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Mom's Basement University

Rational Functions and Asymptotes

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to identify vertical and horizontal asymptotes of rational functions and use them to sketch accurate graphs. You saw rational expressions in Algebra 2. Here we focus on graphing and behavior analysis.

The Concept

Section titled “The Concept”

A rational function is a ratio of two polynomials:

f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}

where P(x) and Q(x) are polynomials and Q(x) cannot equal zero.

The interesting behavior happens near the zeros of the denominator and as x gets very large. That’s where asymptotes come in.

Vertical asymptotes occur where the denominator is zero (and the numerator isn’t zero at that same point). The graph shoots toward infinity or negative infinity near these values.

Horizontal asymptotes describe the long-run behavior as x heads toward infinity. The rule depends on comparing the degrees of the numerator and denominator:

  • Numerator degree less than denominator degree: horizontal asymptote at y = 0
  • Degrees equal: horizontal asymptote at y = (leading coefficient of numerator) / (leading coefficient of denominator)
  • Numerator degree greater: no horizontal asymptote (possibly a slant asymptote)

Holes (removable discontinuities) occur when a factor cancels from both numerator and denominator. The graph has a gap at that point but no asymptote.

Worked Example

Section titled “Worked Example”

Example 1: Full analysis

Analyze and sketch f(x) = (x² - 4) / (x² - 3x - 4).

Factor completely:

f(x)=(x−2)(x+2)(x−4)(x+1)f(x) = \frac{(x - 2)(x + 2)}{(x - 4)(x + 1)}

No common factors, so no holes.

Vertical asymptotes: x = 4 and x = -1 (denominator zeros).

Horizontal asymptote: both numerator and denominator are degree 2, so y = 1/1 = 1.

x-intercepts: x = 2 and x = -2 (numerator zeros).

y-intercept: f(0) = (0 - 4) / (0 - 0 - 4) = -4 / -4 = 1.

The graph has three separate pieces (split by the two vertical asymptotes), approaches y = 1 on both ends, and crosses the x-axis at x = -2 and x = 2.

Example 2: Function with a hole

Analyze g(x) = (x² - 9) / (x² - x - 6).

Factor:

g(x)=(x−3)(x+3)(x−3)(x+2)g(x) = \frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}

The factor (x - 3) cancels, creating a hole at x = 3. After canceling:

g(x)=x+3x+2,x≠3g(x) = \frac{x + 3}{x + 2}, \quad x \neq 3

Vertical asymptote: x = -2. Horizontal asymptote: y = 1 (degrees equal, leading coefficients both 1). Hole at x = 3, where g(3) = 6/5 = 1.2.

Example 3: Numerator degree less than denominator

h(x)=5x2+1h(x) = \frac{5}{x^2 + 1}

No vertical asymptotes (x² + 1 is never zero). Horizontal asymptote: y = 0 (numerator degree 0 is less than denominator degree 2). The graph is a bell-shaped curve centered at x = 0 with maximum value h(0) = 5, approaching 0 as x gets large.

Real-World Application

Section titled “Real-World Application”

Rational functions model many real situations:

  • Average cost per unit: C(x) = (fixed costs + variable costs) / x approaches the variable cost per unit as x grows
  • Drug concentration: the amount of drug in the bloodstream over time often follows a rational model
  • Electrical resistance: parallel resistors combine as 1/R = 1/R₁ + 1/R₂
  • Population models with carrying capacity (logistic-like behavior)
  • Dilution problems: mixing solutions changes concentration as a rational function of volume added

Quiz

Section titled “Quiz”
Vertical asymptotes occur where:
For $f(x) = \frac{2x^2 + 3}{x^2 - 5x + 6}$, the horizontal asymptote is:
If a factor cancels from both numerator and denominator, it creates:
If the numerator degree is less than the denominator degree, the horizontal asymptote is:
The first step when analyzing a rational function is: