Question

I'm having a hard time figuring out a quadratic model for this data.

x y
----------
1 3
2 13
3 31
4 57
5 91
6 133

How do I go about doing this?
Is there a website that could help me?

Answer 1

First show that this is a quadratic relationship by taking differences and second differences:
y dy d²y
3 10 8
13 18 8
31 26 8
57 34 8
91 42 8
133

If the second difference is constant, it is a quadratic relationship.

Assume the quadratic model as:
f(x)=ax² + bx + c
then
f(x+1)=a(x+1)^2+b(x+1)+c
and
f(x+1)-f(x)=2ax+b+a
Similarly,
f(x+2)-f(x+1)=2ax+b+3a
f(x+3)-f(x+2)=2ax+b+5a
....

Thus, the second difference = 2a
2a=8
a=4

By substituting values of x and a into one of the formulas, b can be found
for x=1,
dx=10
2ax+b+a=10
b=-2

Substitute a and b in f(1), we get
2(1)²-2(1)+c=3
c=1
Thus
f(x)=4x²-2x+1

check:
f(1)=3
f(3)=31
f(6)=133