Graphing Rational Functions (Advanced)

In Asymptote Analysis and Oblique Asymptotes, you learned to find the structural features of rational functions. This page ties everything together into a complete graphing strategy that handles any rational function, no matter how complex.

The Seven-Step Graphing Strategy

For any rational function $f(x) = \frac{P(x)}{Q(x)}$:

1. Factor numerator and denominator completely
2. Domain — find all $x$-values where the denominator is zero
3. Holes — cancel common factors, note the coordinates of each hole
4. Asymptotes — find vertical asymptotes (remaining denominator zeros), horizontal or oblique asymptotes (degree comparison or long division)
5. Intercepts — $x$-intercepts (set numerator = 0), $y$-intercept (evaluate $f(0)$)
6. Sign analysis — build a sign chart using all zeros and asymptotes
7. Plot and sketch — combine all information, adding test points as needed

Example 1: A Standard Rational Function

Graph $f(x) = \frac{2x}{x^2 - 4}$.

Step 1 — Factor:

$f(x) = \frac{2x}{(x-2)(x+2)}$

No common factors.

Step 2 — Domain: $x \neq 2$ and $x \neq -2$.

Step 3 — Holes: None (no common factors).

Step 4 — Asymptotes:

• Vertical: $x = 2$ and $x = -2$
• Horizontal: Numerator degree 1, denominator degree 2. Since $1 < 2$, the horizontal asymptote is $y = 0$.

Step 5 — Intercepts:

• $x$-intercept: $2x = 0 \implies x = 0$. So $(0, 0)$.
• $y$-intercept: $f(0) = 0$. Same point: $(0, 0)$.

Step 6 — Sign analysis:

The critical $x$-values are $-2$, $0$, and $2$. Test one value in each interval:

Step 7 — Sketch: The graph passes through the origin, is negative for $x < -2$ (approaching $y = 0$ from below on the far left), positive between $-2$ and $0$, negative between $0$ and $2$, and positive for $x > 2$.

Example 2: With a Hole

Graph $g(x) = \frac{x^2 - 1}{x^2 + x - 2}$.

Step 1 — Factor:

$g(x) = \frac{(x-1)(x+1)}{(x+2)(x-1)}$

Step 2 — Domain: $x \neq -2$ and $x \neq 1$.

Step 3 — Holes: Common factor $(x - 1)$. Hole at $x = 1$.

Simplified: $g(x) = \frac{x+1}{x+2}$, and $g(1) = \frac{2}{3}$. Hole at $(1, \frac{2}{3})$.

Step 4 — Asymptotes:

• Vertical: $x = -2$ (from the remaining denominator)
• Horizontal: Degrees equal (both 1 after simplification, or both 2 before — same result). Leading coefficient ratio: $\frac{1}{1} = 1$. Horizontal asymptote: $y = 1$.

Step 5 — Intercepts:

• $x$-intercept: $x + 1 = 0 \implies x = -1$. So $(-1, 0)$.
• $y$-intercept: $g(0) = \frac{1}{2}$. So $(0, \frac{1}{2})$.

Step 6 — Sign analysis (using simplified form):

Step 7 — Sketch: The graph is above $y = 1$ for $x < -2$ (approaching from above), dips below the $x$-axis between $-2$ and $-1$, crosses the $x$-axis at $-1$, passes through $(0, \frac{1}{2})$, and approaches $y = 1$ from below for large positive $x$. There is an open circle at $(1, \frac{2}{3})$.

Example 3: With an Oblique Asymptote

Graph $h(x) = \frac{x^2 - 2x - 3}{x - 1}$.

Step 1 — Factor:

$h(x) = \frac{(x-3)(x+1)}{x-1}$

No common factors.

Step 2 — Domain: $x \neq 1$.

Step 3 — Holes: None.

Step 4 — Asymptotes:

• Vertical: $x = 1$

• Oblique: Degree 2 over degree 1. Divide $x^2 - 2x - 3$ by $x - 1$:

  Step 1: $x^2 \div x = x$. Multiply: $x^2 - x$. Subtract: $-x - 3$.

  Step 2: $-x \div x = -1$. Multiply: $-x + 1$. Subtract: $-4$.

  $h(x) = x - 1 + \frac{-4}{x - 1}$

  Oblique asymptote: $y = x - 1$.

Step 5 — Intercepts:

• $x$-intercepts: $x = 3$ and $x = -1$. Points: $(3, 0)$ and $(-1, 0)$.
• $y$-intercept: $h(0) = \frac{-3}{-1} = 3$. Point: $(0, 3)$.

Step 6 — Sign analysis:

Step 7 — Sketch: The curve crosses the $x$-axis at $-1$ and $3$, passes through $(0, 3)$, has a vertical asymptote at $x = 1$ (going to $+\infty$ from the left and $-\infty$ from the right), and approaches the oblique asymptote $y = x - 1$ for large $|x|$.

Example 4: Higher Complexity

Graph $r(x) = \frac{3(x+1)(x-2)}{(x+3)(x-1)^2}$.

This function is already factored. Let us apply each step.

Domain: $x \neq -3$ and $x \neq 1$.

Holes: No common factors. No holes.

Asymptotes:

• Vertical: $x = -3$ (odd multiplicity, opposite directions) and $x = 1$ (even multiplicity, same direction)
• Horizontal: Numerator degree 2, denominator degree 3. Since $2 < 3$, horizontal asymptote is $y = 0$.

Intercepts:

• $x$-intercepts: $x = -1$ and $x = 2$
• $y$-intercept: $r(0) = \frac{3(1)(-2)}{(3)(1)} = \frac{-6}{3} = -2$

Sign analysis:

Critical values: $-3, -1, 1, 2$

Notice that the sign does not change at $x = 1$ — because $(x-1)^2$ is always positive. This confirms even-multiplicity behavior: the function stays negative on both sides of $x = 1$.

Behavior near $x = -3$: Odd multiplicity, sign changes from $-$ to $+$ — the function goes from $-\infty$ to $+\infty$.

Behavior near $x = 1$: Even multiplicity, sign stays $-$ — the function goes to $-\infty$ on both sides.

Common Graphing Mistakes

1. Forgetting to cancel before finding vertical asymptotes. Always factor and cancel first. A zero that cancels is a hole, not an asymptote.

2. Drawing the curve crossing a vertical asymptote. The graph never crosses a vertical asymptote — the function is undefined there.

3. Assuming the graph cannot cross a horizontal asymptote. It can and often does in its interior. Only the end behavior follows the horizontal asymptote.

4. Ignoring multiplicity. Even and odd multiplicities in the denominator produce different graphical behaviors. Always check.

5. Skipping sign analysis. Without sign analysis, you cannot determine which direction the curve approaches each asymptote. A sign chart takes two minutes and prevents major errors.

Real-World Application: Lens Optics

The thin lens equation in optics is:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Solving for image distance $d_i$ as a function of object distance $d_o$:

$d_i = \frac{f \cdot d_o}{d_o - f}$

This is a rational function of $d_o$ with a vertical asymptote at $d_o = f$ (when the object is at the focal point, the image is at infinity) and a horizontal asymptote at $d_i = f$ (as the object moves far away, the image approaches the focal point). Engineers use this to design camera systems, telescopes, and projectors.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The seven-step strategy (factor, domain, holes, asymptotes, intercepts, sign analysis, sketch) handles any rational function systematically
• Sign analysis is essential — it tells you which direction the curve approaches each asymptote and which intervals are positive or negative
• Even multiplicity in the denominator means the function approaches the same direction on both sides; odd multiplicity means opposite directions
• Always factor and cancel first before looking for vertical asymptotes — cancelled factors produce holes, not asymptotes
• The strategy combines everything from Asymptote Analysis and Oblique Asymptotes

Return to College Algebra for more topics in this section.