Implicitly Defined Functions
Unit 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 honest branches.
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 honestly, 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 the misconception and moves you to the next.