ACT® · Math: The Modeling Overlay
Improving a Model by Adjusting Its Parameters
A model is improved by changing one parameter at a time — slope, intercept, or base — and checking what that change does to every prediction, not by adding terms until the curve hits every point.
A linear model y = mx + b has two knobs. Raising b shifts every predicted y up by the same amount, no matter what x is — a flat, parallel shift. Raising m tilts the line: predictions far from the pivot point move a lot, predictions near it barely move. That's why the same wrong model can sometimes be rescued by nudging just one of these numbers. If the line is off by a constant amount everywhere, fix the intercept. If it's close for small x but drifts for large x, fix the slope.
For an exponential model y = a(base)t, the base controls growth rate, and because it's raised to a power, even a tiny nudge compounds. Moving the base from 1.1 to 1.15 looks small, but the gap between the two models' predictions grows every time t increases by 1, so the two curves spread apart faster and faster rather than staying a fixed distance apart like the linear case.
This connects straight to graph transformations: changing b in y = mx + b is a vertical shift, changing m is a stretch or tilt about the y-intercept, and changing the base of an exponential rescales how fast the curve bends. Adding more terms, though, isn't a transformation of the same shape — it's a different, more flexible model, and that flexibility can chase noise in the data instead of the real pattern, hurting predictions between and beyond the given points.
Key terms
- Slope
- The rate of change in a linear model; raising it tilts predictions more the farther they are from the pivot.
- Intercept
- The constant term in a linear model; raising it shifts every prediction up by the same fixed amount.
- Base (exponential)
- The number raised to the power of t in y = a(base)t, controlling how fast the model grows.
- Overfitting
- Making a model so flexible it passes through every data point, fitting the noise instead of the underlying trend.
Watch it explained
- How it worksHow does raising the intercept shift every prediction while raising the slope tilts them?
- Common mistakeWhy doesn't piling more terms into a model always make it better?
- Worked exampleFind the new slope that makes y = 2x + 5 pass exactly through the point (10, 31).
- Going furtherWhat happens to the predictions when an exponential base is nudged from 1.1 to 1.15?
- Ties togetherHow does adjusting a model's parameters connect to the graph transformations of that same function?
Check yourself
End of topic.