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):
- Determine the sign of the end behavior (does r approach +∞ or −∞?).
- Explain end behavior by rewriting a rational function with the leading terms factored out.
- Recognize when a rational function has a slant asymptote (a preview of 1.11B).
- Interpret end behavior in an applied / contextual setting.
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).
- r(x) = (x² − 1) / (x³ + 1) → Case ___
- r(x) = (−2x⁴ + 5) / (3x⁴ − x) → Case ___
- r(x) = (x⁵ + 2x) / (x + 1) → Case ___
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.
- r(x) = (6x² + 1) / (2x² − 3x + 1), show lim r(x) as x → ∞ = 3.
- r(x) = (x + 100) / (x² + 1), show lim r(x) as x → ∞ = 0.
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.