Pre-Calculus: Rational Function Graphs

Graphing rational functions is a cornerstone of pre-calculus, bridging your algebraic skills with the visual intuition needed for calculus and engineering. These functions, representing real-world phenomena from circuit design to rate problems, are deceptively complex. Mastering their graphs requires a systematic approach to uncover their hidden features—asymptotes that act like boundaries, holes where points are missing, and intercepts that anchor the curve.

Understanding the Rational Function

A rational function is defined as the ratio of two polynomial functions. Its general form is $r(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero polynomial. The domain of $r(x)$ is all real numbers except those that make the denominator equal to zero. This exclusion is the source of the graph's most interesting behaviors: vertical asymptotes and holes. Before any sketching begins, you must completely factor both the numerator and the denominator. This factored form is not optional; it is essential for accurately identifying every key feature.

Finding Intercepts: Where the Graph Crosses the Axes

Intercepts provide concrete points to plot and help frame the rest of the graph. The y-intercept is found by evaluating the function at $x=0$, provided $0$ is in the domain. Simply compute $r(0)$. The x-intercepts (also called zeros or roots) occur where the function's value is zero. For a rational function $r(x)=\frac{P(x)}{Q(x)}$, this happens when the numerator $P(x)=0$ and the denominator is not zero at those same x-values. Solve $P(x)=0$ and check each solution against the denominator. For example, for $r(x)=\frac{(x-3)(x+1)}{(x-2)}$, the numerator is zero when $x=3$ and $x=-1$. Since neither makes the denominator zero, both are valid x-intercepts.

Identifying Vertical Asymptotes and Holes

This step relies entirely on the factored denominator. Set the factored denominator equal to zero and solve. Each solution represents a place where the function is undefined, but the type of discontinuity depends on whether the same factor is in the numerator.

A vertical asymptote occurs at $x=a$ if the factor $(x-a)$ is in the denominator but not in the numerator after the function is fully simplified. The graph will shoot toward positive or negative infinity as it approaches the line $x=a$ from the left or right. The function never touches or crosses this vertical line.

A hole (or removable discontinuity) occurs at $x=a$ if the factor $(x-a)$ is in both the numerator and the denominator. This common factor cancels out, indicating the function could be simplified to a new function that is defined at $x=a$, but the original $r(x)$ is not. To plot the hole, you must find its y-coordinate. Cancel the common factors to simplify the function to a new function $s(x)$. Then, evaluate $s(a)$ to find the y-coordinate of the hole at $x=a$. For instance, in $r(x)=\frac{(x-2)(x+3)}{(x-2)(x-5)}$, the factor $(x-2)$ cancels. There is a hole at $x=2$. To find its location, simplify to $s(x)=\frac{(x+3)}{(x-5)}$ and find $s(2)=\frac{5}{-3}=-\frac{5}{3}$. So, the hole is at the point $(2,-\frac{5}{3})$.

Determining End Behavior: Horizontal and Oblique Asymptotes

Asymptotes describe the graph's behavior as $x$ approaches positive or negative infinity. The type is determined by comparing the degrees of the numerator ($n$) and denominator ($m$).

1. Case 1: $n<m$. The horizontal asymptote is the line $y=0$ (the x-axis).
2. Case 2: $n=m$. The horizontal asymptote is the line $y=\frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
3. Case 3: $n>m$ by exactly 1. There is no horizontal asymptote. Instead, there is an oblique asymptote (or slant asymptote). To find it, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the oblique line, $y=mx+b$.
4. Case 4: $n>m$ by 2 or more. There is no horizontal or oblique asymptote. The end behavior will mimic the ratio of the leading terms, often producing a parabolic or higher-degree trend.

A critical note: A graph can cross a horizontal or oblique asymptote, but it will never cross a vertical asymptote. The horizontal/oblique asymptote describes behavior at extreme x-values, so crossings can occur in the middle of the graph.

The Sketching Process: Putting It All Together

With all features identified, you can construct an accurate sketch. Follow this workflow:

1. Factor Completely: Factor both the numerator and denominator of $r(x)$.
2. Find Intercepts: Calculate the y-intercept ($r(0)$) and x-intercepts (zeros of the simplified numerator).
3. Find Discontinuities: Solve denominator = 0.

• For factors unique to the denominator: Draw a dashed vertical asymptote line.
• For factors in both: Plot a hole as an open circle. Cancel the factor to simplify the function for later steps.

1. Determine End Behavior: Compare degrees to find the horizontal or oblique asymptote. Draw it as a dashed line.
2. Plot Additional Points & Sketch: Choose test points in the intervals created by your vertical asymptotes and x-intercepts. Evaluate $r(x)$ (using the simplified form where appropriate) to determine whether the graph is positive or negative in that interval. Connect your points, intercepts, and holes with a smooth curve, ensuring the graph approaches all asymptotes correctly.

Worked Example

Let's graph $r(x)=\frac{x^2-9}{x^2-x-6}$.

1. Factor: $r(x)=\frac{(x-3)(x+3)}{(x-3)(x+2)}$.
2. Intercepts:

• y-int: $r(0)=\frac{-9}{-6}=\frac{3}{2}$. Plot $(0,1.5)$.
• x-int: Numerator zeros: $x=3,x=-3$. But $(x-3)$ cancels, so $x=3$ is not an intercept (it will be a hole). Only $x=-3$ is an x-intercept. Plot $(-3,0)$.

1. Discontinuities: Denominator zeros: $x=3,x=-2$.

• $(x-3)$ cancels, so a hole exists at $x=3$. Simplified function: $s(x)=\frac{x+3}{x+2}$. Hole y-coordinate: $s(3)=\frac{6}{5}$. Plot hole at $(3,1.2)$.
• $(x+2)$ remains, so a vertical asymptote exists at $x=-2$.

1. End Behavior: Degrees are equal ($n=2,m=2$). Horizontal asymptote at $y=\frac{1}{1}=1$.
2. Sketch: Draw VA at $x=-2$, HA at $y=1$, hole at $(3,1.2)$, intercepts at $(-3,0)$ and $(0,1.5)$. Test points in intervals $(-\infty ,-3)$, $(-3,-2)$, $(-2,3)$, $(3,\infty )$ to determine if the graph is above or below the HA. Connect with smooth curves approaching all asymptotes.

Common Pitfalls

1. Confusing Holes and Vertical Asymptotes. The most common error is marking a vertical asymptote where a hole should be (or vice versa). Correction: Always factor first. If a factor cancels completely, it's a hole. If it remains in the denominator after simplification, it's a vertical asymptote.
2. Forgetting to Find the y-coordinate of a Hole. Simply stating "there's a hole at $x=2$" is incomplete for graphing. Correction: After canceling the common factor, substitute the x-value into the simplified function to get the precise location $(a,s(a))$ to plot.
3. Misapplying Horizontal Asymptote Rules. Students often try to use the horizontal asymptote formula when an oblique one exists. Correction: Let the degree comparison guide you. If the numerator's degree is exactly one greater, you must perform long division to find the oblique asymptote.
4. Ignoring the Function's Sign Between Asymptotes. Knowing where the graph is above or below the x-axis or its horizontal asymptote is crucial for accuracy. Correction: After finding asymptotes and intercepts, the number line is divided into intervals. Pick one simple test point in each interval and evaluate the sign of $r(x)$ (positive or negative) to know whether to sketch the curve in that region above or below the x-axis.

Summary

• A rational function is a ratio of polynomials, and its graph is characterized by intercepts, asymptotes, and holes.
• Vertical asymptotes occur at non-canceled factors of the denominator, while holes occur at canceled common factors, whose y-coordinate must be calculated using the simplified function.
• End behavior is dictated by comparing the degrees of numerator and denominator: leading to a horizontal asymptote ($y=0$ or $y=\frac{a}{b}$) or, if the numerator degree is one greater, an oblique asymptote found via polynomial long division.
• Successful graphing requires a strict sequence: factor completely, find all intercepts and discontinuities, determine end-behavior asymptotes, use test points to determine the graph's sign in each region, and finally sketch the curve while respecting all identified features.