Graphs of Rational and Quotient Functions

A rational function is defined as the ratio of two polynomials, $P(x)$ and $Q(x)$, such that:
$$f(x) = \frac{P(x)}{Q(x)}$$
The behavior of these functions is characterized by their asymptotes, intercepts, and stationary points. Understanding the relationship between the degrees of $P(x)$ and $Q(x)$ is essential for sketching their graphs.

1. Key Features of Rational Graphs

To sketch a comprehensive graph of a rational function, the following features must be identified and labeled:

• Vertical Asymptotes: Occur at values of $x$ where the denominator $Q(x) = 0$ (provided $P(x) \neq 0$ at those points).
• Non-Vertical Asymptotes: These describe the end behavior of the function as $x \to \pm\infty$. They can be horizontal or oblique (linear).
• Axis Intercepts:
  • $y$-intercept: Evaluate $f(0)$.
  • $x$-intercepts: Solve $P(x) = 0$.
• Stationary Points: Use the quotient rule to find $f’(x) = 0$ to identify local maxima, minima, or stationary points of inflection.
• Points of Inflection: Solve $f’‘(x) = 0$ and check for a change in concavity.

EXAM TIP: VCAA assessors require all asymptotes to be labeled with their equations (e.g., $x = 2$ or $y = 3x - 1$) and all intercepts to be labeled with their coordinates (e.g., $(0, -4)$).

2. Determining Asymptotes

Vertical Asymptotes

A vertical asymptote $x = a$ occurs if the function is undefined at $x = a$ and the limit of the function approaches $\pm\infty$ as $x$ approaches $a$.
* If a factor $(x-a)$ exists in both the numerator and denominator, it may result in a “hole” (removable discontinuity) rather than an asymptote.

Horizontal and Oblique Asymptotes

To find non-vertical asymptotes, perform polynomial division (or use inspection) to rewrite the function in the form:
$$f(x) = S(x) + \frac{R(x)}{Q(x)}$$
where $S(x)$ is the quotient and $R(x)$ is the remainder. As $x \to \pm\infty$, the term $\frac{R(x)}{Q(x)} \to 0$.

KEY TAKEAWAY: For a function like $f(x) = \frac{x^2+1}{x}$, dividing through gives $f(x) = x + \frac{1}{x}$. As $x \to \pm\infty$, $\frac{1}{x} \to 0$, so the oblique asymptote is $y = x$.

3. Techniques for Sketching

Addition of Ordinates

Some rational functions can be viewed as the sum of two simpler functions. For example, $y = x + \frac{1}{x}$ can be graphed by adding the $y$-values of $y_1 = x$ and $y_2 = \frac{1}{x}$.
* The graph will approach the vertical asymptote of the reciprocal part.
* The graph will approach the linear part as $x$ becomes very large.

Reciprocal Graphs

When sketching $y = \frac{1}{f(x)}$:
* Zeros of $f(x)$ become vertical asymptotes of $\frac{1}{f(x)}$.
* Local maxima of $f(x)$ become local minima of $\frac{1}{f(x)}$.
* Where $f(x)$ is increasing, $\frac{1}{f(x)}$ is decreasing.

COMMON MISTAKE: Students often forget to check if the graph crosses its horizontal asymptote. While a graph never crosses a vertical asymptote, it can cross a horizontal or oblique asymptote in the middle of the domain. Solve $f(x) = S(x)$ to check for intersections.

4. Analyzing Regions and Concavity

Understanding the “regions of the plane” involves determining where the graph lies relative to its asymptotes.

1. Sign Diagrams: Test values between intercepts and vertical asymptotes to determine if the curve is above ($y > 0$) or below ($y < 0$) the $x$-axis.
2. Asymptotic Behavior:
   • As $x \to a^+$, does $y \to \infty$ or $y \to -\infty$?
   • As $x \to \infty$, does the graph approach the asymptote from above or below?
   • Example: In $y = 4 - \frac{2}{x^2+1}$, since $\frac{2}{x^2+1}$ is always positive, $y$ will always be slightly less than 4. Thus, the graph approaches $y=4$ from below on both sides.

Calculus in Rational Functions

• First Derivative $f’(x)$: Used to find stationary points. For $f(x) = \frac{x^2+1}{x}$, $f’(x) = 1 - \frac{1}{x^2}$. Setting $f’(x) = 0$ gives $x = \pm 1$.
• Second Derivative $f’‘(x)$: Used to determine concavity and points of inflection. For $f(x) = \frac{x^2+1}{x}$, $f’‘(x) = \frac{2}{x^3}$. Since $f’‘(x) \neq 0$ for all $x$, there are no points of inflection.

VCAA FOCUS: Specialist Math exams often require you to find the coordinates of stationary points for rational functions where the derivative involves solving a quadratic or cubic equation. Practice using the quotient rule accurately.

5. Summary Table: Graphing Process

STUDY HINT: When sketching, always draw the asymptotes as dashed lines. This provides a “skeleton” for the graph and ensures your curves approach the correct lines as $x$ or $y$ approaches infinity.