I am trying to find the rule for a function table with input 2, output 30; input 3, output 48; input 4, output 66.

looks like output grows by 18 when input grows by 1.

So, starting at f(2) = 30 = 36-6,

f(x) = 18x-6

To find the rule for a function table, we need to identify the pattern or relationship between the inputs and outputs. Here's how we can do it:

First, let's observe the difference between consecutive inputs and the difference between consecutive outputs:

Input: 2, 3, 4
Output: 30, 48, 66

The difference between consecutive inputs is constant: 3 - 2 = 1, and 4 - 3 = 1.

The difference between consecutive outputs, however, is not constant: 48 - 30 = 18, and 66 - 48 = 18.

Since the difference between the inputs is constant, but the difference between the outputs is not, it indicates that the relationship is likely not linear (as the differences in a linear relationship would be constant).

To further analyze the relationship, let's look for a pattern in the differences between the inputs and outputs:

Input: 2, 3, 4
Output: 30, 48, 66

Input difference: 1
Output difference: 18

If we examine the output difference, we can notice that it is six times the input difference.

So, we can conclude that the rule for this function table is likely a quadratic relationship, where the output is a function of the input raised to the power of two, multiplied by some constant factor.

To find the specific rule, we can use the formula for a quadratic function, which is often written as: f(x) = ax^2 + bx + c.

Let's substitute the values from our table into the formula:

30 = a(2^2) + b(2) + c -- (1)
48 = a(3^2) + b(3) + c -- (2)
66 = a(4^2) + b(4) + c -- (3)

Now we have a system of three equations with three unknowns (a, b, and c), which we can solve to find the specific values for the function:

Expanding the equations, we get:

4a + 2b + c = 30 -- (1')
9a + 3b + c = 48 -- (2')
16a + 4b + c = 66 -- (3')

By solving this system of equations, we can find the values of a, b, and c, and hence the rule for the function.