In mathematics there is a whole cycle of identities, among which quadratic equations occupy a significant place. Similar equalities can be solved both separately and for plotting on the coordinate axis. The roots of the quadratic equations are the intersection points of the parabola and the line oh.
General form
The quadratic equation in general has the following structure:
ax 2 + bx + c = 0
In the role of "X" can be considered as individual variables or entire expressions. For instance:
2x 2 + 5x-4 = 0;
(x + 7) 2 +3 (x + 7) + 2 = 0.
In the case when the expression plays the role of x, it is necessary to represent it as a variable and find the roots of the equation. After that, equate the polynomial to them and find x.
So, if (x + 7) = a, then the equation takes the form a 2 + 3a + 2 = 0.
D = 3 2 -4 * 1 * 2 = 1;
a 1 = (- 3-1) / 2 * 1 = -2;
and 2 = (- 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 x-coordinate of the point of intersection of the parabola with the abscissa. In principle, their significance is not so important if the task is only to find the top of the parabola. But for plotting, roots play an important role.
How to find the top of the parabola
Let's go back to the initial equation. To answer the question of how to find the top of the parabola, you need to know the following formula:
x cn = -b / 2a,
where x vp is the x-coordinate of the desired point.
But how to find the vertex of a parabola without a y-coordinate value? We substitute the obtained value of x into the equation and find the desired variable. For example, solve the following equation:
x 2 + 3x-5 = 0
We find the x-coordinate value for the vertex of the parabola:
x bp = -b / 2a = -3 / 2 * 1;
x cn = -1.5.
Find the value of the y-coordinate for the vertex of the parabola:
y = 2x 2 + 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 connection of points having a vertical
axis of symmetry. For this reason, its construction itself is not particularly difficult. The most difficult thing is to make the correct calculations of the coordinates of the points.
It is worth paying special attention to the coefficients of the quadratic equation.
Coefficient a affects the direction of the parabola. In the case when it has a negative value, the branches will be directed down, and with a positive sign - up.
The coefficient b shows how wide the parabola sleeve will be. The greater its value, the wider it will be.
Coefficient c indicates the displacement of the parabola along the axis of the OS relative to the origin.
We already learned how to find the top of the parabola, and to find the roots, you should be guided by the following formulas:
D = b 2 -4ac,
where D is the discriminant that is necessary to find the roots of the equation.
x 1 = (- b + V - D) / 2a
x 2 = (- bV - D) / 2a
The resulting x values will correspond to zero y values, because they are the points of intersection with the axis OX.
After that, we mark the vertex of the parabola and the obtained values on the coordinate plane . For a more detailed graph, you need to find a few more points. To do this, select any value of x allowed by the domain of definition, and substitute it in the equation of the function. The result of the calculations will be the coordinate of the point along the axis of the op-amp.
To simplify the process of plotting, you can draw 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, you can designate the second, equidistant from the drawn line.