Twelfth Grade · Math · week 9

Rational functions: domain, holes, and vertical asymptotes

Find the domain of a rational function, and distinguish a removable hole from a vertical asymptote by factoring numerator and denominator.

A rational function is one polynomial divided by another, and its domain excludes every x that makes the DENOMINATOR zero -- full stop, before you simplify anything. The interesting question is what happens graphically at each excluded value, and the answer depends on whether that factor also appears in the numerator.

If a factor cancels between numerator and denominator, the graph has a HOLE at that x-value (a single missing point) rather than a vertical asymptote -- the function is undefined there, but nearby values behave normally. If a factor in the denominator does NOT cancel, the graph has a true vertical asymptote there, and the function shoots off toward positive or negative infinity as x approaches it.

📋 For the teacher
Teach it (10–15 min):
  • Insist on factoring BOTH the numerator and denominator completely before saying anything about domain, holes, or asymptotes -- an unfactored rational function hides its real behavior.
  • Physically cross out a canceling pair of factors together, and say out loud "this makes a hole, not an asymptote" every time it happens.
  • To find a hole's y-coordinate, always substitute into the SIMPLIFIED expression, never the original (the original is undefined there, by definition).
  • Keep a running list of every value that makes ANY factor of the original denominator zero -- that full list is the domain restriction, whether or not each one turns into a hole.
Talk about it:
  • Why does the domain restriction come from the ORIGINAL denominator, even after a factor cancels?
  • What is the graphical difference between a hole and a vertical asymptote, as you trace the curve with a finger?
  • Could a rational function have two holes and no vertical asymptotes? What would that require?

Mastery looks like: They factor a rational function completely, correctly separate holes from vertical asymptotes, and find a hole's exact coordinates using the simplified expression.

Common stumbles: Plugging into the original (unsimplified) expression to find a hole's y-coordinate; assuming every zero of the denominator is a vertical asymptote without checking for cancellation.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

FractionsSlice bars and circles into equal parts and compare them.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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