Identifying the Right Model for a Relationship
Look at how outputs change step by step: constant differences mean linear, constant second differences mean quadratic, and constant ratios mean exponential. A plain table of numbers tells you the model before you ever see a graph.
Every model leaves a fingerprint in a table of evenly spaced x-values. Take consecutive outputs and subtract: if those first differences are all the same number, the function is linear, because a constant difference is just a constant rate of change written out in a list. If the first differences aren't constant but the second differences are (the differences of the differences), the function is quadratic. If instead you divide each output by the one before it and keep getting the same ratio, the function is exponential, since that's a constant multiplier applied again and again.
The trap is judging a table by its shape alone. A curve that climbs faster and faster looks dramatic, and it's tempting to call that exponential on sight. But quadratics also climb faster and faster, since their first differences grow by a fixed amount each time. Growing jumps only rule out linear. To split quadratic from exponential you have to check the actual numbers: take second differences for quadratic, or take ratios of consecutive terms for exponential. Only a constant ratio confirms exponential growth.
This skill sits right before writing the actual function once you know the family. For a linear model you read off the constant difference as the slope and use one point to find the intercept. For a quadratic you work with the constant second difference to pin down the leading coefficient. For an exponential you read the constant ratio as the base and the starting output as the coefficient out front. Spotting the pattern first keeps you from forcing the wrong equation onto numbers that don't fit it.
Key terms
- First difference
- The result of subtracting each output from the next one in a table with evenly spaced inputs.
- Second difference
- The difference between consecutive first differences, used to detect quadratic patterns.
- Constant ratio
- The same value obtained by dividing each output by the previous output, the signature of exponential growth.
- Rate of change
- How much the output changes per unit increase in input, constant for linear functions.
Watch it explained
- How it worksHow do constant differences, constant second differences, and constant ratios each fingerprint one model?
- Common mistakeWhy isn't a curve that climbs faster and faster automatically an exponential one?
- Worked exampleFind which model fits the values 3, 6, 12, 24, 48 and write out its function.
- Worked exampleFind which model fits the values 2, 8, 18, 32, 50 and write out its function.
Check yourself
End of topic.