5.3 Rational Function Graphs, Asymptotes, and Discontinuities

5.3 Rational Function Graphs, Asymptotes, and Discontinuities

[!NOTE] [!NOTE] Curricular Context & Standards Alignment: The Next Generation Algebra II blueprint lists F-IF.7(c, e) only - polynomial, exponential, logarithmic, and trigonometric graphing. Graphing rational functions is F-IF.7d, a plus standard, so you will not be asked to sketch a rational function on this examination.

The analysis in this section is still directly useful, because it is the machinery behind two standards that are tested. Excluded values and the difference between a canceling factor and a surviving factor are what make an extraneous root extraneous under AII-A.REI.2, and the quotient-plus-remainder rewriting that produces a slant asymptote is precisely AII-A.APR.6. Learn the structure; do not expect a graphing prompt.

A rational function is any function of the form: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero polynomial. The domain comprises all real numbers except the real roots of $Q(x)$. Graphing a rational function requires identifying where the curve breaks, how it behaves near those breaks, and where its values stabilize as $x$ increases or decreases without bound.

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1. Discontinuities: Removable Holes vs. Vertical Asymptotes

When an input value $x = c$ produces $Q(c) = 0$, the function is discontinuous at $x = c$. Discontinuities are partitioned into two distinct structural classifications based on whether the corresponding binomial factor cancels with the numerator.

Removable Discontinuities (Holes)

A removable discontinuity, commonly called a hole, occurs at $x = c$ if $(x - c)$ is a factor of both the numerator $P(x)$ and the denominator $Q(x)$.

• Algebraic Mechanism: Canceling $(x - c)$ eliminates the indeterminate form $\frac{0}{0}$.
• Determining Coordinates of the Hole: To find the exact coordinates of the hole:
  1. Divide out the common factor $(x - c)$ to produce the simplified function $f_{\text{simplified}}(x)$.
  2. Evaluate the simplified function at $x = c$ to obtain the y-coordinate: $\text{Hole Coordinates: } \left(c, \, f_{\text{simplified}}(c)\right)$
• Visual Representation: On a hand-drawn Regents graph or graphing display, a hole is plotted as an open circle ($\circ$) at those exact coordinates.

Non-Removable Discontinuities (Vertical Asymptotes)

A vertical asymptote is a vertical line $x = a$ where function values approach positive or negative infinity ($f(x) \to \pm\infty$) as $x$ approaches $a$ from the left or right.

• Algebraic Mechanism: A vertical asymptote occurs at any value $x = a$ where the denominator of the simplified rational function equals zero, while the numerator is nonzero ($\frac{k}{0}$ where $k \neq 0$).
• Equation Notation: Always write vertical asymptotes as linear equations: $x = a$. Writing solely "$a$" forfeits credit on Regents exams because an asymptote is a geometric line, not a scalar value.
• Uncrossable Barrier: A graph can never cross or touch a vertical asymptote because the function is strictly undefined at that input value.

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2. Horizontal Asymptotes, Slant Asymptotes, and End Behavior

While vertical asymptotes govern localized behavior near domain exclusions, horizontal and slant asymptotes describe the global end behavior of the function as $x \to \infty$ and $x \to -\infty$.

Let $P(x) = a_m x^m + \dots$ have degree $m$ and leading coefficient $a_m$, and let $Q(x) = b_n x^n + \dots$ have degree $n$ and leading coefficient $b_n$.

The Three End-Behavior Classifications

1. Case 1: Denominator Degree Greater ($m < n$, "Bottom-Heavy"):

   • As $x$ grows large, the denominator outpaces the numerator, driving values toward zero.
   • Horizontal Asymptote: The line $y = 0$ (the x-axis).
   • End behavior: $\lim_{x \to \pm\infty} f(x) = 0$.

2. Case 2: Equal Degrees ($m = n$, "Balanced"):

   • The leading terms grow at identical rates, rendering lower-degree terms negligible.
   • Horizontal Asymptote: The line $y = \frac{a_m}{b_n}$ (the ratio of leading coefficients).
   • End behavior: $\lim_{x \to \pm\infty} f(x) = \frac{a_m}{b_n}$.

3. Case 3: Numerator Degree Greater ($m > n$, "Top-Heavy"):

   • The numerator outgrows the denominator, so function values diverge to $\pm\infty$. There is no horizontal asymptote.
   • Oblique (Slant) Asymptote ($m = n + 1$): When the numerator degree is exactly one greater than the denominator degree, the graph approaches a slanted line $y = mx + b$ as $x \to \pm\infty$.
   • Perform polynomial long division or synthetic division: $\frac{P(x)}{Q(x)} = (mx + b) + \frac{R(x)}{Q(x)}$
   • Because the remainder fraction $\frac{R(x)}{Q(x)} \to 0$ as $x \to \pm\infty$, the slant asymptote is the linear quotient $y = mx + b$.

[!TIP] Crossing Asymptotes: Students often assume graphs can never cross any asymptote. While a graph can never cross a vertical asymptote (where $x$ is undefined), a graph can cross a horizontal or slant asymptote in the interior of its domain! Horizontal asymptotes govern only end behavior ($x \to \pm\infty$), not local behavior.

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3. Discontinuity and Asymptote Classification Guide

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4. Behavior Near Vertical Asymptotes and Sign Analysis

To sketch rational functions accurately without a graphing calculator, use a sign chart around critical values:

1. Plot all real zeros of the simplified numerator and denominator on a number line to establish test intervals.
2. Pick a test point in each interval to determine whether the function is positive ($f(x) > 0$) or negative ($f(x) < 0$).
3. Multiplicity Rules at Vertical Asymptotes:
   • If the factor $(x - a)$ has an odd multiplicity (such as $(x - a)^1$), the function changes sign across the asymptote: one branch approaches $+\infty$ while the opposite branch plunges to $-\infty$.
   • If the factor $(x - a)$ has an even multiplicity (such as $(x - a)^2$), the function maintains the same sign on both sides: both branches head toward $+\infty$ or both head toward $-\infty$.

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5. Worked Regents-Style Examples

Example 1: Comprehensive Rational Function Analysis

Problem: Analyze $f(x) = \frac{2x^2 - 8}{x^2 - x - 6}$. Determine all discontinuities, asymptotes, and intercepts.

• Step 1: Factor numerator and denominator: $f(x) = \frac{2(x - 2)(x + 2)}{(x - 3)(x + 2)}$
• Step 2: Identify hole: Common factor $(x + 2)$ cancels $\implies$ hole at $x = -2$. Simplified function: $f_{\text{simp}}(x) = \frac{2(x - 2)}{x - 3}$. $y\text{-coordinate}: f_{\text{simp}}(-2) = \frac{2(-2 - 2)}{-2 - 3} = \frac{2(-4)}{-5} = \frac{8}{5} = 1.6$. Hole coordinates: $(-2, 1.6)$.
• Step 3: Vertical asymptote: Remaining denominator factor $(x - 3) = 0 \implies$ line $x = 3$.
• Step 4: Horizontal asymptote: Both degrees equal 2 ($m = n = 2$). $y = \frac{2}{1} = 2 \implies \text{Line } y = 2$
• Step 5: Intercepts:
  • $x$-intercept: Set simplified numerator to zero: $2(x - 2) = 0 \implies (2, 0)$.
  • $y$-intercept: Evaluate at $x = 0$: $f(0) = \frac{-8}{-6} = \frac{4}{3} \implies \left(0, \frac{4}{3}\right)$.

Example 2: Determining a Slant Asymptote

Problem: Determine the equation of the slant asymptote for $g(x) = \frac{x^2 + 3x - 1}{x - 1}$.

• Step 1: Compare polynomial degrees: Numerator degree is 2 and denominator degree is 1 ($2 = 1 + 1$). A slant asymptote exists.
• Step 2: Perform polynomial long division: Divide $x^2 + 3x - 1$ by $x - 1$: $\frac{x^2 + 3x - 1}{x - 1} = (x + 4) + \frac{3}{x - 1}$
• Step 3: Formulate asymptote equation: As $x \to \pm\infty$, $\frac{3}{x - 1} \to 0$. Slant Asymptote: $y = x + 4$.

Example 3: Sign Analysis Near Asymptotes

Problem: Analyze the behavior of $h(x) = \frac{x - 1}{(x - 3)^2}$ as $x \to 3^-$ and $x \to 3^+$.

• Step 1: Evaluate the numerator at $x = 3$: $3 - 1 = 2 > 0$ (strictly positive).
• Step 2: Evaluate denominator signs near $x = 3$:
  • Because $(x - 3)^2$ is squared, it is strictly positive for all $x \neq 3$.
  • As $x \to 3^-$: $\frac{\text{positive}}{\text{small positive}} \to +\infty$.
  • As $x \to 3^+$: $\frac{\text{positive}}{\text{small positive}} \to +\infty$.
  • Conclusion: Both branches ascend toward $+\infty$ due to the even multiplicity of $(x - 3)^2$.