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.