The parabola represents a function graph of a type of y = A·x \+ B·x + C. Branches of a parabola can be directed up or down. Comparing A coefficient at x² to zero, it is possible to define the direction of branches of a parabola.
Instruction
1. Let some square function y = A·x \+ B·x + by C, A≠0 is set. The condition of A≠0 is important for a task of square function since at A=0 it degenerates in linear y = B·x + C. Not the parabola, but a straight line will be the schedule of the linear equation any more.
2. In expression A·x \+ B·x + C compare the senior coefficient of A to zero. If it positive, branches of a parabola are directed up if negative − down. At an analytical research of function before creation of the schedule paint this moment.
3. Find parabola top coordinates. On abscissa axis the coordinate is on formula x0 = - B/2A. That find top coordinate on ordinate axis, substitute the received value for x0 in function. Then you receive y0 = y(x0).
4. If the parabola is directed up, its top will be the lowermost point on graphics. If branches of a parabola "look" down, the top will be the topmost point of the schedule. In the first case of x0 is a function minimum point, in the second − a maximum point. y0, respectively, the smallest and greatest value of function.
5. For creation of a parabola of one point and knowledge of where branches are directed, it is not enough. Therefore find coordinates of several more additional points. Remember that a parabola - a symmetric figure. Through top carry out the symmetry axis perpendicular axes of Ox and Oy parallel to an axis. It is enough to look for points only on one side from an axis, and on other side to construct symmetrically.
6. Find "zero" function. Equate to zero x, count y. So you receive a point in which the parabola crosses Oy axis. Further equate to zero y and find at what x equality A·x \+ B·x + by C = 0 is carried out. It will give you parabola points of intersection with Ox axis. Depending on a discriminant, such points two or one, and can not be at all.
7. A discriminant of D = B² - 4·A·C. It is necessary for search of roots of a quadratic equation. If D> 0, to the equation two points satisfy; if D = 0 − one. At D
Having coordinates of top of a parabola and knowing the direction of its branches, it is possible to draw a conclusion on a set of values of function. The set of values − are that range of numbers which is run by the f (x) function on all range of definition. And square function on all numerical straight line is defined if additional conditions are not set.
Let, for example, top is the point with coordinates (K, Q). If branches of a parabola are directed up, a set of values of the E(f) function = [Q;+ ∞), or, in the form of inequality, y(x)> Q. If branches of a parabola are directed down, then E(f) = (-∞; Q] or y (x)
8. Having coordinates of top of a parabola and knowing the direction of its branches, it is possible to draw a conclusion on a set of values of function. The set of values − are that range of numbers which is run by the f (x) function on all range of definition. And square function on all numerical straight line is defined if additional conditions are not set.
9. Let, for example, top is the point with coordinates (K, Q). If branches of a parabola are directed up, a set of values of the E(f) function = [Q;+ ∞), or, in the form of inequality, y(x)> Q. If branches of a parabola are directed down, then E(f) = (-∞; Q] or y (x)