How to find a parabola?
A parabola is a graph of a quadratic function. This line has a significant physical meaning. To make it easier to find the vertex of a parabola, you need to draw it. Then on the chart you can easily see its top. But in order to build a parabola, it is necessary to know how to find the points of a parabola and how to find the coordinates of a parabola.
We 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. If a> 0, its branches are directed upwards, and for a <0 - down. To construct a parabola on a graph, it is necessary to know three points if it passes along the ordinate axis. Otherwise, four points of construction should be known.
When finding the abscissa (x), it is necessary to take the coefficient of (x) from the given polynomial formula, and then divide by twice the coefficient at (x2), then multiply by a 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), first multiplying it by 4.
Further, substituting numerical values, it is calculatedvertex of a parabola. For all calculations it is desirable to use an engineering calculator, and when drawing charts and parabolas to use a ruler and a lumo-graph, this will significantly increase the accuracy of your calculations.
Consider the following example, which will help us understand how to find the vertex 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 vertex
After you have learned how to find a parabola, and you can calculate the points of its intersection with the coordinate axis (x), you can easily calculate the abscissa of the vertex.
Suppose that (x1) their2) are the roots of a parabola. The roots of the parabola are the points of its intersection with the abscissa axis. These values turn to zero a quadratic equation of the following form: ax2 + bx + c.
In this case | x2| | > | x1|, then the vertex 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 a figure on a coordinateyou 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 a Cartesian coordinate system.
First, according to the method described above, thecoordinate of the vertex of the axis (x), then the axis (y), after which the vertex of the parabola is located. Now we need to determine the limits of integration. As a rule, they are indicated in the condition of the problem using variables (a) and (b). These values should be placed in the upper and lower parts of the integral, respectively. Next, enter the value of the function in a general form and multiply it by (dx). In the case of a parabola: (x2) dx.
Then we need to calculate in a general form the antiderivativethe value of the function. To do this, use a special table of values. Substituting there the limits of integration, there is a difference. This difference will be the area.
As an example, consider the system of equations: y = x2+1 and x + y = 3.
There are abscissas of points of intersection: x1= -2 and x2= 1.
We assume that y2= 3, while y1= x2 + 1, substitute the values in the above formula and get a value of 4.5.
Now we have learned how to find a parabola, and also, based on these data, calculate the area of the figure, which it restricts.
