How to find the vertex of a parabola and build it

В математике есть целый цикл тождеств, среди which occupy a significant place quadratic equations. Similar equalities can be solved both separately and for plotting graphs on the coordinate axis. The roots of the quadratic equations are the points of intersection of the parabola and the straight line ox.

General form

The quadratic equation has the following general structure:

Oh² + bx + c = 0

In the role of "X" can be considered as separate variables, and entire expressions. For example:

2x²+ 5x-4 = 0;

(x + 7)²+3 (x + 7) + 2 = 0.

In the case when an expression appears in the role of x, it is necessary to represent it as a variable and find the roots of the equation. After that, equate them with a polynomial and find x.

So, if (x + 7) = a, then the equation takes the form a²+ 3a + 2 = 0.

D = 3²-4 * 1 * 2 = 1;

a₁= (- 3-1) / 2 * 1 = -2;

a₂= (- 3 + 1) / 2 * 1 = -1.

With roots equal to -2 and -1, we get the following:

x + 7 = -2 and x + 7 = -1;

x = -9 and x = -8.

The roots are the value of the x-coordinate of the pointthe intersection of the parabola with the abscissa axis. In principle, their importance is not so important if the task is only to find the vertex of a parabola. But for plotting the roots play an important role.

How to find the vertex of a parabola

Let us return to the initial equation. To answer the question of how to find the vertex of a parabola, it is necessary to know the following formula:

from_in= -b / 2a,

where x_inis the value of the x-coordinate of the desired point.

But how to find the vertex of a parabola without the y-coordinate value? We substitute the obtained value of x into the equation and find the required variable. For example, we solve the following equation:

x²+ 3x-5 = 0

We find the value of the x-coordinate for the vertex of the parabola:

x_in= -b / 2a = -3 / 2 * 1;

x_in= -1.5.

We find the value of the y-coordinate for the vertex of the parabola:

y = 2x²+ 4x-3 = (- 1.5)²+3 * (-1.5) -5;

y = -7.25.

As a result, we get that the vertex of the parabola is at the point with coordinates (-1.5, -7.25).

Building a parabola

A parabola is a union of points,having a vertical axis of symmetry. For this reason, its very construction is not very difficult. The most difficult thing is to make correct calculations of the coordinates of points.

It is worth paying special attention to the coefficients of the quadratic equation.

The coefficient a affects the direction of the parabola. In the case when it has a negative value, the branches will be directed downwards, and with a positive sign - up.

The coefficient b shows how wide the sleeve of the parabola is. The larger its value, the wider it will be.

The coefficient c indicates the displacement of the parabola along the OY axis relative to the origin.

How to find the vertex of a parabola, we have already learned, and in order to find the roots, one should follow the following formulas:

D = b²-4ac,

where A is the discriminant required to find the roots of the equation.

from₁= (- in + B^-D) / 2a

from₂= (- b-V^-D) / 2a

The obtained values ​​of x will correspond to zero values ​​of y, since they are points of intersection with the axis OX.

After this, we mark in the coordinate planethe vertex of the parabola and the values ​​obtained. For a more detailed graph, you need to find a few more points. To do this, we select any value of x that is allowed by the domain of definition, and substitute it into the equation of the function. The result of the calculations is the coordinate of the point along the OY axis.

To simplify the process of building a graph, you candraw a vertical line through the top of the parabola and perpendicular to the axis OX. This will be the axis of symmetry, with the help of which, having one point, it is possible to designate the second one equidistant from the drawn line.