AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.7B — Rational Functions and End Behavior
Notes — Deeper End Behavior, Sign of the Limit, and Slant Previews
💡 Learning Objectives By the end of this lesson you will be able to (AP CED 1.7.A):
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1. Quick Recap of the Three Cases
In 1.7A we sorted rational functions by comparing the degree of the numerator (n) to the degree of the denominator (m). Case 1 (n < m) gives horizontal asymptote y = 0; Case 2 (n = m) gives y = aₙ/bₘ; Case 3 (n > m) gives no horizontal asymptote. In this lesson we refine that picture: we determine whether the outputs approach +∞ or −∞, and we look at the special sub-case when n = m + 1.
💡 Quick Check Before reading further, test yourself: for each function, just name the case (1, 2, or 3).
Answers: Case 1 ; Case 2 ; Case 3. |
2. Sign of the End Behavior in Case 3
When numerator wins (deg N > deg D), there is no horizontal asymptote — but we still need to say whether the graph shoots up to +∞ or down to −∞ on each end. The answer comes from looking at the ratio of the leading terms alone.
📘 Example Example 1 — Determining the sign of the end behavior Describe the end behavior of r(x) = (−2x³ + x) / (x² − 4). Leading-term ratio: (−2x³) / (x²) = −2x. As x → +∞, −2x → −∞. As x → −∞, −2x → +∞. So lim r(x) as x → +∞ = −∞ and lim r(x) as x → −∞ = +∞. The graph behaves like the line y = −2x at the far ends — slanted downward to the right, upward to the left. |
⚠️ Common Mistake Do not forget the leading-coefficient sign in Case 3. A negative leading-coefficient ratio flips the end behavior upside-down. The shape of x² and −x² are mirror images; the same logic applies here. |
3. A More Rigorous Method: Factoring Out the Leading Power
Here is why the three rules work. Take r(x) = (3x² + 5x + 1) / (2x² − 4). Divide numerator and denominator by x² (the highest power in the denominator).
r(x) = (3 + 5/x + 1/x²) / (2 − 4/x²). As x → ±∞, every term of the form (constant / xᵏ) approaches 0. So r(x) → 3/2. The three cases are simply what happens to this limit: Case 1 the numerator vanishes, Case 2 a finite nonzero value appears, Case 3 the numerator grows without bound.
📘 Try This Use the factoring-out method (divide top and bottom by the highest power of x) to verify each limit.
Each calculation confirms the degree rule without requiring you to memorize it. |
4. When n = m + 1 — A Slant Preview
Case 3 splits into two sub-cases. When the numerator's degree is exactly one more than the denominator's (n = m + 1), the leading-term ratio is a linear expression ax + b, and the graph has a slant (oblique) asymptote y = ax + b. We study how to find a and b using polynomial long division in Topic 1.11B — for now, simply recognize when a slant exists.
📘 Example Example 2 — Recognizing a slant r(x) = (x² − x − 2) / (x − 1). n = 2, m = 1, so n = m + 1 ✓. The graph has a slant asymptote. The exact line is found in 1.11B by dividing. |
5. End Behavior in Context
Rational functions often model real-world quantities like average cost, concentration, or population density. The horizontal asymptote in these models has a meaningful interpretation — the long-run steady value.
📘 Example Example 3 — Interpreting a horizontal asymptote The average cost of producing x items is C(x) = (50x + 1200) / x dollars per item. What does the end behavior tell us? Rewrite as C(x) = 50 + 1200/x. As x → +∞, C(x) → 50. So the horizontal asymptote y = 50 means the long-run average cost per item approaches $50 as production scales up. Fixed costs are spread over more items. |
🎯 AP Tip When an applied problem asks about long-run behavior, name the horizontal asymptote AND interpret it in context with appropriate units. 'As production grows, the average cost per item approaches $50' is a complete response; 'HA at y = 50' is not. |
6. Summary
- In Case 3, the sign of the leading-coefficient ratio determines whether outputs head to +∞ or −∞.
- Factoring out the highest power of x makes every rule provable, not just memorized.
- When n = m + 1, the graph has a slant asymptote instead of a horizontal one.
- In applied problems, the horizontal asymptote is the long-run equilibrium value — interpret it with units.