Nonlinear Parameters and Number of Intersections

Objective: Compare a horizontal level with a parabola extremum.

Prerequisite skills: quadratic discriminants and nonlinear graphs. For y=(x−h)²+k, a horizontal line y=c has two intersections if c>k, one if c=k, and none if c<k. This follows from the number of real square roots of c−k. Common error: counting two distinct intersections when the squared term equals zero. Restrictions and interpretation: Intersection counts must respect domains; tangency corresponds to one distinct solution rather than two repeated labels.

Objective: Use a discriminant to identify a line touching a parabola.

Equate a line with the quadratic and set the resulting discriminant to zero for exactly one real intersection. A graphical near-touch can be misleading; use the exact condition to establish tangency. Common error: equating a graph display near a vertex with an exact tangency proof. Restrictions and interpretation: Intersection counts must respect domains; tangency corresponds to one distinct solution rather than two repeated labels.

Objective: Count only intersections on an allowed interval.

A full algebraic graph can have more intersections than a context permits. Solve the equality first, then apply the stated domain separately to every root. A restricted interval can remove a mathematically valid crossing. Common error: counting every real root even when the domain removes some. Restrictions and interpretation: Intersection counts must respect domains; tangency corresponds to one distinct solution rather than two repeated labels. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.