To determine a quadratic function means to compute it's coefficients. Indeed, the system of equations method is the most usually applied since it is quite easy to substitute the given values of coordinates within the quadratic resulting a linear equation with 3 variables (a,b,c-coefficients) for each given point. To determine the solution of this system we need three points to generate three equation, since there are 3 coefficients to find out.
But there are many other methods, as well.
One of them is vertex method that allows us to write the equation of parabola in "vertex form":
`y = a(x - h)^2 + k`
Other method is to use a math software that helps you to find the function. For instance, you can use Excel to generate the parabola, plugging the given values of the points in the specified cells.