Function model construction and application

Before any algebra, name the input, the output, and the units of each. A gym membership with a 20 dollar signup fee and 5 dollars per month is not yet a formula; it is two quantities (months in, dollars out) and two roles. The per-month number multiplies the input; the one-time number stands alone: $C(m) = 5m + 20$.

Role errors, not arithmetic errors, are what break constructed models. Merging the fee into the rate ($25m$), or swapping the roles ($20m + 5$), produces formulas that agree with the story at exactly one point or none. The check is always the same: feed the model two easy cases from the story, month 0 and month 1, and see whether it reproduces them.

Often the new model is an old model with a dial turned, and the whole construction is choosing the right transformation. Conditions that scale the output attach outside: wet-road braking distances 1.5 times as long turn $B(s)$ into $1.5\,B(s)$. Conditions that re-index the input attach inside, and in reverse: re-zeroing a clock 7 days later turns $H(t)$ into $H(t + 7)$, because the new day $t$ is the old day $t + 7$.

The transformation rules from Topic 1.12 carry over verbatim, including the traps. Adding 1.5 instead of multiplying, or writing $H(t - 7)$ because the new clock starts later, are the same inside-outside and direction errors wearing a word problem's clothes.

A constructed model repays you by letting every number be interpreted. In $V(t) = 500 - 40t$ (gallons, minutes): 500 is the starting volume, the value at $t = 0$; 40 is the rate, gallons lost per minute; and the zero of the function, $t = 12.5$, is when the tank runs dry. In a factored model like $R(n) = -2(n - 10)(n - 50)$, the zeros 10 and 50 are the inputs where the output dies, and the vertex between them is where it peaks.

Units ride along with every parameter. A rate is output units per input unit; an average rate of change over an interval is the total output change divided by the total input change, carrying those same units. Saying "7" is half an answer; "7 square centimeters per centimeter of side" is the model speaking in full sentences.

A model that predicts 340 when reality delivers 352 is not broken; it is off by 12, and the question is whether 12 matters for the purpose at hand. Model error is information. What discredits a model is not the existence of error but error with a pattern, or error too large for the decision the model serves.

The sharpest edge is the data window. A model fit on inputs from 0 to 10 has earned no trust at 50. The formula will happily compute there, and the computation means nothing, because the construction assumed the observed behavior continues, and that assumption was never tested past 10. Interpolation borrows the data's authority; extrapolation borrows only the modeler's hope.