Rational Functions and End Behavior
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalFor large inputs a rational function collapses to one fraction: leading term over leading term. Comparing degrees settles everything: bottom-heavy ratios flatten onto y = 0, balanced degrees flatten onto the ratio of leading coefficients, and top-heavy ratios follow their polynomial quotient, a slant line when the degrees differ by one. Limit notation states each case precisely, and the same line governs both directions.
The mistakes here are recited rules detached from their reasons: dividing leading coefficients when the degrees do not match, claiming a slant asymptote for every top-heavy case, and treating a horizontal asymptote as a fence the graph may never cross. Each rule is rebuilt from the leading-term ratio in the lesson, then drilled.
The work
3 ways in · any order
Lesson
Rational Functions and End Behavior
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Replace top and bottom with their leading terms and the ends fall out: the three degree cases, slant asymptotes from the quotient, and why crossing a horizontal asymptote is perfectly legal. Ten scenarios drill the cases, the limits, and the rules students misremember.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 1.7 misconceptions: asymptote rules applied to the wrong degree case, and end behavior judged from constants or low-order terms instead of the leading-term ratio. 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.