Exponential Functions at the 700-800 Level

Hard-tier exponential items are exercises in growth factors and time units, not formula plugging. Three archetypes cover nearly all of them. Rebasing the time unit: a model like 60(0.5)^(t/3) states its factor per 3 hours; the per-hour rate requires the cube root of 0.5, and the planted distractors are the linear split (50%/3) and the conversion run backward (0.5 cubed). Compound vs. linear: effective annual rates from monthly compounding, and fixed-dollar vs. fixed-percent depreciation plans — for decay, compounding always loses less than the linear stack; for growth it gains more, and knowing that direction eliminates choices before computing. Counting doublings and halvings: think in doublings, not amounts — a quarter full is exactly two doublings before full; half-life back-solving steps in half-lives (multiply by 2 per step back). Recognize powers of 2 up to 4096 on sight.

The distractor set is rigid: the percent-remaining complement, the linear stack, the reversed conversion, the off-by-one period on threshold crossings, and intermediate amounts (original quantity, the other plan's value). The module drills naming the archetype in ten seconds, predicting which side of the linear estimate the truth sits on, and checking both integers around any threshold crossing.