I have to find the function rule that goes with this table...

X Y
2 8
4 18
7 33
9 43
12 57

I've been trying to figure it out for an hour and can't find anything that works for all of them.

To find the function rule that goes with the table, we need to look for a pattern or relationship between the input values (X) and output values (Y). Let's examine the differences between consecutive pairs of X and Y values to see if we can identify a pattern.

First, let's calculate the differences between consecutive X values:
4 - 2 = 2
7 - 4 = 3
9 - 7 = 2
12 - 9 = 3

These differences do not have a consistent pattern. Let's try calculating the differences between consecutive Y values:
18 - 8 = 10
33 - 18 = 15
43 - 33 = 10
57 - 43 = 14

Looking at the differences between Y values, we can see that there is a pattern. The differences alternate between 10 and 15. Now, let's examine the differences between consecutive pairs of differences:
15 - 10 = 5
10 - 15 = -5
5 - 10 = -5

Here we can observe that the differences between differences are constant. The differences between pairs of differences is -5.

Since the differences between Y values have a consistent pattern, we can conclude that the function rule is likely quadratic. A quadratic function can be written in the form Y = ax^2 + bx + c, where a, b, and c are constants.

To find the specific function rule, we can use the given data points (X, Y) to solve a system of equations. We will derive three equations by substituting the X and Y values into the quadratic equation:

Equation 1: 8 = a(2^2) + b(2) + c
Equation 2: 18 = a(4^2) + b(4) + c
Equation 3: 33 = a(7^2) + b(7) + c

By solving this system of equations simultaneously, we can determine the values of a, b, and c, which will give us the specific quadratic function rule that matches the table.

Alternatively, you can use software, such as spreadsheet programs or online tools, to fit a quadratic function to the given data points to find the function rule. These tools often have built-in regression analysis functions that can determine the best-fitting quadratic equation for the given data.