Can somebody tell me a method(s) of graphing quadratic equations by hand and give me an example and a non-example.
One technique that you should practice with is completing the square. For example, if y = x^2 + 3x - 1, then after completing the square we get y = (x +3/2)^2 - 13/4. The graph of y is simply the graph of x^2 translated 3/2 units left and 13/4 units down. A handy formula you might sometimes use is if y = ax^2 + bx + c, then the x-coordinate for the vertex of y is -b/2a.