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