How-to · Updated October 2, 2026
How do you find the function from a table of values?
Direct answer
If the outputs change by the same amount for equal steps in x, the rule is a line: slope = change in y ÷ change in x. Outputs 9, 14, 19, 24 for x = 0, 1, 2, 3 give y = 5x + 9. Inputs 4, 6, 7 with outputs 16, 24, 28 give y = 4x, so the output for n is 4n.
Four tables
| Inputs | Outputs | Rule |
|---|---|---|
| 0, 1, 2, 3 | 9, 14, 19, 24 | y = 5x + 9 |
| 4, 6, 7, n | 16, 24, 28, ? | y = 4x |
| 1, 2, 3, 4, 5 | 1, 4, 9, 16, ? | y = x² |
| 0, 1, 2, 3, 4 | 3, 6, 12, 24, ? | y = 3 · 2^x |
The missing outputs are 4n for n, 25 at x = 5, and 48 at x = 4.
How the linear rule is built
- Rate of change: (14 − 9) ÷ (1 − 0) = 5 — the same between every pair of points, so the table is linear.
- When x = 0, y = 9, so the starting value (y-intercept) b = 9.
- Rule: y = 5x + 9.
Limits
- Two points always determine a line. A third point is what confirms the pattern continues.
- Three points always fit some parabola. A constant second difference, or a fourth point, is the check.
- One input with two different outputs is not a function. Measured data that only comes close to a line needs a regression, not an exact rule.
Use the calculator
Start from the rule instead: how to complete a function table. Fit a table in the function table calculator.