Implicitly Defined Functions
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalUnit 4 is not assessed on the AP exam; schools include it based on local requirements, and everything here still builds directly toward calculus. An implicitly defined curve is the set of points satisfying an equation in x and y. Membership is a substitution check, intercepts come from setting one variable to zero, and solving for y may split the curve into two branches, each a function in its own right.
The mistakes orbit one misreading: treating "fails the vertical line test" as "invalid or ungraphable," when the test only decides whether y is a function of x. Its companions are keeping a single square-root branch and losing half the curve, and pattern-matching the equation's constant into an intercept it never was. The lesson drills the substitution-first discipline.
The work
3 ways in · any order
Lesson
Implicitly Defined Functions
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An equation in x and y draws a curve whether or not y is a function of x. The lesson runs the substitution test, splits square-root solves into both branches, reads intercepts off the equation instead of its constant, and puts the vertical line test back in its lane. Ten scenarios close it out.
Diagnostic
10-item topic check
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Ten items spanning the Topic 4.5 misconception cluster: vertical-line-test verdicts promoted into nonexistence claims, dropped square-root branches, and constants misread as intercepts. 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.