Systematic way of getting to the above answer
You know that the answer is going to have the form "y = ax^2 + bx + c" by looking at the way that y is increasing.
So a, b and c are just some numbers whose values we don't know. But we know that the statement "y = ax^2 + bx + c" is always true for any of the pairs of numbers in the table.
So by plugging in some of those numbers we can figure out what a, b and c are.
Let x = 1 and y = 1, and the statement gives us 1 = a + b + c. (Equation 1)
Let x = 2 and y = 3, and the statement gives us 3 = 4a + 2b + c. (Equation 2)
Let x = 3 and y = 6, and the statement gives us 6 = 9a + 3b + c. (Equation 3)
Now we have a system of 3 equations and 3 unknowns. You can solve it to figure out what a, b and c are. The answer ends up being a = 1/2, b = 1/2, c = 0.
Thus y = (x^2 + x) / 2.