The surest way to get many of the points needed to plot a quadratic is to use the quadratic formula. This will give the roots (real or imaginary). It will give you the completed square form also called the vertex form (if you know how to use the discriminant). It can easily give you the y intercept (which you can find before you use the quadratic formula). It gives the max or min upon solution.
The easiest one to use if it is available to you, is factoring. The quadratic may not be factorable. But if it is and you can see it, then this gives you 2 points immediately (the roots) and a third without much trouble (the y intercept). Factoring will also give you the x value of the vertex. (Find the average between the 2 roots)
This needs an example
Suppose you have y = (x - 5)(x - 9) The roots are 5 and 9, correct? So the x value of the vertex is (5 + 9)/2 = 7 It always works.
Completing the square always gives you the minimum or maximum right away. For example if you have y = (x - 2)^2 - 5 it means you have the vertex at (2,-5) You can get the roots easily enough. So this form is useful, but not as sure as the quadratic equation or as simple as factoring.
Graphing is the most certain way to check your answer. I find it the most useful thing to do with modern computers. There are all sorts of things that a graph will reveal that algebra by itself might be laborious and prone to leading you to mistakes. Graphing tends to correct that problem.
