What is rule for (0,0), (1,4), (2,16), (3,36), (4, 64)

Here the set of input values is {0, 1, 2, 3, 4} and the set of output values is {0, 4, 16, 36, 64).  

Because of the rapid growth in the set of output values (0 to 4, 4 to 16, 16 to 36, 36 to 64), we suspect that exponential growth is illustrated here.

Let's start with the basic equation for exponential growth:  f(x) = ar^(n-1), where a is the value of the first term, n is the subscript of the term in question (e.g., n = 2 would indicate the second term), r is the common factor (if there is one).

The first y-value is 0; the second is 4, and the third is 16.  Is 4 a common ratio here?  Multiplying the first y value (0) by 4 results in 0.  Is that the same as the second given y-value (4)?  No.  So 4 is not a common ratio, even though 4 times 4 = 16 (the third y-value).

If you are in an advanced algebra class, one approach to try next would be to "fit" the given points to a 2nd, 3rd, 4th or 5th order polynomial.  For example, we could begin with given point (1,4) and attempt a 2nd order (quadratic) fit to (1,4) as a first step towards finding the "rule" in question.

The general form of a quadratic function is y=ax^2+bx+c.  What happens if we let x=1 and y=4, to fit (1,4)?     y = 4 = a(1)^2 + b(1) + c, which has 3 unknown coefficients.  You must repeat this process until you have three equations in three unknowns {a, b, c}, and then you must solve for a, b and c.  If you do this properly, you'll end up with a quadratic formula for the "rule" in question.  

I'd suggest you double-check to ensure that you have copied the five given data points properly.

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