An amount doubles every three days. If t counts days, its model can be written 20·2^(t/3). If h counts hours, the same physical model is 20·2^(h/72). These are not two growth processes. They use different numerical descriptions of elapsed time, so the exponent coefficient must change with the unit.
Predict
Predict whether a yearly factor can be divided by twelve to obtain a monthly factor. Factors multiply across periods, so the monthly factor must be a twelfth root. Predict whether twelve monthly increases of eight percent produce exactly ninety-six percent yearly growth. Compounding changes the base amount each month, so that simple sum is not exact.
Experiment
Graph H: exponential amount. Exact landmarks are (0 days,20 units),(3 days,40 units),(6 days,80 units); the stated model is exponential.
Supplied Graph H is an ideal exponential model with exact plotted landmarks; the curve is not merely an arbitrary connection of observations.
Elapsed days
Amount, units
0
20
3
40
6
80
Read the vertical origin and three-day horizontal tick spacing. Recover an initial amount and a three-day factor from the graph, then write a rule. Compare the same physical time expressed in days, hours and weeks.
Understand
An exponential model A b^(ct) has initial amount A when t=0, provided the model is anchored at that time. The base b is positive and different from one for nonconstant exponential behavior. The coefficient c adjusts how much exponent accumulates per stated unit of t. The per-unit-time factor is b^c, not necessarily b itself.
If t measures days and h measures hours, t=h/24. Substitute this relation into the original exponent rather than guessing a new growth percentage. Similarly one week is seven days, so t=7w. Equivalent expressions must agree whenever their inputs represent the same elapsed time. Checking one or two such times is a useful error detector, though the substitution proves the identity for every time.
To fit A q^t from two positive exact model values at distinct times, their ratio determines q raised to the time difference. For the graph’s ratio 40/20 over three days, q³=2, so q is the positive cube root of two. The initial point identifies A=20. The third point tests that the same factor works across the next equal interval. Without an exponential-family assumption, three points alone do not establish future growth.
The distinction between linear and exponential models also works for arbitrary interval lengths. For L(t)=A+mt, subtracting L(t) from L(t+h) gives mh. For E(t)=Cb^t with C≠0, dividing E(t+h) by E(t) gives b^h. Each expression is independent of the starting time t, but depends on the interval length h. Therefore unequal table gaps should be compared using their actual lengths rather than expecting raw differences or ratios to match.
A percentage increase r corresponds to factor 1+r, with r expressed as a decimal. Across twelve equal periods the factor is (1+r)^12. The percentage increase over the combined interval is that factor minus one, then multiplied by 100 percent.
Apply
Translate a monthly model into a yearly one and interpret each parameter in words. State whether the model applies at all real times or only at recorded monthly indices. A smooth exponential curve is an extension when the original rule only updates discrete balances; use it only with an explicit interpretation.
Continue
Equivalent time-unit expressions preserve predictions, not the appearance of the exponent. Next, compare exponential growth with a polynomial over a wider window. A slow exponential can remain smaller for a long time before eventually overtaking, so an early graph can be misleading.
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Supplied Graph H has values 20 at day 0 and 40 at day 3 under an exponential model. What is its multiplicative factor per three-day interval?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
An ideal exponential model multiplies by 1.21 per year. What is its equivalent factor per half-year?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Infer the exponential rule for supplied Graph H in days from its exact landmarks (0,20),(3,40),(6,80). Interpret the parameters and predict day 9 under the same ideal model.
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Rewrite P(t)=20·2^(t/3), t in days, using h in hours and w in weeks. Verify the same prediction at six days.
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Prove equal differences for L(t)=A+mt and equal ratios for E(t)=Cb^t over any interval length h, with C nonzero and b>0. Explain why unequal gaps require adjustment.
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A quantity grows by 8 percent each month. Give its model from initial value 50, its exact yearly factor and approximate yearly percentage; explain why 96 percent is not exact.
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