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Rational Functions

Overview

Breaking the rules.

What is it?

A ratio of two polynomials. Using x in the denominator.

History

Studied to understand asymptotes—places where math explodes to infinity.

Key Idea

If the bottom is zero, the graph hits a brick wall (Vertical Asymptote).

Practice This Topic

Concept Guide

Plain English: A rational function is a fraction made of x's. Because you can NEVER divide by zero, these graphs have 'forbidden zones' called Asymptotes. The line gets closer and closer but never touches.

Real-world example: Photography. The relationship between aperture and shutter speed is an inverse rational function.

How to do it

  1. Find Vertical Asymptotes: Set the bottom equal to zero and solve.
  2. Find Horizontal Asymptotes: Compare the degree (power) of top vs bottom.
  3. If Top < Bottom, y=0.
  4. If Top = Bottom, y = ratio of coefficients.
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Common Pitfall

Canceling terms illegally. In (x+2)/(x+3), you cannot cross out the x's! You can only cancel factors (multiplication), not terms (addition).

Word Problem
"A party costs $100 fee plus $10 per person. What is the average cost per person (C) if x people come?"
Reasoning: Total Cost = 100 + 10x. Average = Total / x. Function: C(x) = (100 + 10x) / x.

Practice Examples

Asymptote
y = 1 / (x-2)
Wall at x=2.