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Rational Functions

The Hook

Breaking the rules.

What is it?

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

Time Travel (History)

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

Cheat Code

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

Practice This Now!

The Field Guide

In 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.

In The Real World: 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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Booby Trap!

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

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

Training Drills

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