Rational Functions and Holes
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA hole is one missing point on an otherwise smooth curve. It happens where numerator and denominator share a zero and the numerator has at least as many copies of the shared factor, so the whole break cancels away: near that input the function behaves like its reduced form, and the reduced form evaluated there names the hole's exact height. The function is still undefined at the hole; the graph just misses one dot instead of blowing up.
The traps: calling the hole a vertical asymptote (or an ordinary zero when it happens to sit on the x-axis), believing the pattern of nearby outputs defines a value at the missing input, and cancelling terms instead of factors to manufacture holes that are not there. The lesson drills the factor bookkeeping that settles all of it.
The work
3 ways in · any order
Lesson
Rational Functions and Holes
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Shared factor, numerator supply holds: the break collapses to a puncture. The lesson finds holes and their exact coordinates from the reduced form, contrasts bounded puncture behavior with asymptote blow-up, then closes with ten scenarios of locating, measuring, and defending holes.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.10 misconceptions: holes promoted to asymptotes, hole heights read as function values, and term-by-term cancellation inventing breaks that do not exist. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.