Rational Functions Characteristics

Why This Matters

Rational functions show up everywhere in Algebra 2—from modeling real-world scenarios to serving as the foundation for calculus concepts you'll encounter later. You're being tested on your ability to analyze these functions systematically: finding where they're undefined, predicting their long-term behavior, and identifying the features that shape their graphs. The key concepts here—domain restrictions, asymptotic behavior, discontinuities, and intercepts—form a toolkit you'll use repeatedly on exams.

Here's the thing: rational functions aren't random. Every characteristic connects back to the relationship between the numerator and denominator polynomials. The degrees of these polynomials determine asymptotes, shared factors create holes, and zeros of each polynomial give you intercepts and restrictions. Don't just memorize procedures—understand why each feature exists based on the function's structure, and you'll be able to tackle any rational function problem thrown at you.

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Building the Foundation: Definition and Domain

Before you can analyze any rational function, you need to understand what you're working with and where the function actually exists. The denominator controls everything about restrictions.

Definition of a Rational Function

• A rational function is the ratio of two polynomials—written as $f(x) = \frac{P(x)}{Q(x)}$ where both $P(x)$ and $Q(x)$ are polynomials
• The numerator $P(x)$ controls zeros; the denominator $Q(x)$ controls restrictions and vertical behavior
• Behavior depends on the degrees of $P(x)$ and $Q(x)$—this relationship determines asymptotes and end behavior

Domain and Restrictions

• The domain excludes all x-values where $Q(x) = 0$—these create either vertical asymptotes or holes
• Set the denominator equal to zero and solve to find all restricted values; factor completely to catch all restrictions
• Express domain in interval notation—for example, if $x = 2$ is restricted, write $(-\infty, 2) \cup (2, \infty)$

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Asymptotic Behavior: What Happens at Extremes

Asymptotes describe where the function wants to go but can never reach. These invisible boundary lines are determined entirely by the polynomial structure.

Vertical Asymptotes

• Occur at x-values where $Q(x) = 0$ and the factor doesn't cancel with $P(x)$
• The function approaches $\pm\infty$ as x approaches the asymptote from either side—check signs to determine direction
• Write as a vertical line equation $x = a$ where $a$ is the restricted value

Horizontal Asymptotes

• Compare the degrees of numerator and denominator to find horizontal asymptotes as $x \to \pm\infty$
• Three cases to memorize: degree of $P <$ degree of $Q$ gives $y = 0$; equal degrees give $y = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q}$; degree of $P >$ degree of $Q$ means no horizontal asymptote
• The function approaches but may cross a horizontal asymptote—unlike vertical asymptotes, crossing is allowed for finite x-values

Slant (Oblique) Asymptotes

• Occur when degree of $P(x)$ is exactly one more than degree of $Q(x)$—no horizontal asymptote exists in this case
• Perform polynomial long division of $P(x) \div Q(x)$; the quotient (ignoring remainder) is the slant asymptote equation
• The function approaches this diagonal line as $x \to \pm\infty$, giving the graph a tilted end behavior

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Discontinuities and Key Points

These features tell you where the function breaks, crosses axes, or has special behavior. Finding intercepts and holes requires solving equations with the numerator and denominator.

Holes (Removable Discontinuities)

• A hole occurs when $P(x)$ and $Q(x)$ share a common factor that cancels out completely
• Factor both polynomials fully to identify shared factors; the x-value that makes the shared factor zero is the hole location
• Find the y-coordinate by substituting the x-value into the simplified function—this gives the hole's position on the graph

X-Intercepts and Y-Intercept

• X-intercepts occur where $P(x) = 0$ (numerator equals zero) and $Q(x) \neq 0$—set numerator to zero and solve
• Y-intercept is $f(0)$, found by substituting $x = 0$ into the function, provided $Q(0) \neq 0$
• These intercepts anchor your graph—plot them first along with asymptotes before sketching curves

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Analyzing Overall Behavior

These characteristics help you see the big picture of how the function behaves across its entire domain.

End Behavior

• End behavior describes $f(x)$ as $x \to \pm\infty$—determined by the leading terms of $P(x)$ and $Q(x)$
• Simplify to leading terms only: $\frac{ax^m}{bx^n}$ behaves like $\frac{a}{b}x^{m-n}$ for large $|x|$
• Connects directly to horizontal/slant asymptotes—the asymptote equation is the end behavior

Symmetry

• Even symmetry means $f(-x) = f(x)$—the graph mirrors across the y-axis
• Odd symmetry means $f(-x) = -f(x)$—the graph has 180° rotational symmetry about the origin
• Test by substituting $-x$ into the function and simplifying; if neither condition holds, the function has no symmetry

Graphing Techniques

• Start with domain, asymptotes, and intercepts—these create the framework before you draw any curves
• Use test points in each interval created by vertical asymptotes to determine if the function is positive or negative
• Plot holes as open circles and connect behavior smoothly, respecting all asymptotes and intercepts

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Quick Reference Table

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Self-Check Questions

1. For $f(x) = \frac{(x-3)(x+2)}{(x-3)(x-1)}$, identify which feature occurs at $x = 3$ and which occurs at $x = 1$. How do you determine the difference?

2. Compare the end behavior of $f(x) = \frac{2x^2 + 1}{x^2 - 4}$ and $g(x) = \frac{x + 1}{x^2 - 4}$. What horizontal asymptote does each have, and why?

3. A rational function has a vertical asymptote at $x = -2$ and a hole at $x = 4$. Write a possible equation for this function and explain how the structure creates each feature.

4. How would you determine whether $f(x) = \frac{x^3}{x^2 + 1}$ has a horizontal asymptote, slant asymptote, or neither? What is the asymptote equation?

5. Compare and contrast finding x-intercepts versus finding vertical asymptotes. Both involve setting something equal to zero—what's different about what you're solving, and what does each tell you about the graph?