Rational Function Characteristics

Why This Matters

Rational functions show up everywhere in Honors Algebra II—and they're the perfect testing ground for your ability to analyze function behavior. When you're working with $f(x) = \frac{P(x)}{Q(x)}$, you're really being tested on how well you understand domains and restrictions, asymptotic behavior, discontinuities, and end behavior. These concepts don't just apply to rational functions; they're foundational for calculus and advanced mathematics.

Here's the key insight: every characteristic of a rational function tells you something about the relationship between the numerator and denominator. Vertical asymptotes reveal where the denominator creates undefined values. Horizontal and slant asymptotes emerge from comparing polynomial degrees. Holes appear when factors cancel. Don't just memorize rules—understand why each feature exists based on the structure of the polynomials involved.

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Domain and Discontinuities

The denominator controls where a rational function can and cannot exist. Any value that makes $Q(x) = 0$ creates either a vertical asymptote or a hole, depending on whether the numerator shares that factor.

Domain and Restrictions

• Set $Q(x) = 0$ to find excluded values—these x-values make the function undefined and must be removed from the domain
• Express domain in interval notation by breaking the real number line at each restriction, using parentheses to show exclusion
• All other real numbers are valid inputs—the domain includes everything except your identified restrictions

Vertical Asymptotes

• Occur where $Q(x) = 0$ but $P(x) \neq 0$—the function approaches infinity near these x-values
• Find them by factoring the denominator and identifying roots that don't cancel with numerator factors
• Graph behavior: the function shoots toward $+\infty$ or $-\infty$ on either side of the asymptote

Holes (Removable Discontinuities)

• Occur where both $P(x) = 0$ and $Q(x) = 0$—this means a common factor exists in numerator and denominator
• Factor completely and cancel to simplify; the cancelled factor reveals the hole's x-coordinate
• Find the y-coordinate by plugging the x-value into the simplified function—graph this as an open circle

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Asymptotic Behavior and End Behavior

Asymptotes describe what happens at the extremes—either near restricted values or as x heads toward infinity. The degrees of $P(x)$ and $Q(x)$ determine which type of asymptote governs the function's long-run behavior.

Horizontal Asymptotes

• Compare degrees of numerator and denominator to determine the horizontal asymptote:
  • If $\deg(P) < \deg(Q)$: horizontal asymptote at $y = 0$
  • If $\deg(P) = \deg(Q)$: horizontal asymptote at $y = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q}$
  • If $\deg(P) > \deg(Q)$: no horizontal asymptote
• Describes behavior as $x \to \pm\infty$—the function levels off and approaches this y-value
• The graph can cross a horizontal asymptote in the middle of the graph, unlike vertical asymptotes

Slant (Oblique) Asymptotes

• Occur when $\deg(P) = \deg(Q) + 1$—the numerator is exactly one degree higher than the denominator
• Use polynomial long division of $P(x) \div Q(x)$; the quotient (ignoring remainder) is your slant asymptote
• Cannot have both a horizontal and slant asymptote—it's one or the other based on degree comparison

End Behavior

• Determined by leading terms of $P(x)$ and $Q(x)$—everything else becomes negligible as $|x|$ grows large
• Matches the asymptote type: approaches horizontal asymptote, follows slant asymptote, or grows without bound
• Use arrow notation to describe: as $x \to \infty$, $f(x) \to$ [asymptote value or $\pm\infty$]

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Intercepts and Key Points

Intercepts anchor your graph to specific coordinates. The numerator controls x-intercepts; evaluating at zero gives you the y-intercept.

X-Intercepts

• Set $P(x) = 0$ and solve—x-intercepts occur where the numerator equals zero (and denominator doesn't)
• Factor the numerator to find all zeros; each gives an x-intercept of the form $(a, 0)$
• Check that $Q(a) \neq 0$—if both equal zero, you have a hole, not an intercept

Y-Intercept

• Evaluate $f(0) = \frac{P(0)}{Q(0)}$—substitute zero for x in the entire function
• Only exists if $Q(0) \neq 0$—if the denominator equals zero at $x = 0$, there's no y-intercept
• Gives you the point $(0, f(0))$—a reliable anchor point for sketching your graph

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Symmetry and Graphing Strategy

Recognizing patterns in rational functions can dramatically simplify your analysis. Symmetry reduces your workload; a systematic graphing approach ensures accuracy.

Symmetry

• Even symmetry (y-axis): check if $f(-x) = f(x)$—graph is mirror image across the y-axis
• Odd symmetry (origin): check if $f(-x) = -f(x)$—graph has 180° rotational symmetry about the origin
• Most rational functions have neither—but when symmetry exists, you only need to analyze half the graph

Graphing Techniques

• Start with domain, asymptotes, and intercepts—these create the framework before plotting any curves
• Use test points between vertical asymptotes to determine if the function is positive or negative in each interval
• Connect the pieces respecting all asymptotes and discontinuities—the function approaches but never crosses vertical asymptotes

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Definition and Structure

Understanding what makes a function "rational" helps you recognize these functions and predict their behavior.

Definition of a Rational Function

• A ratio of two polynomials: $f(x) = \frac{P(x)}{Q(x)}$ where both $P(x)$ and $Q(x)$ are polynomials
• The denominator $Q(x)$ cannot be zero—this is what creates all the interesting behavior (asymptotes, holes, domain restrictions)
• Behavior depends on degree comparison—the relationship between $\deg(P)$ and $\deg(Q)$ determines asymptotes and end behavior

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Quick Reference Table

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Self-Check Questions

1. For $f(x) = \frac{(x-2)(x+3)}{(x-2)(x-5)}$, does $x = 2$ create a vertical asymptote or a hole? How do you determine which?

2. Compare the end behavior of $f(x) = \frac{3x^2 + 1}{x^2 - 4}$ and $g(x) = \frac{2x + 1}{x^2 - 4}$. What type of asymptote does each have, and why?

3. A rational function has vertical asymptotes at $x = -1$ and $x = 4$, and a horizontal asymptote at $y = 2$. What can you conclude about the degrees of the numerator and denominator?

4. How would you find the slant asymptote of $f(x) = \frac{x^2 + 3x - 2}{x - 1}$? What conditions must be met for a slant asymptote to exist?

5. If a rational function has no y-intercept, what must be true about its denominator? Give an example of such a function and identify all its other key characteristics.