Sketching Rational Functions: A Step-by-Step Curve Sketching Algorithm

The Curve Sketching Algorithm

To sketch a graph accurately, you should follow a consistent checklist of key features.

1. Removable Discontinuities (Holes)

Before finding asymptotes, factor both the numerator and denominator completely. If a factor appears in both the top and the bottom, it creates a removable discontinuity, commonly known as a "hole" in the graph. You must note the coordinates of this hole, but it does not act as an asymptote.

2. Intercepts

• x-intercepts: These occur where the numerator equals zero (after any common factors have been removed).
• y-intercepts: This is found by evaluating the function at $x = 0$.

3. Vertical Asymptotes

Vertical asymptotes occur at the $x$-values that make the denominator equal to zero (after removing any common factors). These are vertical lines that the graph approaches but never crosses.

4. Horizontal or Oblique Asymptotes

As discussed in the categorization section, use the degrees of the polynomials to determine if the graph approaches a horizontal line ($y = c$) or a slanted line ($y = mx + b$) as $x$ approaches infinity or negative infinity.

Analyzing Behavior

Once the "skeleton" (asymptotes and intercepts) is drawn, you must determine the behavior around the asymptotes and the end behavior.

• Behavior around Vertical Asymptotes: As the graph approaches a vertical asymptote, does it shoot up toward $+\infty$ or down toward $-\infty$? You can determine this by testing a value very close to the asymptote on either side.
• End Behavior: This describes what happens to the $y$-values as $x$ becomes extremely large ($x \to \infty$) or extremely small ($x \to -\infty$). This is dictated by your horizontal or oblique asymptotes.

Exam Tip: Always check for common factors first! If you forget to identify a removable discontinuity (a hole), you might mistakenly label it as a vertical asymptote, which will change the entire shape of your graph.

Worked Example

Let's apply the algorithm to a function: $g(x) = \frac{6x - 4}{3x + 1}$

Step 1: Categorize The degree of the numerator is 1 and the degree of the denominator is 1. This is a balanced function.

Step 2: Asymptotes

• Horizontal Asymptote: Since it is balanced, we use the leading coefficients: $y = \frac{6}{3}$, so $y = 2$.
• Vertical Asymptote: Set the denominator to zero: $3x + 1 = 0 \implies x = -\frac{1}{3}$.

Step 3: Intercepts

• x-intercept: Set numerator to zero: $6x - 4 = 0 \implies x = \frac{4}{6} = \frac{2}{3}$. The intercept is $(\frac{2}{3}, 0)$.
• y-intercept: Evaluate $g(0)$: $g(0) = \frac{6(0)-4}{3(0)+1} = -4$. The intercept is $(0, -4)$.

Step 4: Sketching Plot the lines $x = -\frac{1}{3}$ and $y = 2$. Plot the points $(\frac{2}{3}, 0)$ and $(0, -4)$. By observing the intercepts, we can see the graph passes through the fourth quadrant and approaches the asymptotes accordingly.