Give a quadratic function that would result in the values given below. Compute the missing value.
| \(x\) | \(f(x)\) |
|---|---|
| 1 | 2 |
| 2 | 6 |
| 3 | 12 |
| 4 | 20 |
| 5 | 30 |
| 100 | 10,100 |
\(f(x) = x(x+1) = x^2 + x \)
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 | 19,317 |
\(g(x) = x(x+4) = x^2 + 4x\)
Give a quadratic function that would result in the values given below. Compute the missing value.
| \(x\) | \(h(x)\) |
|---|---|
| 1 | -4 |
| 3 | 1 |
| 5 | 12 |
| 7 | 29 |
| 9 | 52 |
| 11 | 81 |
| 141 | 14,836 |
\(h(x) = \frac{3}{4} x^2 - \frac{1}{2} x - \frac{17}{4} \)
Give a linear function that would result in the values given below.
| \(x\) | \(j(x)\) |
|---|---|
| 3 | 25 |
| 4 | 23 |
| 5 | 21 |
| 6 | 19 |
| 7 | 17 |
| 8 | 15 |
Linear function general form \(ax + b\), answer: \(j(x) = -2 x + 31 \).
Unfortunately this method will not always work. Why won't it work in the example below?
| \(x\) | \(k(x)\) |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 4 | 81 |
| 5 | 243 |
Differences never become constant. Function is not polynomial but is exponential in this case, \(k(x) = 3^x \).
Give a polynomial function that would result in the values given below. Compute the missing value.
| \(x\) | \(m(x)\) |
|---|---|
| 0 | 2 |
| 1 | 3 |
| 2 | 6 |
| 3 | 12 |
| 4 | 22 |
| 100 | 171,702 |
\(m(x) = \frac{1}{6} x^3 + \frac{1}{2} x^2 + \frac{1}{3} x + 2\)
| x | p(x) |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 4 |
| 4 | 7 |
| 5 | 11 |
| 100 | 4951 |
Coefficient answers for general quadratic: a = 1/2, b = -1/2, c = 1