Quadratic forms and graphs

Connect quadratic algebra with the shape and features of a parabola. This practice set explores expanded, factored and vertex forms, showing how each representation makes different information visible. You will identify intercepts and the axis of symmetry, read a labelled coordinate graph, complete the square and reconstruct a quadratic from sufficient constraints.

Questions progress from recognizing useful forms to combining roots with an extra point, analyzing restricted domains and distinguishing a repeated zero from a crossing. The set also separates factorization over the rational numbers from factorization over the real numbers. A quadratic can have irrational real zeros even when it has no rational linear factors; another can have no real zeros at all. Complex arithmetic is reserved for the following topics.

Useful prerequisites are expanding and factoring expressions, solving quadratic equations and interpreting function transformations. Work with exact values unless a question says otherwise. Both single-select and multiselect tasks include explanations that connect calculations to the graph and address common sign, scale and endpoint errors. For fuller practice, sketch each parabola from your own reasoning before checking the choices. The questions develop flexibility between representations without treating a graph window as the entire domain of a quadratic function.