Real data rarely comes with a formula attached. Choosing a model means asking what kind of change you see, using everything from Exponential and Logarithmic Models, rational functions and polynomials.
Key idea
Equal steps in $x$ with equal differences in $y$: linear. Equal second differences: quadratic. Equal ratios: exponential. Equal differences in $y$ when $x$ is multiplied by a constant: logarithmic. A quantity that shares out a fixed total (like time per worker) suggests a rational model.
Why it works
Each family has its own signature. A line adds the slope for each unit step. A quadratic's differences grow by a constant amount, because its first differences are themselves linear. An exponential multiplies by $b$ for each unit step. A logarithm is the inverse of an exponential, so it adds a constant each time $x$ is multiplied by the base. Checking which pattern fits the data tells you the family.
Worked examples
Worked example: Reading a table
$x$: 0, 1, 2, 3, 4; $y$: 3, 6, 12, 24, 48. Which model?
- Differences: 3, 6, 12, 24 (not constant).
- Ratios: $\frac{6}{3} = \frac{12}{6} = \dots = 2$.
Answer: Exponential: $y = 3 \cdot 2^x$
Worked example: Another table
$x$: 1, 10, 100, 1000; $y$: 5, 7, 9, 11. Which model?
- Each time $x$ is multiplied by 10, $y$ goes up by 2.
Answer: Logarithmic: $y = 5 + 2\log x$
Worked example: A situation
A phone plan costs \$20 a month plus \$0.10 per text. Which model fits the monthly bill?
- Each extra text adds the same \$0.10.
Answer: Linear
Common mistake
Deciding from a picture of a few points. Many curves pass through any two points, as the visual shows. Compute differences and ratios, or think about the process that produced the data: a fixed percent change is exponential, a fixed amount is linear.