How to find a parabola?

A parabola is a graph of a quadratic function. This line has a significant physical value. In order to make it easier to find the top of a parabola, you need to draw it. Then on the chart you can easily see its top. But to build a parabola, you need to know how to find the points of the parabola and how to find the coordinates of the parabola.

Find the points and the vertex of the parabola

In the general representation, the quadratic function has the following form: y = ax2+ bx + c. The graph of this equation is a parabola. With a value of ›0, its branches are directed upwards, and with a value of a‹ 0, they are directed downwards. To build a parabola on a graph, it is necessary to know three points if it passes along the ordinate axis. Otherwise, four construction points should be known.

When finding the abscissa (x), it is necessary to take the coefficient at (x) from the given polynomial formula, and then divide by the doubled coefficient at (x2), then multiply by the number - 1.

In order to find the ordinate, it is necessary to find the discriminant, then multiply it by - 1, then divide by the coefficient at (x2), after multiplying it by 4.

Further, substituting numerical values, the vertex of the parabola is calculated. For all calculations, it is advisable to use an engineering calculator, and when drawing graphs and parabolas to use a ruler and a lyumograph, this will significantly improve the accuracy of your calculations.

Consider the following example that will help us understand how to find the top of a parabola.

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

Find the abscissa of the top

After you have learned how to find a parabola, and can calculate its intersection points with the coordinate axis (x), you can easily calculate the abscissa of the vertex.

Let (x1) their2) are the roots of the parabola. The roots of a parabola are the points of its intersection with the abscissa axis. These values ​​translate into zero quadratic equation of the following form: ax2+ bx + c.

In this case, | x2| > | x1|, then the top of the parabola is located in the middle between them. Thus, it can be found by the following expression: x0= ½ (| x2| - | x1|).

Find the area of ​​the figure

To find the area of ​​the shape on the coordinate plane, you need to know the integral.And to apply it, it is enough to know certain algorithms. In order to find the area bounded by parabolas, it is necessary to produce its image in the Cartesian coordinate system.

First, by the method described above, the coordinate of the vertex of the axis (x) is determined, then the axis (y), after which the vertex of the parabola is found. Now it is necessary to determine the limits of integration. As a rule, they are specified in the problem statement with the help of variables (a) and (b). These values ​​should be placed in the upper and lower parts of the integral, respectively. Next, you should write in general the value of the function and multiply it by (dx). In the case of a parabola: (x2) dx.

Then you need to calculate in general the primitive value of the function. To do this, use a special table of values. Substituting there the limits of integration, is the difference. This difference will be the area.

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

The abscissas of the intersection points are: x1= -2 and x2=1.

We believe that2= 3, and1= x2+ 1, we substitute the values ​​in the above formula and we get a value of 4.5.

Now we have learned how to find a parabola, and also, based on this data, calculate the area of ​​the figure that it limits.