Roses, lemniscates and intersections

Roses, lemniscates and intersections

Analyze polar roses, lemniscates and curve intersections with careful attention to the geometric point set. This precalculus extension practice covers petal counts, petal directions, tracing intervals, valid angles and intersections that an equal-angle calculation can miss. Fourteen single-select and multiselect questions combine exact trigonometric reasoning, a labelled rose graph and algebraic checks. Detailed explanations state the positive-integer assumptions behind petal rules, check the sign of squared radii and test the pole separately. Later problems require all intersections, including those represented with opposite radius signs. No calculus-based area or arc-length formulas are used.