Drill 1 · Math · Polynomial End Behavior
AP Precalculus: Polynomial End Behavior (Drill 5) is a Math practice drill covering Polynomial End Behavior. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.
This AP® Precalculus drill focuses on polynomial end behavior, using limit notation to describe what happens to a function's output as x approaches positive or negative infinity. Master the role of degree and leading coefficient in determining tail direction.
Question 1. Which of the following correctly describes the end behavior of \( f(x) = -3x^4 + 7x^2 - 2 \)?
Explanation: Choice A is correct. End behavior is determined entirely by the leading term, which is −3x4. Since the degree is 4 (even), both tails point in the same direction. Since the leading coefficient is −3 (negative), both tails point downward: as x → ∞, −3x4 → −∞, and as x → −∞, −3x4 → −∞. Choice B is incorrect because a negative leading coefficient causes both tails to fall, not rise; this would describe a polynomial with a positive leading coefficient such as +3x4. Choice C is incorrect because opposite-tail behavior (one up, one down) is a feature of odd-degree polynomials; an even-degree polynomial always has matching tails. Choice D is incorrect for the same reason as C, opposite tails require odd degree, and the leading coefficient would need to be positive to produce this particular pattern.
Question 2. A polynomial p(x) has the property that \( \lim_{x \to \infty} p(x) = \infty \) and \( \lim_{x \to -\infty} p(x) = -\infty \). Which of the following must be true?
Explanation: Choice D is correct. Because the two tails point in opposite directions (right tail up, left tail down), the degree must be odd. Among odd-degree polynomials, the right tail rises (→ +∞) and the left tail falls (→ −∞) when the leading coefficient is positive. This matches the given limits exactly. Choice A is incorrect because even-degree polynomials always have matching tails; a negative even-degree leading coefficient gives both tails pointing down. Choice B is incorrect because an even degree with a positive leading coefficient gives both tails pointing up, again, matching, not opposite. Choice C is incorrect because an odd-degree polynomial with a negative leading coefficient has its right tail pointing down and its left tail pointing up, which is the reverse of what is described.
Question 3. The table below shows selected values of a polynomial function g(x).
| x | −1000 | −100 | 100 | 1000 |
|---|---|---|---|---|
| g(x) | 8.1 × 108 | 8.1 × 104 | 8.1 × 104 | 8.1 × 108 |
Explanation: Choice A is correct. The table shows g(−1000) and g(1000) are both large positive values (8.1 × 108), and g(−100) and g(100) are both positive (8.1 × 104). Both tails are rising toward positive infinity, which means the end behavior matches an even-degree polynomial with a positive leading coefficient. Choice B is incorrect because it describes opposite tails (one up, one down), which contradicts the table: both extreme values are positive. Choice C is incorrect because it predicts both tails falling toward −∞, but the table shows large positive values at both extremes. Choice D is incorrect because it also describes opposite tails, and with the wrong orientation relative to the table data.
Question 4. Let f(x) = 2x3 − 5x2 + x − 7 and h(x) = −f(x). Which of the following correctly describes the end behavior of h(x)?
Explanation: Choice D is correct. Since h(x) = −f(x) = −(2x3 − 5x2 + x − 7) = −2x3 + 5x2 − x + 7, the leading term of h is −2x3. The degree is 3 (odd) and the leading coefficient is −2 (negative). For an odd-degree polynomial with negative leading coefficient: as x → ∞, −2x3 → −∞, and as x → −∞, −2x3 → +∞. This matches Choice D. Choice A is incorrect because it describes an odd-degree polynomial with a positive leading coefficient; this would apply to f(x) itself, not h(x) = −f(x). Choice B is incorrect because matching tails both going to +∞ require an even degree, but the leading term is odd (degree 3). Choice C is incorrect because matching tails both going to −∞ would require an even degree with a negative leading coefficient.
Question 5. A scientist models the net population growth rate of a species (in thousands per year) using the polynomial P(t) = −t5 + 3t3 + 10t, where t is measured in decades after 1900. A colleague claims: "Because the model includes positive terms, the growth rate must eventually become positive for large values of t." Which of the following best evaluates this claim?
Explanation: Choice A is correct. The colleague's claim is incorrect. End behavior is determined solely by the leading term, which is −t5. As t → +∞, −t5 → −∞. The lower-degree positive terms 3t3 and 10t grow much more slowly than t5 and become negligible in comparison for large values of t, the leading term always dominates. Therefore the end behavior can still be determined: as \( t \to \infty \), \( P(t) \to -\infty \), so the growth rate eventually becomes arbitrarily negative regardless of the positive terms. Choice B is incorrect because no lower-degree terms can outweigh the leading term as t → ∞; this reflects the common misconception that positive terms can "overcome" a negative leading term. Choice C is incorrect because it correctly identifies that an odd-degree polynomial has opposite tails, but misidentifies which tail is which: with a negative leading coefficient (−t5), the right tail (t → +∞) goes to −∞ and the left tail (t → −∞) goes to +∞. Choice D is incorrect because end behavior is determined entirely by the leading term and does not depend on the locations of the zeros.