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.
\( ax^2 + bx + c \) | \(x\) | \(f(x)\) | |||
---|---|---|---|---|---|
\( a + b + c \) | 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 | |||
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 |
\(g(x) = x^2 + 4x; g(137) = 19,317\)
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 |
\(h(x) = 3x^2 + 2x - 4; h(142) = 60,772 \)
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 |
Differences never become constant. Function is not polynomial but is exponential in this case, \(j(x) = 2^x \).
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 |
\(k(x) = \frac{x^3}{6} + \frac{x^2}{2} + \frac{x}{3} + 2; k(100) = 171,702\)