The Vertical Line Test

Pre-Calculus · Axiom Academy

A graph is a function only when every vertical line crosses it at most once — watch the test sweep across each curve and decide. Picture a vertical line as a single input — one fixed value of x . Sweep it across the parabola y = x^2 . No matter where it stands, it crosses the curve in exactly one place : that one input has one output. Because every vertical line meets the graph at most once, y = x^2 is a function . Every vertical line hits at most once → passes Now run the same sweep across the circle x^2 + y^2 = 16 . Almost everywhere between x = -4 and x = 4 , the line stabs the curve in two places at once — an upper point and a lower point. That one input is sending out two different outputs, and , so the circle is not a function . At x = 0 the line hits y = 4 and y = -4 — two outputs for a single input. A function may not assign two outputs to the same input, so the test fails. — the is the second output the line catches. Only at does it touch once; one tangent point doesn't rescue the graph. Why a single failure is enough The graph is disqualified the instant one vertical line crosses it twice — it does not have to fail everywhere. Curves that open sideways or wrap around (circles, ellipses, sideways parabolas) fail for exactly this reason. 3. Reading Any Graph at a Glance

This is the written version of the interactive lesson above. See the full Pre-Calculus course.