Practice maths class test — Year 3: the parabola.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

The test on the parabola falls in November-December of Year 3, the first of the three big analytic geometry chapters of the year (parabola, then circle, then ellipse and hyperbola) — and it uses the same tangent method that will come back identical for the other two. Besides vertex, axis and intersections, the test almost always has a family of parabolas and a problem with a parameter to discuss.

The first mistake concerns parabolas with an axis parallel to the x axis, of the form x = ay² + by + c: anyone who has learned the formulas for the “normal” parabola by heart applies them without swapping the roles of x and y, and systematically gets the vertex and axis wrong. In this form the vertex has y-coordinate −b/2a (the same formula, but in y) and x-coordinate c − b²/4a — the coordinates swapped compared with the case you study first.

The second is about families of parabolas: when the equation contains a parameter k, the point (or points) that all the parabolas of the family have in common is found by eliminating k from the equation, isolating the terms with k and those without. Intersecting two particular parabolas of the family, for example the one with k = 0 and the one with k = 1, gives a point of the family but does not guarantee you have found all the base points if there is more than one.

The third, in problems with a parameter: if the parameter multiplies the coefficient of x², there is a value that makes that coefficient zero — and in that case the “parabola” degenerates into a line. Discussing the problem without checking this case first leaves out a solution the teacher expects.

To prepare, redo a problem with a parameter all the way through, writing explicitly for which value the coefficient of x² becomes zero, before even calculating discriminants.

What a maths test on the parabola looks like

Vertex, axis and intersections, the parabola through three points or with mixed conditions, tangents from an external point with the condition Δ = 0, and almost always a family of parabolas with a parameter to discuss.

The mistake that costs the grade

Confusing the condition for tangency with the condition for intersection: tangent means Δ = 0, not Δ ≥ 0. And in the vertex, forgetting the minus sign in −b/2a, which shifts the whole rest of the exercise.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.