AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.7A — Rational Functions and End Behavior
Notes — Horizontal Asymptotes from Comparing Degrees
💡 Learning Objectives By the end of this lesson you will be able to (AP CED 1.7.A):
|
1. What Is a Rational Function?
A rational function is a quotient of two polynomials. We write it as r(x) = N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. The domain of r is every real number except the zeros of D(x) — at those inputs the function is undefined.
In this lesson we are not solving for domain or asymptote locations. We are asking a different question: what does the graph of r(x) do way out on the far left and far right — its end behavior?
💡 Quick Check Before you read further, decide: which of these is a rational function?
Answer: (a) and (c) are rational — both are polynomial over polynomial. (b) is not — √x is not a polynomial. |
2. End Behavior — Which Polynomial Wins?
When x is very large in magnitude (far from zero, positive or negative), only the leading terms of N(x) and D(x) really matter. All the lower-degree terms become negligible compared with the leading term. So the end behavior of r(x) matches the end behavior of the ratio of its leading terms.
There are exactly three cases, depending on how the degree of the numerator compares to the degree of the denominator. Memorize this trio — it unlocks every end-behavior problem in this unit.
Three cases for horizontal asymptotes. Compare the degree of the numerator (n) to the degree of the denominator (m).
💡 Definitions Let r(x) = N(x)/D(x) with N and D sharing no common factors. Let n = deg N and m = deg D.
|
3. Case 1 — Denominator Wins (n < m)
When the denominator has a higher degree than the numerator, the denominator grows faster for large |x|. The quotient shrinks toward zero in both directions. The horizontal asymptote is the x-axis, y = 0.
📘 Example Example 1 — Denominator dominates Find the horizontal asymptote of r(x) = (2x + 5) / (x² − 4). Degrees: n = 1, m = 2, so n < m. By Case 1, the horizontal asymptote is y = 0. Numerical sanity check: r(1000) = 2005 / 999 996 ≈ 0.002, and r(−1000) ≈ −0.002. Both tiny — consistent with y = 0. |
4. Case 2 — Same Degree (n = m)
When the numerator and denominator have the same degree, neither polynomial wins the race. The ratio of the leading coefficients sets the horizontal asymptote. Everything else in the polynomials becomes a small correction for large |x|.
📘 Example Example 2 — Same degree Find the horizontal asymptote of r(x) = (3x² − x + 1) / (5x² + 2x − 7). Both degrees equal 2. Leading coefficients: aₙ = 3, bₘ = 5. By Case 2, the horizontal asymptote is y = 3/5. Check: r(100) = (30000 − 100 + 1) / (50000 + 200 − 7) ≈ 29901 / 50193 ≈ 0.5957 ≈ 3/5. ✓ |
5. Case 3 — Numerator Wins (n > m)
When the numerator has a higher degree than the denominator, the quotient grows without bound in magnitude. There is no horizontal asymptote. The graph shoots off to ±∞ on each end — much like a polynomial does.
📘 Example Example 3 — Numerator dominates Describe the end behavior of r(x) = (x³ + 2) / (x − 1). Degrees: n = 3, m = 1. Since n > m, there is no horizontal asymptote. The leading-term ratio is x³ / x = x², which is a polynomial. So for large |x|, r behaves like x² — it rises to +∞ on both ends. |
6. The Three Cases, Drawn
Three different end-behavior patterns for rational functions. Left: denominator dominates, HA y = 0. Middle: same degree, HA at the coefficient ratio. Right: numerator dominates, no HA.
⚠️ Common Mistake Comparing degrees is about the exponents, not the coefficients. r(x) = (100x) / (x²) still has y = 0 as its horizontal asymptote — the coefficient 100 does not change the fact that x² eventually wins. |
7. Writing End Behavior with Limit Notation
On the AP Exam you will be expected to communicate end behavior using limit notation. There are exactly two end-behavior limits per function:
- lim r(x) as x → +∞ — what happens to the outputs as inputs grow without bound in the positive direction.
- lim r(x) as x → −∞ — what happens to the outputs as inputs grow without bound in the negative direction.
When the graph has a horizontal asymptote y = b, both limits equal b: lim r(x) as x → +∞ = b AND lim r(x) as x → −∞ = b. When there is no horizontal asymptote, the limits are ±∞.
Limit notation captures what the outputs approach as x moves to either end. Here both limits equal 3, so the graph has the horizontal asymptote y = 3.
🎯 AP Tip On AP responses, write the full limit statement with both sides equal and a specific value or symbol: lim r(x) as x → ∞ = 3. A claim like "the function goes to 3" without limit notation will not earn the communication point. |
📘 Try This Practice — classify each by case and state the horizontal asymptote if one exists.
Answers: Case 2, y = 2 ; Case 1, y = 0 ; Case 3, no HA. |
8. Summary
- A rational function is a ratio N(x)/D(x) of two polynomials.
- End behavior is governed by the leading terms of N and D.
- Case 1: deg N < deg D → HA is y = 0.
- Case 2: deg N = deg D → HA is y = (leading coefficient ratio).
- Case 3: deg N > deg D → no HA.
- Express end behavior on the AP Exam with limit notation: lim r(x) as x → ±∞.