Quadratic Function Essay

In the variety of mathematical functions studied in colleges and universities there is one of particular interest. It is quadratic function. On the graph it looks like letter “U”. Depending on the configuration of the function it can be inverted upside down and look like a horseshoe. Mathematicians call this function a parabola. The general form of quadratic function is

y = ax2 + bx +c, where a ≠ 0

On the plane parabola may lie in any part of the plane and intersect any reference axis or do not intersect them at all. This paper explains the behavior of quadratic function with respect to X axis.

Evidently quadratic function can intercept with X axis or not. But the point is that function can intercept X axis in two ways: it can have 1 or 2 intercepts. The first case implies that the curve is a tangent to X axis and therefore there is a single point of their interception. So the vertex of parabola lies on the axis directly; the curve can have different slope configuration and position. The second case implies that the vertex of parabola does not lie on the axis but can be anywhere else on the plane. But the direction of parabola should be specific: if the vertex lies under the X axis then the branches of parabola should be up-directed and vice versa (if the vertex lies above axis – the branches should be down-directed. In that case the curve intersects with X axis in two points that are equidistant from the vertex. But we should not forget about the third case – when parabola does not intersects with X axis at all. It is possible if the curve’s branches are up-directed and the vertex lies above the horizontal axis, or the branches are down-directed and the vertex lies below the axis simultaneously.

The points of intersection of parabola with X axis are called the roots. They can be found by equating the quadratic function to zero. But if we need to determine the number of intersections of quadratic function with horizontal axis we do not actually need to solve the equation. So as to determine the number of intersections we need to find the vertex of the function and the direction of its branches. The direction of the curve’s branches can be easily determined by comparing the sign of coefficient a of the function. If a > 0 then curve’s branches are up-directed, otherwise they are down-directed (if a < 0). The vertex of parabola has coordinates (q; w) which can found by the following formulas: q = -b / 2a, w = c – b2/4a Finally we compare the position of curve’s vertex in respect to horizontal axis (if w > 0 then parabola is above horizontal axis, otherwise – below the axis), determine the direction of parabola and make conclusion about the number of parabola’s intersects with X axis.