Rational Functions- Graphs and Analysis

What Rational Functions Actually Are

A rational function is simply one polynomial divided by another polynomial. That's it. No fancy definitions needed.

The basic form is:

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

where P(x) and Q(x) are polynomials, and Q(x) ≠ 0. The denominator cannot be zero because division by zero doesn't exist.

Real-world examples include:

The Anatomy of a Rational Function

Numerator and Denominator

The numerator tells you where the function equals zero. These are your x-intercepts.

The denominator tells you where the function is undefined. These are your vertical asymptotes (or holes, if factors cancel).

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

Degrees Matter

The degree of the numerator compared to the denominator determines the horizontal asymptote:

Vertical Asymptotes: Where the Function Blows Up

Vertical asymptotes occur at x-values that make the denominator zero—but only if those factors don't cancel with the numerator.

Critical rule: If a factor cancels completely (appears in both numerator and denominator), you get a hole instead of an asymptote.

Example with a hole:

f(x) = (x² - 4) / (x - 2) = [(x-2)(x+2)] / (x-2)

The (x-2) cancels, leaving f(x) = x + 2, except at x = 2 where the original function is undefined. That's a hole at (2, 4).

Horizontal and Oblique Asymptotes

These tell you the function's end behavior—what happens as x goes to positive or negative infinity.

For horizontal asymptotes, compare degrees:

To find a slant asymptote, divide the numerator by the denominator using polynomial long division. The quotient (ignoring the remainder) is your slant asymptote.

How to Graph a Rational Function: Step by Step

Here's the actual process, in order:

Step 1: Find the y-intercept

Plug in x = 0. Calculate f(0). That's your y-intercept.

Step 2: Find x-intercepts

Set the numerator equal to zero. Solve for x. These are your x-intercepts—if the denominator doesn't also equal zero at those points.

Step 3: Find vertical asymptotes

Set the denominator equal to zero. Solve for x. These are potential vertical asymptotes. Check if any factors cancel—if they do, mark a hole instead.

Step 4: Find horizontal or slant asymptote

Compare degrees. Use polynomial division if needed.

Step 5: Test regions

Pick test values in each region divided by asymptotes. Plug in to see if the function is positive or negative in that region.

Step 6: Sketch

Plot intercepts, draw asymptotes as dashed lines, and sketch the curve approaching but never crossing them.

Working Example

Graph: f(x) = (2x) / (x² - 4)

Step 1: y-intercept → f(0) = 0 → (0, 0)

Step 2: x-intercept → 2x = 0 → x = 0 → (0, 0)

Step 3: Vertical asymptotes → x² - 4 = 0 → x = ±2

Step 4: Horizontal asymptote → degree of numerator (1) < degree of denominator (2) → y = 0

Step 5: Test regions:

Now you know the curve's behavior in each region. Sketch accordingly.

Analysis: What to Look For

When analyzing a rational function, answer these questions:

This systematic approach works every time. No guessing.

Common Mistakes That Ruin Your Graph

Mistake What Should Happen
Crossing a vertical asymptote Curves approach but never cross VA lines
Ignoring holes Always check for cancelled factors
Drawing through asymptotes Asymptotes are boundaries—don't cross them
Wrong sign in test regions Double-check your arithmetic when plugging in test values
Forgetting the y-intercept Always check f(0)

Comparing Rational Function Analysis Methods

Method Best For Speed
Factoring and canceling Finding holes and intercepts Fast
Polynomial long division Finding slant asymptotes Medium
Test point substitution Determining sign in each region Reliable but slower
Graphing calculator Checking work, complex functions Fastest

Quick Reference: Key Takeaways

That's the full picture. Graph rational functions by following the steps in order, and you'll never get lost. The asymptotes divide the coordinate plane into regions—test each one, and the graph practically draws itself.