Help me find the function behind this data?

Question

I have a function $f(x, y)$ and I have another (non-mathematical) algorithm capable of inefficiently generating the exact same results as in in a 'brute force' manner. Since I have been able to find no good 3-D graphing software at the time, I have used my algorithm to generate several tables. of the function. In each table I have a fixed $y$ value in the function, noted with each, and I input the $x$ values 1 - 10. Using Desmos Graphing Calculator, I have graphed each of these tables here. The higher-y lines increase much much faster than the lower-y lines, which makes it difficult to see it all at once. You can change the scale of the x and y axes in the top left, and clicking on the icon of a table turns it off or on. The $y$ of each line is as follows: black - 1, green - 2, blue - 3, red - 4, yellow - 5, purple - 6. So what I'm looking for is to find a $f(x)$-type function in each line, and then using those the overall f(x, y) function for all the lines. So far I have not been able to find patterns except in 1 and 2 where they are obviously $f(x) = 1$ and $f(x) = x$. If anyone is able to find an $f(x)$ for any of the functions that I haven't or is able to find the overall $f(x, y)$ that would be great. Also, if needed I can provide arbitrary data from my brute force algorithm, just ask if it's needed and tell me the range of x and y.

Answer 1

FOR REFERENCE ONLY THIS IS ONLY AN ANSWER BECAUSE A COMMENT CANNOT CONTAIN AS MANY CHARACTERS

For #3, the interpolating polynomial= $$\frac{-x^9}{5670}+\frac{11 x^8}{1260}-\frac{5 x^7}{27}+\frac{11 x^6}{5}-\frac{4333 x^5}{270}+\frac{1111 x^4}{15}-\frac{121634 x^3}{567}+\frac{469633 x^2}{1260}-\frac{15536 x}{45}+128$$.

For #4:$$\frac{2x^3+x}{3}$$

For #5:$$-\frac{167 x^9}{45360}+\frac{517 x^8}{2880}-\frac{5665 x^7}{1512}+\frac{20999 x^6}{480}-\frac{676811 x^5}{2160}+\frac{1362551 x^4}{960}-\frac{9137729 x^3}{2268}+\frac{4931819 x^2}{720}-\frac{1964948 x}{315}+2272$$

For #6:$$\frac{11x^5+5x^3+4x}{20}$$