AP Precalculus Function Model Selection Drill 1

Question 1. The table below shows values of a function f.

x	0	1	2	3	4
f(x)	3	6	12	24	48

Which of the following best justifies that f is best modeled by an exponential function?

• A) The first differences of f(x) are constant.
• B) The second differences of f(x) are constant.
• C) The average rate of change of f over each unit interval is constant.
• D) The ratios of successive f(x) values are constant. ✓

Explanation: Choice D is correct. Computing successive ratios: 6/3 = 2, 12/6 = 2, 24/12 = 2, 48/24 = 2. A constant ratio between successive output values is the defining characteristic of an exponential function, each output is the previous output multiplied by the same factor. Choice A is incorrect because constant first differences indicate a linear model, not exponential. Choice B is incorrect because constant second differences indicate a quadratic model. Choice C is incorrect because a constant average rate of change over equal intervals is equivalent to constant first differences, which also indicates a linear model, not exponential.

Question 2. The table below shows values of a function g.

x	0	1	2	3	4
g(x)	2	5	10	17	26

Which of the following best describes the function type that models g?

• A) Linear, because the first differences are constant
• B) Exponential, because the ratios of successive values are constant
• C) Quadratic, because the second differences are constant ✓
• D) Rational, because the rate of change is decreasing

Explanation: Choice C is correct. First differences: 5 − 2 = 3, 10 − 5 = 5, 17 − 10 = 7, 26 − 17 = 9. These are not constant, so the model is not linear. Second differences: 5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2. The second differences are constant, which is the defining characteristic of a quadratic model. Choice A is incorrect because the first differences (3, 5, 7, 9) are not constant; they increase by 2 each step. Choice B is incorrect because the ratios 5/2 = 2.5, 10/5 = 2, 17/10 = 1.7 are not constant, ruling out an exponential model. Choice D is incorrect because a non-constant rate of change does not imply a rational function; the constant second differences here specifically confirm a quadratic model.

Question 3. A population of bacteria doubles every 3 hours. At time t = 0, there are 500 bacteria. Which of the following correctly expresses the number of bacteria after t hours?

• A) \(P(t) = 500 \cdot 2^{t}\)
• B) \(P(t) = 500 \cdot 2^{t/3}\)  (where the exponent is \(t/3\), the number of 3-hour doubling periods elapsed) ✓
• C) \(P(t) = 500 \cdot \left(\tfrac{2}{3}\right)^{t}\)
• D) \(P(t) = 500 + 2t\)

Explanation: Choice B is correct. The population doubles every 3 hours, so after t hours, t/3 doubling periods have elapsed, giving the model \(P(t) = 500 \cdot 2^{t/3}\) (the exponent is the fraction t/3, not t divided separately by 3 after the power is applied). Checking: at t = 0, P = 500 · 1 = 500 ✓; at t = 3, P = 500 · 2 = 1,000 ✓; at t = 6, P = 500 · 4 = 2,000 ✓. Choice A is incorrect because \(500 \cdot 2^{t}\) doubles every 1 hour, not every 3, this error comes from using t directly as the exponent without accounting for the 3-hour doubling period. Choice C is incorrect because \(\left(\tfrac{2}{3}\right)^{t}\) has a base less than 1, modeling a decaying population rather than a growing one. Choice D is incorrect because 500 + 2t is a linear model with a constant rate of change of 2 bacteria per hour, which does not reflect doubling behavior.

Question 4. The table below shows the value V (in dollars) of a car t years after purchase.

t	0	1	2	3	4
V(t)	24,000	19,200	15,360	12,288	9,830

Which of the following best describes the function type that models V, and what feature of the table justifies this choice?

• A) Exponential, because successive values have an approximately constant ratio ✓
• B) Linear, because the first differences are approximately constant across the values shown in the table
• C) Quadratic, because the rate of change of V is decreasing
• D) Rational, because V approaches zero but never reaches it

Explanation: Choice A is correct. Computing successive ratios: 19,200/24,000 = 0.8, 15,360/19,200 = 0.8, 12,288/15,360 = 0.8, 9,830/12,288 ≈ 0.80. Note that the exact model value at t = 4 is 24,000 · (0.8)⁴ = 9,830.4, which is rounded to 9,830 in the table; this is why Choice A correctly describes the ratios as approximately (rather than exactly) constant. The approximately constant ratio of 0.8 identifies an exponential decay model V(t) = 24,000 · (0.8)^t, representing 20% depreciation per year. Choice B is incorrect because the first differences (−4,800; −3,840; −3,072; −2,458) are not constant; they become less negative each year, ruling out a linear model. Choice C is incorrect because while the rate of change of V is decreasing in magnitude, this alone does not distinguish exponential decay from quadratic; the approximately constant ratio specifically identifies exponential. Choice D is incorrect because approaching zero alone does not identify the model type; the approximately constant ratio is the key evidence for exponential decay, not the asymptotic behavior.

Question 5. A company’s monthly revenue R (in thousands of dollars) over the first several months is shown in the table below.

Month (t)	1	2	3	4	5
R(t)	4	7	11	16	22

A student concludes: “Because the revenue is always increasing and the rate of increase is also increasing, the data must be modeled by an exponential function.” Is the student’s conclusion correct?

• A) Yes, because increasing output with an increasing rate of change always indicates exponential growth.
• B) No. The data could be quadratic, the second differences are constant, which is consistent with a quadratic model, not necessarily exponential. ✓
• C) No. The data must be linear because the first differences are nearly constant.
• D) No. The data must be rational because the rate of change is not constant.

Explanation: Choice B is correct. First differences: 3, 4, 5, 6, increasing, confirming the rate of change is growing. Second differences: 1, 1, 1, constant, the defining feature of a quadratic model. Checking ratios: 7/4 = 1.75, 11/7 ≈ 1.57, 16/11 ≈ 1.45, not constant, ruling out exponential. The student’s reasoning is flawed: an increasing rate of change is consistent with both quadratic and exponential growth; checking second differences and ratios is required to distinguish between them. Choice A is incorrect because increasing output with an increasing rate of change does not exclusively indicate exponential growth, quadratic functions with positive leading coefficients also have increasing rates of change. Choice C is incorrect because the first differences (3, 4, 5, 6) are clearly not constant, ruling out a linear model. Choice D is incorrect because a non-constant rate of change does not imply a rational function; both quadratic and exponential functions have non-constant rates of change.