Guide
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Let some quadratic function y = A · x² + B · x + C, A ≠ 0.Conditions A ≠ 0, it is important to define a quadratic function, becausewith A = 0 it degenerates into a linear y = B · x + C. The graph of a linear equation is not a parabola and a straight line.
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In terms of A · x² + B · x + C compare with zero leading coefficient A. If it is positive, the branches of the parabola will be directed upwards, if negative - down.The analytic functions before plotting shall describe this moment.
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find the coordinates of the vertex of the parabola.On the x-axis coordinate is given by x0 = -B / 2A.To find the coordinate of the top of the vertical axis, substitute the resulting value of x0 the function.Then you get y0 = y (x0).
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If a parabola is directed upward, its top is the lowest point on the chart.If the branches of the parabola "look" down to the top of the uppermost point of the schedule.In the first case x0 is a point of minimum of the function, the second - the maximum point.y0, respectively, the smallest and largest value of the function.
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To build a single point of the parabola, and the knowledge of where the branches are directed, is not enough.So find the coordinates of a few additional points.Remember that a parabola - a symmetrical figure.Swipe across the top of the symmetry axis perpendicular to the axis parallel to the axis Ox and Oy.It suffices merely to search for points on one side of the axis, and on the other side to construct a symmetric.
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find the "zero" function.Equating to zero x, count y.So you get the point where the parabola intersects the axis Oy.Next equate to zero y and get at any x the equality of A · x² + B · x + C = 0. This will give you the point of intersection of the parabola with the axis Ox.Depending on the discriminant of two points or one, and can not be at all.
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discriminant D = B² - 4 · A · C.It is necessary to find the roots of a quadratic equation.If D & gt;0, the equation is satisfied by two points;if D = 0 - one.When D
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With vertex coordinates of the parabola, and knowing the direction of its branches, it can be concluded on a set of values ​​of the function.The set of values ​​- is the range of numbers that runs the function f (x) over the entire domain.A quadratic function is defined on the whole real line, if you do not set additional conditions.
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Suppose, for example, the apex is the point with coordinates (K, Q).If the branches of the parabola are directed upwards, set of values ​​of E (f) = [Q; + ∞), or in the form of inequality, y (x) & gt;Q. If the branches of the parabola are directed downwards, then E (f) = (-∞; Q] or y (x)
Sources:
• parabola formula
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