Asymptote Analysis

A rational function has the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials. Unlike polynomial functions, rational functions can have breaks, gaps, and lines they approach but never reach. Understanding these features — vertical asymptotes, horizontal asymptotes, and holes — is the key to analyzing rational function behavior.

Vertical Asymptotes

A vertical asymptote is a vertical line $x = a$ where the function values grow without bound as $x$ approaches $a$. To find vertical asymptotes:

1. Factor both the numerator and denominator completely
2. Cancel any common factors (these create holes, not asymptotes)
3. Set the remaining denominator equal to zero and solve

The zeros of the denominator that are not cancelled by the numerator produce vertical asymptotes.

Example 1: Find the vertical asymptotes of $f(x) = \frac{x^2 - 1}{x^2 - 3x + 2}$.

Factor both:

$f(x) = \frac{(x - 1)(x + 1)}{(x - 1)(x - 2)}$

The factor $(x - 1)$ cancels, leaving:

$f(x) = \frac{x + 1}{x - 2}, \quad x \neq 1$

The remaining denominator is zero when $x = 2$. So $x = 2$ is a vertical asymptote. The point $x = 1$ is a hole (discussed below), not a vertical asymptote.

Behavior Near Vertical Asymptotes

Near a vertical asymptote, the function goes to $+\infty$ or $-\infty$ on each side. Sign analysis determines which direction.

For $f(x) = \frac{x + 1}{x - 2}$ near $x = 2$:

• As $x \to 2^-$ (from the left): numerator $\to 3$ (positive), denominator $\to 0^-$ (small negative), so $f(x) \to -\infty$
• As $x \to 2^+$ (from the right): numerator $\to 3$ (positive), denominator $\to 0^+$ (small positive), so $f(x) \to +\infty$

This tells you the graph drops to $-\infty$ on the left side of $x = 2$ and rises to $+\infty$ on the right side.

Multiplicity Matters

If the remaining denominator factor has an even multiplicity, the function goes the same direction on both sides of the asymptote. If it has an odd multiplicity, the function goes opposite directions.

• $(x - 2)^1$ in the denominator: odd multiplicity, so opposite signs on each side
• $(x - 2)^2$ in the denominator: even multiplicity, so same sign on each side

Horizontal Asymptotes

A horizontal asymptote is a horizontal line $y = L$ that the function approaches as $x \to \pm\infty$. The rule depends entirely on comparing the degree of the numerator ($n$) to the degree of the denominator ($m$):

Example 2: Find the horizontal asymptote of $g(x) = \frac{3x^2 + 5x - 1}{2x^2 - 7}$.

Both numerator and denominator have degree 2, so $n = m$. The leading coefficients are 3 and 2.

$\text{Horizontal asymptote: } y = \frac{3}{2}$

Example 3: Find the horizontal asymptote of $h(x) = \frac{4x + 1}{x^3 + 2}$.

The numerator has degree 1, the denominator has degree 3. Since $n < m$:

$\text{Horizontal asymptote: } y = 0$

Crossing a Horizontal Asymptote

Unlike vertical asymptotes, a function can cross its horizontal asymptote. The horizontal asymptote describes end behavior — what happens far out on the $x$-axis — but in the middle of the graph, the function may cross the line $y = L$.

To find where $f(x)$ crosses its horizontal asymptote $y = L$, solve $f(x) = L$.

Example: For $f(x) = \frac{2x^2 + x}{x^2 + 1}$ the horizontal asymptote is $y = 2$. Setting $f(x) = 2$:

$\frac{2x^2 + x}{x^2 + 1} = 2 \implies 2x^2 + x = 2x^2 + 2 \implies x = 2$

So the function crosses its horizontal asymptote at $x = 2$.

Holes (Removable Discontinuities)

A hole occurs at $x = a$ when both the numerator and denominator have $(x - a)$ as a factor. After cancellation, the function is defined everywhere except $x = a$, but the graph has a missing point.

To find the $y$-coordinate of a hole, substitute $x = a$ into the simplified (cancelled) function.

Example 4: Find the hole in $f(x) = \frac{x^2 - 1}{x^2 - 3x + 2}$ (from Example 1).

We found the common factor $(x - 1)$, giving a hole at $x = 1$. Substituting into the simplified form:

$f(x) = \frac{x + 1}{x - 2} \implies f(1) = \frac{1 + 1}{1 - 2} = \frac{2}{-1} = -2$

The hole is at the point $(1, -2)$.

Complete Asymptote Analysis Strategy

For any rational function $f(x) = \frac{P(x)}{Q(x)}$:

1. Factor numerator and denominator completely
2. Identify common factors — each gives a hole; find the $y$-value from the simplified function
3. Cancel common factors to get the simplified form
4. Vertical asymptotes: set the remaining denominator to zero
5. Horizontal asymptote: compare degrees ($n$ vs $m$)
6. Sign analysis: test a value on each side of every vertical asymptote to determine up/down behavior

Example 5 (Complete Analysis): Analyze $f(x) = \frac{2x^2 - 2}{x^3 - x^2 - 2x}$.

Step 1 — Factor:

$f(x) = \frac{2(x^2 - 1)}{x(x^2 - x - 2)} = \frac{2(x - 1)(x + 1)}{x(x - 2)(x + 1)}$

Step 2 — Common factors: $(x + 1)$ cancels. Hole at $x = -1$.

Hole $y$-value: $\frac{2(-1 - 1)}{(-1)(-1 - 2)} = \frac{2(-2)}{(-1)(-3)} = \frac{-4}{3}$

Hole: $(-1, -\frac{4}{3})$

Step 3 — Simplified form:

$f(x) = \frac{2(x - 1)}{x(x - 2)}, \quad x \neq -1$

Step 4 — Vertical asymptotes: $x = 0$ and $x = 2$

Step 5 — Horizontal asymptote: Original numerator degree 2, denominator degree 3. Since $n < m$, horizontal asymptote is $y = 0$.

Step 6 — Sign analysis near $x = 0$:

• $x \to 0^-$: numerator $\approx 2(-1) = -2$, denominator $\approx (0^-)(0^- - 2) = (0^-)(-2) = 0^+$, so $f(x) \to -\infty$
• $x \to 0^+$: numerator $\approx -2$, denominator $\approx (0^+)(-2) = 0^-$, so $f(x) \to +\infty$

Real-World Application: Drug Concentration

In pharmacology, a drug’s concentration in the bloodstream is often modeled by a rational function. For example:

$C(t) = \frac{50t}{t^2 + 4}$

where $C$ is the concentration in mg/L and $t$ is time in hours after administration.

• Horizontal asymptote: Degree of numerator (1) is less than degree of denominator (2), so $y = 0$. This means the drug concentration approaches zero as time increases — the body metabolizes the drug.
• No vertical asymptotes: $t^2 + 4 = 0$ has no real solutions, so the concentration is defined for all $t \geq 0$.
• Peak concentration: Occurs at $t = 2$ hours (found by calculus or by testing values), where $C(2) = \frac{100}{8} = 12.5$ mg/L.

This type of model helps nurses and pharmacists determine dosing schedules.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Vertical asymptotes occur at zeros of the denominator after cancelling common factors with the numerator
• Horizontal asymptotes depend on comparing degrees: $n < m$ gives $y = 0$, $n = m$ gives $y = \frac{a_n}{b_m}$, and $n > m$ gives no horizontal asymptote
• Holes occur where a factor cancels between numerator and denominator — find the $y$-value by substituting into the simplified form
• The multiplicity of a denominator factor determines whether the function goes in the same or opposite directions on each side of the asymptote
• Functions can cross horizontal asymptotes in their interior — the asymptote only describes end behavior

Return to College Algebra for more topics in this section.