Practice a College Board-style free response question on an exponential growth model. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 2 · Exponential Growth Modeling
A biologist studies a bacteria culture. The number of bacteria (in thousands) is modeled by the exponential function P(t) = 200(1.15)^t, where t is the time in hours since the study began (t ≥ 0). Selected values of the model are shown below.
t (hours)
P(t) (thousands)
0
200
2
≈ 264.5
5
≈ 402.3
10
≈ 809.1
A
Identify the initial value and the growth factor of P, and interpret the growth factor in the context of the bacteria culture.
✓ Model answer (earns the point)
The initial value is P(0) = 200 thousand bacteria, and the growth factor is the base 1.15. Because 1.15 = 1 + 0.15, the growth factor means the population increases by 15% each hour — every hour the number of bacteria is 1.15 times what it was the hour before.
Why it scores: Correctly names the initial value (200 thousand) AND the growth factor (1.15), and interprets 1.15 as a 15% hourly increase in context. Stating the base without connecting it to a percent rate would be incomplete.
B
Using logarithms, determine the time t at which the population reaches 500 thousand bacteria. Show the steps that lead to your answer and include units.
✓ Model answer (earns the point)
Set P(t) = 500: 200(1.15)^t = 500, so (1.15)^t = 500/200 = 2.5. Taking a logarithm of both sides, t = log_(1.15)(2.5) = ln(2.5) / ln(1.15) ≈ 6.56 hours. The population reaches 500 thousand bacteria after about 6.56 hours.
Why it scores: Sets up the equation, isolates the exponential (divides to get 2.5), applies a logarithm to solve for the exponent, AND reports the answer with units. An answer with no supporting work may not receive full credit on the free-response section.
C
The biologist wants a function that gives the time t as a function of the population P. Find the inverse of P, state what its input and output represent, and explain why P has an inverse function.
✓ Model answer (earns the point)
Starting from P = 200(1.15)^t, divide to get P/200 = (1.15)^t, then take a logarithm: t = log_(1.15)(P/200) = ln(P/200) / ln(1.15). In this inverse function the input is a population P (in thousands) and the output is the time t (in hours) at which that population is reached. P has an inverse because it is one-to-one — the exponential is strictly increasing, so each population value corresponds to exactly one time, and it passes the horizontal line test.
Why it scores: Correctly solves for t using a logarithm, states that the inverse takes population to time (with units), AND justifies the existence of an inverse by the one-to-one / strictly increasing property. Just writing the formula without the interpretation or the one-to-one justification would lose points.
How to score points on AP Precalculus FRQs
Always include units and context. Numerical answers need units, and "interpret" prompts need a sentence tying the number to the scenario.
Isolate the exponential before taking a log. Divide or subtract first so the base with the unknown exponent stands alone, then apply a logarithm.
Show the log step explicitly. Write "take ln of both sides" and the change-of-base form — answers without supporting work may not earn full credit.
Justify inverses with one-to-one. A function has an inverse because it is one-to-one (passes the horizontal line test); say so.
Name the mathematics precisely. "Growth factor," "one-to-one," "inverse function" — precise terms earn points that vague descriptions do not.