From a Picture to a Rule
Look for 1x1, 2x2, and 3x3 squares in the pattern. Then look for rows that are matching the term number (a row of 1 in the first, row of 2 inthe second, row of 3 inthe third....), then find the constant that doesn't change.
From a Table to a Rule
Look for a pattern in the y column. If it is quadratic there is a pattern in the second difference. Divide this number by 2 to find the coefficient to x squared. Calculate the x squared value to figure out what is still to do (subtract your x squared value from y). Find the linear rule for what remains.