How to determine the vertex of a parabola

Parabola – one of the curves of the second order, it is constructed in accordance with a quadratic equation. Most importantly in the construction of this curve is to find the vertex of a parabola. This can be done in several ways.

Instruction

1. To find the coordinates of the vertex of the parabola, use the following formula: x=-b/2A, where a is the coefficient in front of x squared, and b is the coefficient in front of X. Substitute your values and calculate its value. Then substitute the resulting value is x in the equation and calculate the y coordinate of the vertex. For example, if you are given the equation y=2x^2-4x+5, then find the abscissa in the following way: x=-(-4)/2*2=1. Substituting x=1 into the equation, calculate the value of y for the vertex of a parabola: =2*1^2-4*1+5=3. Thus, the vertex of the parabola has coordinates (1;3).
2. The value of the ordinates of the parabola can be found without preliminary calculation of the abscissa. Use the formula y=-b^2/4 ° C+S.
3. If you are familiar with the concept of the derivative, find the vertex of a parabola using derivatives, using the following property of any functions: the first derivative equal to zero indicates extreme points. Since the vertex of the parabola, regardless of the sent its branches upwards or downwards, is a point of extremum, calculate derivative for your function. In General it will be of the form f(x)=2ах+b. Paranaita it to zero and get the coordinates of the vertex of the parabolacorresponding to your function.
4. Try to find the vertex of a parabolausing such property as symmetry. To do this, find the point of intersection of the parabola with the axis ox, equating the function to zero (substituting y=0). Solving the quadratic equation, you will find x1 and x2. Since the parabola is symmetric with respect to the directrix passing through the vertex, those points will be equidistant from the abscissa of the vertex. To find it, divide the distance between points in half: x=(Іх1-х2І)/2.
5. If any of the coefficients is zero (except a), calculate the coordinates of the vertex of a parabola on a lightweight formula. For example, if b=0, the equation has the form y=Ah^2+C, the vertex will lie on the axis Oy and its coordinates will be zero (0;off). If not only the coefficient b=0 but C=0, the vertex of the parabola is at the origin, the point (0;0).