Key Concepts of Critical Rational Functions

Why This Matters

Rational functions are the workhorses of AP Pre-Calculus—they combine everything you've learned about polynomials and push you to think about function behavior in new ways. You're being tested on your ability to analyze asymptotic behavior, discontinuities, and end behavior all at once. These concepts form the foundation for understanding limits in calculus and appear constantly in real-world modeling scenarios, from physics to economics.

Here's the key insight: rational functions are really about what happens when polynomials divide. The relationship between the numerator and denominator degrees determines everything—asymptotes, end behavior, and overall shape. Don't just memorize rules about asymptotes; understand why the degree comparison matters. When you see a rational function on the exam, your first move should always be comparing degrees and factoring to find common factors.

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Asymptotic Behavior: What Happens at the Extremes

Asymptotes tell you where a function can't go or where it's heading. The key principle: asymptotes arise from division by zero (vertical) or from the limiting ratio of leading terms (horizontal/slant).

Vertical Asymptotes

• Occur where $Q(x) = 0$ but $P(x) \neq 0$—these are true discontinuities where the function shoots toward $\pm\infty$
• Multiplicity matters for behavior—odd multiplicity means the function changes sign across the asymptote; even multiplicity means it stays the same sign
• Find them by factoring the denominator and checking that numerator factors don't cancel at those zeros

Horizontal Asymptotes

• Determined by comparing degrees of $P(x)$ and $Q(x)$—this is the most testable concept for rational functions
• 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
• Describes end behavior as $x \to \pm\infty$, not behavior near the origin

Slant (Oblique) Asymptotes

• Occur when degree of $P(x)$ is exactly one more than degree of $Q(x)$—no other degree difference produces a slant asymptote
• Found using polynomial long division—the quotient (ignoring remainder) is your slant asymptote equation
• Function approaches this line as $x \to \pm\infty$, making it a linear end-behavior model

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Discontinuities: Where Functions Break

Not all "problem points" are created equal. Discontinuities occur where the denominator equals zero, but the type depends on whether the numerator shares that zero.

Holes (Removable Discontinuities)

• Occur where both $P(x) = 0$ and $Q(x) = 0$—a common factor cancels, leaving a "hole" in the graph
• Find the location by setting the common factor equal to zero—the $x$-value is where the hole appears
• Find the $y$-coordinate by evaluating the simplified function at the hole's $x$-value (after canceling the common factor)

Domain Restrictions

• Domain excludes all $x$-values where $Q(x) = 0$—both vertical asymptotes AND holes are excluded from the domain
• Write domain in interval notation by removing all denominator zeros from $(-\infty, \infty)$
• Range requires deeper analysis—use asymptotes, intercepts, and function behavior to determine which $y$-values are achievable

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Intercepts and Key Points

Intercepts anchor your graph to the coordinate axes. Finding intercepts requires setting appropriate values to zero and solving.

X-Intercepts

• Found by solving $P(x) = 0$—the numerator's zeros become the function's zeros (as long as they don't cancel with denominator factors)
• Multiplicity affects crossing behavior—odd multiplicity crosses the axis, even multiplicity touches and turns back
• Check that your zeros aren't holes—if a numerator zero matches a denominator zero, it's a hole, not an intercept

Y-Intercept

• Found by evaluating $f(0) = \frac{P(0)}{Q(0)}$—simply plug in $x = 0$
• Only exists if $Q(0) \neq 0$—if the denominator equals zero at $x = 0$, there's no $y$-intercept
• Quick calculation for graphing—often the easiest point to plot when sketching

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Graphing Strategy: Putting It All Together

Graphing rational functions requires a systematic approach. The key is gathering all critical features before sketching.

Step-by-Step Graphing Process

• Start with asymptotes and holes—these create the "skeleton" that guides your curve; sketch vertical asymptotes as dashed lines
• Plot intercepts next—x-intercepts from numerator zeros, y-intercept from $f(0)$
• Use sign analysis between critical x-values—determine whether the function is positive or negative in each region

End Behavior Analysis

• Horizontal/slant asymptotes guide the "wings"—as $x \to \pm\infty$, the function approaches these lines
• Leading coefficients determine direction—positive leading ratio means function approaches asymptote from above on the right
• Test points at extreme values if you're unsure—plug in $x = 100$ or $x = -100$ to check behavior

Critical Points and Turning Points

• Critical points occur where $f'(x) = 0$ or undefined—these indicate potential local maxima or minima
• Turning points show direction changes—analyze sign of derivative to determine if function is increasing or decreasing
• Not always tested in Pre-Calc—but understanding that rational functions can have local extrema helps with sketching accuracy

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Quick Reference Table

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Self-Check Questions

1. Given $f(x) = \frac{(x-2)(x+3)}{(x-2)(x-5)}$, identify all vertical asymptotes and holes. What's the difference in how each appears on the graph?

2. Compare the end behavior of $f(x) = \frac{3x^2 + 1}{x^2 + 4}$ and $g(x) = \frac{x + 1}{x^2 + 4}$. Which has a horizontal asymptote at $y = 0$, and why?

3. A rational function has a vertical asymptote at $x = 3$ with even multiplicity. Describe the function's behavior as $x$ approaches 3 from both sides.

4. How would you find the slant asymptote of $f(x) = \frac{x^3 - 2x + 1}{x^2 + 1}$? Write out the first step and explain why this function has a slant asymptote rather than a horizontal one.

5. (FRQ-style) A rational function $r(x)$ has x-intercepts at $x = -1$ and $x = 4$, a vertical asymptote at $x = 2$, and a horizontal asymptote at $y = 1$. Write a possible equation for $r(x)$ and justify each component of your answer.