# Finite differences

Goal: Complete the table below for the Handshake problem.

$$x$$ $$f(x)$$
1 0
2 1
3 3
4 6
5 10
6
137

Subgoal: Create a polynomial function by looking at differences and differences of differences.

Function values and differences
difference of differences differences $$ax^2 + bx + c$$ $$x$$ $$f(x)$$ differences difference of differences
$$a + b + c$$ 1 0
$$3a + b$$ 1
$$2a$$ $$a \cdot 4 + b \cdot 2+ c$$ 2 1 1
$$5a + b$$ 2
$$2a$$ $$a \cdot 9 + b \cdot 3 + c$$ 3 3 1
$$7a + b$$ 3
$$a \cdot 16 + b \cdot 4 + c$$ 4 6

## Questions for Handshake problem

1. Where does $$4a + 2b + c$$ come from? What does it equal?
2. Where does $$5a + b$$ come from? What does it equal?
3. Where does $$2a$$ come from? What does it equal?
4. State $$f(x)$$.
5. Find $$f(137)$$.

## Function g

Give a quadratic function that would result in the values given below. Compute the missing value.

$$x$$ $$g(x)$$
0 0
1 5
2 12
3 21
4 32
137

## Function h

Give a quadratic function that would result in the values given below. Compute the missing value.

$$x$$ $$h(x)$$
0 -4
1 1
2 12
3 29
4 52
5 81
142

## Function j

Unfortunately this method will not always work. Why won't it work in the example below?

$$x$$ $$j(x)$$
0 1
1 2
2 4
3 8
4 16
5 32

## Challenge: Function k

Give a polynomial function that would result in the values given below. Compute the missing value.

$$x$$ $$k(x)$$
0 2
1 3
2 6
3 12
4 22
100