The schedule of square function is called a parabola. This line has powerful physical value. On parabolas some celestial bodies move. The antenna in the form of a parabola focuses the beams going parallel to an axis of symmetry of a parabola. The bodies thrown up at an angle reach the top point and fall down, also describing a parabola. It is obvious that it is always useful to know coordinates of top of this movement.
Instruction
1. Square function in a general view registers the equation: y = ax² + bx + with. The schedule of this equation is the parabola which branches are directed up (at a> 0) or down (at a <0). School students just are offered to remember a formula of calculation of coordinates of top of a parabola. The top of a parabola lies in x0 point = - b/2a. Having substituted this value in a quadratic equation, receive y0: y0 = a (-b/2a)² - b²/2a + c = - b²/4a + with.
2. To the people familiar with a concept of a derivative, it is easy to find parabola top. Irrespective of the provision of branches of a parabola its top is an extremum point (a minimum if branches are directed up, or a maximum when branches are directed down). To find points of an estimated extremum of any function, it is necessary to calculate its first derivative and to equate it to zero. The derivative of square function is equal in a general view to f' (x) = (ax² + bx + c)' = 2ax + b. Having equated to zero, you receive 0 = 2ax0 + b => x0 = - b/2a.
3. A parabola - the symmetric line. The axis of symmetry passes through parabola top. Knowing parabola points of intersection with an axis of coordinates of X, it is possible to find x0 top abscissa easily. Let x1 and x2 - parabola roots (so call parabola points of intersection with abscissa axis as these values turn a quadratic equation of ax² + bx + with into zero). At the same time let |x2|> |x1|, then top of a parabola lies in the middle between them and can be found from the following expression: x0 = ½ (|x2| - |x1|).