Rational Function Examples- Key Concepts and Graphs

What Is a Rational Function?

A rational function is simply a function where one polynomial is divided by another polynomial. The general form looks like this:

f(x) = P(x) / Q(x)

Where P(x) and Q(x) are polynomials, and Q(x) ≠ 0. That's it. Nothing fancy.

The denominator is what makes these functions interesting—and tricky. When the denominator equals zero, you get a vertical asymptote or a hole in the graph. That's the core of what makes rational functions different from regular polynomials.

Key Parts You Need to Know

Before looking at examples, memorize these terms:

• Numerator — the polynomial on top
• Denominator — the polynomial on bottom (can't be zero)
• Vertical Asymptote — a vertical line where the function approaches infinity
• Horizontal Asymptote — a horizontal line the graph approaches as x goes to infinity
• Hole — a point where both numerator and denominator are zero
• x-intercept — where the graph crosses the x-axis (numerator = 0, but denominator ≠ 0)
• y-intercept — where the graph crosses the y-axis (plug in x = 0)

Simple Rational Function Examples

Example 1: The Basic Reciprocal

f(x) = 1/x

This is the simplest rational function you'll encounter. The domain excludes x = 0 because you can't divide by zero. The graph has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The curve approaches both axes but never touches them.

Notice how the function values get huge positive or negative depending on which side of zero you're on. From the right (0.1, 0.01, 0.001), the function rockets upward. From the left, it plunges downward.

Example 2: Linear Over Linear

f(x) = (x + 2) / (x - 3)

This is the most common type you'll see in homework and exams. Let's break it down:

• Vertical asymptote at x = 3 (where denominator = 0)
• Hole at x = -2? No. The numerator equals zero at x = -2, but that's fine—it just means the function crosses the x-axis there
• x-intercept at (-2, 0)
• y-intercept at (0, -2/3)
• Horizontal asymptote at y = 1 (coefficients of leading terms)

Example 3: With a Hole

f(x) = (x² - 4) / (x - 2)

This looks like it has a vertical asymptote at x = 2. But factor the numerator:

(x + 2)(x - 2) / (x - 2)

The (x - 2) terms cancel. This means the function is actually f(x) = x + 2 with a hole at x = 2. The hole exists because the original function was undefined there.

Always factor before you decide whether you have an asymptote or a hole. This is where students lose points.

Example 4: Degree of Numerator Greater Than Degree of Denominator

f(x) = (x³ + 1) / (x + 1)

When the numerator's degree is higher, you get an oblique (slanted) asymptote, not a horizontal one. You find it by polynomial long division.

Divide x³ + 1 by x + 1. You get x² - x + 1 with a remainder of 0 (since x³ + 1 = (x + 1)(x² - x + 1)).

The asymptote is the quotient: y = x² - x + 1. The graph approaches this parabola as x goes to ±∞.

How to Graph a Rational Function

Here's the step-by-step process that actually works:

Step 1: Find the Domain

Set the denominator equal to zero. Those x-values are excluded. Write them down.

Step 2: Simplify by Factoring

Factor numerator and denominator completely. Cancel any common factors—these create holes, not asymptotes.

Step 3: Find the Holes

For each canceled factor, plug the excluded x-value into the simplified function to get the y-coordinate of the hole.

Step 4: Find the Asymptotes

Vertical asymptotes occur at x-values where the simplified denominator equals zero and no cancellation happened.

Horizontal asymptotes depend on degrees:

Step 5: Find Intercepts

x-intercepts: Set numerator = 0 (in the original function, not simplified). Make sure the denominator isn't also zero there.

y-intercept: Plug x = 0 into the function.

Step 6: Plot Points and Sketch

Pick x-values near the asymptotes and plug them in. You need to know which direction the graph is heading on each side of each asymptote.

Common Mistakes That Cost You Points

• Forgetting to check for holes after simplifying. Many students see a canceled factor and assume it's gone. It's not. The point still doesn't exist.
• Drawing horizontal asymptotes as actual lines the graph touches. Asymptotes are guides, not part of the function. The graph approaches them but never crosses horizontal asymptotes (unless the function crosses them at a specific point, which is rare).
• Not testing points between asymptotes. The behavior between asymptotes determines the shape. Don't skip this.
• Confusing holes with vertical asymptotes. If factors cancel, it's a hole. If they don't cancel, it's an asymptote.

Comparing Basic Rational Function Types

Quick Reference: What to Do When

See a rational function on a test? Here's your checklist:

• Factor everything first
• Cancel common factors → holes
• Remaining denominator zeros → vertical asymptotes
• Compare degrees → find horizontal or oblique asymptote
• Find intercepts from the original (uncanceled) form
• Plot points between asymptotes to get the shape right

If you follow these steps every time, you'll get the right graph. There's no trick that replaces doing the work.

Bottom Line

Rational functions are just polynomials divided by each other. The denominator being zero creates the interesting behavior—holes and asymptotes. Factor before you do anything else. Cancel carefully. Plot enough points to see the shape. That's the whole game.