What is the formula for this table...

X-0 1 2 3 4 5 6 7 8 9
Y-0 8 14 18 20 20 18 14 8 0
What is the formula and how did u find it be descriptive please

Got it.
y = ax² + bx + c
we know, c = 0

so,
y = ax² + bx

simultaneous equations

20 = 16a + 4b
=> 5 = 4a + b
b= 5 - 4a (eqn 1)

20 = 25a + 5b
4 = 5a + b
b = 4 - 5a

merging equations,
5 - 4a = 4 - 5a
a = -1

now,
b = 4 - 5(-1)
b = 4 + 5
b = 9

hence, we get:
²
y = -x² + 9x

This one turned out to be really easy. The first thing I do when trying to figure out the formula for a sequence is to look at the differences between terms. If the differences are constant, then the formula is linear--i.e., the point are on a line. If the differences aren't constant I look at 2nd differences. If those are constant, then the formula is a quadratic, i.e., a polynomial in x of degree 2. That turned out to be the case for your data.

(If it wasn't the case, I'd keep going to see if 3rd or higher degree polynomial worked, so long as there were enough data points to work with.)

Finding the formula is easy since you are given both zeroes for f(x) = y right in your data. They are 0 and 9, the first and last data points. The data is looking like a parabola opening downward, with axis of symmetry x=4 1/2. You can see that since the maximums are at 4 and 5 and the data is symmetric.

So the formula is going to be f(x) = kx(x-9). Since f(1) = k(1)(-8) = 8, we very quickly see that the constant k is -1.

I did this all in Excel, and in the last column entered the formula -x(x-9) to verify that it reproduces all the data:

(I added the _ character to help make the numbers line up better. Yahoo Answers messes up columns of numbers.)

x _y d1 d2 -x*(x-8)
0 _0 __ __ 0
1 _8 _8 __ 8
2 14 _6 -2 14
3 18 _4 -2 18
4 20 _2 -2 20
5 20 _0 -2 20
6 18 -2 -2 18
7 14 -4 -2 14
8 _8 -6 -2 8
9 _0 -8 -2 0

If you want another problem to practice on, try this one:
0 32
1 14
2 0
3 -10
4 -16
5 -18
6 -16
7 -10
8 0
9 14