Finite differences

Goal: Complete the table below.

\(x\) \(f(x)\)
1 2
2 6
3 12
4 20
5
100

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

Function values and differences
difference of differences differences \( f(x) = ax^2 + bx + c \) \(x\) \( f(x)\) differences difference of differences
    \( a + b + c \) 1 2    
\( 3a + b \) 4
\( 2a \) \( a \cdot 4 + b \cdot 2+ c \) 2 6 2
\( 5a + b \) 6
\( 2a \) \( a \cdot 9 + b \cdot 3 + c \) 3 12 2
\( 7a + b \) 8
  \( a \cdot 16 + b \cdot 4 + c \) 4 20  
   


Comprehension questions

  1. Where does \(4a + 2b + c\) come from? What does it equal?
  2. Where does \(3a + b\) come from? What does it equal?
  3. Where does \(2a\) come from? What does it equal?
  4. State \(f(x)\).
  5. Find \(f(100)\).

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