🌌 Grade 11 · Lesson 3 of 12
Asymptote Navigator: Rational Functions and Discontinuities
Wherever the denominator hits zero the graph tears open into a vertical wall, and far from the origin the curve settles onto an invisible horizontal line it never quite reaches.
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The big idea: A rational function is one polynomial divided by another, and dividing by numbers close to zero is what makes it interesting. Where the denominator is zero the output blows up, creating a vertical asymptote — unless a matching factor in the numerator cancels, which instead punches a single removable hole. Far away from the origin the leading terms take over, pinning the graph to a horizontal (or slant) asymptote. Reading those pieces tells you the whole skeleton of the curve.
Dividing one polynomial by another
A rational function has the form f(x) = p(x) ÷ q(x), where p and q are polynomials and q is not the zero polynomial. The whole personality of these graphs comes from one fact: you can never divide by zero, so the x-values that make the denominator zero are forbidden — and near them the function does something dramatic.
\[ f(x) = \dfrac{p(x)}{q(x)}, \qquad q(x) \neq 0 \]
A rational function is a ratio of polynomials; its domain excludes every x that makes the denominator zero.
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Vertical asymptotes come from the denominator
Where the denominator is zero but the numerator is not, the fraction's size grows without bound — the graph shoots up toward +∞ or down toward −∞. That forbidden x-value is a vertical asymptote: an invisible vertical line the curve races toward but never touches.
Holes: when a factor cancels
There is one exception. If the same factor appears in both the top and bottom, it cancels, and instead of a vertical wall you get a single missing point — a removable discontinuity, or hole. The curve looks continuous there except for that one pinprick gap, because the x-value is still outside the domain.
So always factor first: a shared factor means a hole, a leftover denominator factor means a vertical asymptote.
🎮 Asymptote Navigator LIVE
Find vertical asymptotes where the denominator is zero and the horizontal asymptote far away.
Horizontal asymptotes: the view from far away
Zoom far out and the highest-degree terms of top and bottom dominate everything else. Comparing their degrees tells you where the graph levels off as x heads to ±∞.
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The degree comparison rule
Let n = degree of the numerator and d = degree of the denominator. If n < d, the horizontal asymptote is y = 0. If n = d, it is y = (ratio of leading coefficients). If n > d, there is no horizontal asymptote — and if n is exactly one more than d, the graph follows a slant (oblique) asymptote instead.
\[ y = \dfrac{a_n}{b_d} \quad \text{when } \deg p = \deg q \]
When top and bottom have equal degree, the horizontal asymptote is the ratio of their leading coefficients.
📝 Worked example: Find all asymptotes and holes of f(x) = (x² − 9) / (x² − 2x − 3).
- Factor both: numerator \( x^2 - 9 = (x-3)(x+3) \); denominator \( x^2 - 2x - 3 = (x-3)(x+1) \).
- The shared factor \( (x-3) \) cancels, so there is a hole at x = 3 (not an asymptote).
- The leftover denominator factor \( (x+1) \) gives a vertical asymptote at x = −1.
- Degrees are equal (2 and 2), so the horizontal asymptote is the ratio of leading coefficients: \( 1/1 = 1 \).
✓ Vertical asymptote <strong>x = −1</strong>, horizontal asymptote <strong>y = 1</strong>, and a <strong>hole at x = 3</strong>.
📝 Worked example: What is the horizontal asymptote of f(x) = (3x + 5) / (x² + 1)?
- Degree of numerator is 1; degree of denominator is 2.
- Since the numerator's degree is smaller (\( 1 < 2 \)), the denominator grows faster as \( x \to \pm\infty \).
- A larger, faster-growing denominator drives the fraction toward 0.
✓ The horizontal asymptote is <strong>y = 0</strong>.
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Cancel before you conclude
A common error is to declare a vertical asymptote at every zero of the original denominator. Always factor and cancel first: a factor shared with the numerator produces a hole, not an asymptote. And a graph may cross its horizontal asymptote in the middle — the “never touch” rule is only assured for the far tails, never for vertical asymptotes.
Check your understanding
1. A rational function has a denominator that is zero at x = 4, and the numerator is not zero there. What happens at x = 4?
Denominator zero with a nonzero numerator makes the value blow up — a vertical asymptote.
2. For f(x) = (2x² + 1) / (5x² − x), what is the horizontal asymptote?
Equal degrees (2 and 2), so the asymptote is the ratio of leading coefficients, 2/5.
3. In f(x) = (x − 5)(x + 2) / (x − 5)(x − 3), what occurs at x = 5?
The (x − 5) factor cancels top and bottom, leaving a removable hole at x = 5, not an asymptote.
4. For a rational function, when is the horizontal asymptote y = 0?
A smaller numerator degree means the denominator wins as x → ±∞, pushing the ratio toward 0.
5. If the numerator degree is exactly one more than the denominator degree, the graph has…
Degree one higher on top produces a slant asymptote, found by polynomial long division.
✅ Key takeaways
- A rational function is one polynomial divided by another; its domain excludes zeros of the denominator.
- A denominator zero with a nonzero numerator gives a vertical asymptote; a factor shared with the numerator gives a removable hole.
- Compare degrees for the horizontal asymptote: n < d → y = 0; n = d → ratio of leading coefficients; n > d → none (slant if n = d + 1).
- Always factor and cancel before classifying discontinuities.
- Graphs may cross a horizontal asymptote in the middle, but never cross a vertical asymptote.