How to find a parabola ?

How to find a parabola ?

parabola is the graph of a quadratic function.This line has a significant natural value.In order to make it easier to find the vertex of the parabola, it is necessary to draw.Then on the chart with ease can be seen its peak.But to build a parabola, it is necessary to know how to find the point of the parabola and how to find the coordinates of the parabola.

find the point and the top of the parabola

In general presentation quadratic function has the following form: y = ax2 + bx + c.The graph of this equation is a parabola.If the value of a> 0, its branches are directed upwards, and a value of a <0 - down.To construct the graph of the parabola is necessary to know the three-point when it passes along the ordinate.Otherwise, it must be known four point construction.

When finding abscissa (x) it is necessary to take the coefficient of (x) of the polynomial given formula, and then divided by twice the coefficient of (x2), and then multiply by the number of - 1.

To find ordinate need to find a discriminantthen multiply by - 1, and then divided by the coefficient of (x2), pre-multiplying it by 4.

Further, substituting numerical values, the vertex of the parabola is calculated.For all the calculations it is desirable to use a scientific calculator, and when drawing graphs of parabolas and use a ruler and lyumografom, it will significantly improve the accuracy of your calculations.

Consider the following example, which will help us to understand how to find the vertex of the parabola.

x2-9 = 0.In this case, the vertex coordinates are calculated as follows: Point 1 (-0 / (2 * 1), point 2 - (0 ^ 2-4 * 1 * (- 9)) / (4 * 1)).Thus, the values ​​of the coordinates of the vertices are (0, 9).

find the abscissa apex

Once you've learned how to find the parabola, and it can calculate the point of intersection with the coordinate axis (x), we can easily calculate the abscissa of the vertex.

Let (x1) and (x2) are the roots of the parabola.The roots of the parabola - this is the point of intersection with the x-axis.These values ​​vanish quadratic equation of the following form: ax2 + bx + c.

This | x2 |& Gt;| X1 |, then the vertex of the parabola is situated in the middle between them.Thus, it can be found from the following expression: x0 = ½ (| x2 | - | x1 |).

find the area of ​​the figure To find the area of ​​the figure on the coordinate plane you need to know the integral.But to apply it, to know certain algorithms enough.To find the area bounded by the parabolas, it is necessary to make the image in the Cartesian coordinate system.

First, according to the method described above, is determined by the coordinate axis of the vertex (x), then the axis (y), then there is a vertex of the parabola.Now we have to define the limits of integration.Typically, these are shown in the problem using the variables (a) and (b).These values ​​should be placed in the upper and lower parts respectively integral.Then it should be entered in the general form of the function value and multiply it by (dx).In the case of a parabola: (x2) dx.

then need to calculate in general antiderivative of the function value.To do this, use a special table of values.Substituting to the limits of integration, it is the difference.This difference will be a square.

As an example, consider the system of equations: y = x2 + 1, and x + y = 3.

Are abscissa intersection points x1 = 2 and x2 = 1.

believe that v2 = 3, and y1 = x2 + 1, substitute the values ​​in the above formula, and we get a value of 4.5.

Now we know how to find a parabola, and based on these data, calculate the area of ​​the figure, which it limits.