Establish what distinguishes an exponential function from a power function
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 9-point free response. What you recorded goes to the grader with your writing, so a conclusion that does not follow from your own numbers will cost you the point — exactly as it would with a real reader.
Predict before you look
- An exponential function multiplies by a constant factor per unit input, while a power function raises the input to a fixed power. Which of the two grows in proportion to how large it already is?
- If a population grows by 3% per year, the number of new individuals each year depends on the current population. Write that dependence as a ratio of growth rate to current size, and say whether you expect the ratio to be constant.
Nothing to submit here — these are to think through, so the prediction below is an informed one rather than a guess.
Commit to an answer now. It is not graded and being wrong costs nothing — the point is to have something specific to reconcile against once you have the data.
Answer every prediction to unlock the lab. A sentence is enough.
Run the investigation
Predictions first
The procedure and the simulation unlock once you have committed above. Observing before predicting is how a wrong intuition survives a lab intact.
Record what you measured
These are your numbers, not ours. The grader sees them, so your conclusions have to follow from what you actually recorded.
| f(x) at x = 0.00 on f(x) = eˣ/4 | |
|---|---|
| Slope f′(x) at x = 0.00 on f(x) = eˣ/4 | |
| f(x) at x = 1.00 on f(x) = eˣ/4 | |
| Slope f′(x) at x = 1.00 on f(x) = eˣ/4 | |
| f(x) at x = 2.00 on f(x) = eˣ/4 | |
| Slope f′(x) at x = 2.00 on f(x) = eˣ/4 | |
| Ratio (slope)/(value) for eˣ/4 at x = 2.00 | |
| f(x) at x = 2.00 on f(x) = x² | |
| Slope f′(x) at x = 2.00 on f(x) = x² | |
| f(x) at x = 3.00 on f(x) = x² | |
| Slope f′(x) at x = 3.00 on f(x) = x² | |
| Ratio (slope)/(value) for x² at x = 3.00 |
Answer the free response
This investigation isolates the property that defines exponential change. (a) Report your three pairs of readings for f(x) = eˣ/4 and state what you notice about the slope compared with the value at each x. (b) Compute the ratio (slope)/(value) at each of your three x values for eˣ/4 and state what it equals. (c) Report the ratio for f(x) = x² at x = 2 and at x = 3. One of those two ratios equals 1, matching the exponential exactly. Explain why that agreement is a coincidence rather than evidence that x² is exponential, and identify which single reading settles the question. (d) A population model is proposed in which the number of new individuals added each year is a fixed percentage of the current population. Explain which of the two behaviors you measured this describes, and state one thing this lab cannot tell you about whether such a model suits a real population.
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