Graphing Rational Functions

Graphing Rational and Quotient Functions

Systematic Approach

To sketch $f(x) = P(x)/Q(x)$, work through these steps in order:

1. Simplify: Cancel common factors (note holes).
2. Domain: Exclude zeros of $Q$.
3. Intercepts: $y$-intercept at $f(0)$; $x$-intercepts where $P(x)=0$ (in domain).
4. Vertical asymptotes: Zeros of $Q$ not cancelled.
5. Horizontal/oblique asymptote: Compare degrees.
6. Sign diagram: Determine where $f > 0$ and $f < 0$.
7. Stationary points: $f’(x) = 0$ (for accurate sketching).
8. Sketch: Use all gathered information.

Example 1: Simple Rational

Sketch $f(x) = \dfrac{2x}{x^2 - 4}$.

• Factorise denominator: $(x-2)(x+2)$.
• No common factors.
• Domain: $\mathbb{R} \setminus {-2, 2}$.
• $y$-intercept: $f(0) = 0$.
• $x$-intercept: $x = 0$.
• VAs: $x = 2$ and $x = -2$.
• HA: $\deg P = 1 < \deg Q = 2$, so $y = 0$.
• Sign: $f$ is odd ($f(-x) = -f(x)$), so symmetric about the origin.
• Near $x=2^+$: $f \to +\infty$; near $x=2^-$: $f \to -\infty$.
• Near $x=-2^+$: $f \to +\infty$; near $x=-2^-$: $f \to -\infty$.

Example 2: Oblique Asymptote

Sketch $f(x) = \dfrac{x^2 - 1}{x}$.

Divide: $f(x) = x - \dfrac{1}{x}$.
- Domain: $\mathbb{R} \setminus {0}$.
- VA: $x = 0$.
- Oblique asymptote: $y = x$.
- $x$-intercepts: $x^2 - 1 = 0 \Rightarrow x = \pm 1$.
- $f’(x) = 1 + x^{-2} > 0$ for all $x \neq 0$ (strictly increasing on each branch).

Example 3: Repeated Factor

Sketch $f(x) = \dfrac{x^2}{(x-1)^2}$.

• Domain: $\mathbb{R} \setminus {1}$.
• VA: $x = 1$.
• HA: $y = 1$ (equal degrees, leading coefficients both 1).
• $x$-intercept: $x = 0$.
• $y$-intercept: $f(0) = 0$.
• $f(x) \geq 0$ everywhere (numerator and denominator both squares).
• Near $x=1$: $f \to +\infty$ from both sides (repeated factor means no sign change).

Regions of the Plane

A region defined by a rational inequality, e.g., $f(x) \geq k$, is found by:
1. Solving $f(x) - k = 0$ for boundary $x$-values.
2. Using a sign diagram for $f(x) - k$.
3. Combining with domain restrictions.

Example: Solve $\dfrac{2x}{x^2-4} \geq 0$.

From Example 1 sign analysis:
- $f > 0$ when $x \in (-2, 0) \cup (2, \infty)$.
- $f = 0$ when $x = 0$.
- Answer: $x \in (-2, 0] \cup (2, \infty)$.

KEY TAKEAWAY: A systematic step-by-step approach — simplify, domain, intercepts, asymptotes, sign, sketch — avoids errors and ensures all key features are captured.

EXAM TIP: On VCAA Paper 1, show the equations of all asymptotes and label coordinates of all intercepts. Marks are allocated for each piece of information.

COMMON MISTAKE: Forgetting that a horizontal asymptote can be crossed by the function graph (it only describes end behaviour). A function can equal its HA at some finite $x$ value.