, but I have made some progress. Here is what I have so far: the b in standard form moves the graph's location on a parabola equal to the opposite of a (the vertex is the point following the parabola) (ex: 3x^2+2x aligns with f(x)=-3x^2, -2x^2-3x aligns with a point on f(x)=2x^2) Obviously, the c value would cause the graph to move off course by however mush the c value is. Additionally, I have found that you can find the vertex of a parabola who's a is equal to 1 by using these coordinates: (b/2,x^2) and a parabola who's a is 2 would be: (b/a^2,x(1/2b)). This one is a more recent development, so I have not worked out how negative signs work with it. Does anyone have any ideas as to how to find the x coordinate for the vertex location?