Learning goal and prerequisites
Recover a per-unit factor and test an alternative model using an extra observation. Before starting, review exponential growth.
Method and conditions
For y=A b^t with positive outputs, divide two observations to eliminate A. The ratio spans the difference between their input times; take the corresponding positive root to obtain b. Then recover A and check a withheld observation.
Worked example: show the decisions
If y(2)=36 and y(4)=324, dividing gives b²=9, so b=3 and A=4. An exponential prediction at t=3 is 108. A linear model through the endpoints predicts 180 there, so a new observation can distinguish them.
Contrast: why a plausible shortcut fails
A ratio of nine over two hours is not a factor of nine each hour. Likewise, passing through two data points does not by itself establish an exponential mechanism. State that the chosen model assumes a constant proportional change.
Transfer: try before revealing the answer
An exponential rule has y(1)=10 and y(3)=40. Find y(0) and y(2), then compare y(2) with the straight-line interpolation.
Transfer solution and check
The two-step ratio is 4, so b=2, y(0)=5 and y(2)=20. Linear interpolation gives 25. Both models match the supplied endpoints, but their interior predictions differ.
A positive exponential model y=A·b^t satisfies y(2)=12 and y(4)=48. What is A?
- The two-step ratio is four, so the one-step factor is two; reverse two factors from y(2).
- Using these quantities: 12/√(48/12)^2 = 3.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Compare the other answer choices
Reverse only one factor of two from the observation at t=2.
Treat the observation at t=2 as the initial value.
Use the two-step factor four as the one-step factor.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance