In mathematics, there is a whole series of identities, among which an important place occupied by the quadratic equation.Such equality can be addressed separately, and charting on the coordinate axes.The roots of quadratic equations are the points of intersection of a parabola and a straight oh.
quadratic equation in general has the following structure:
ax2 + bx + c = 0
In the role of "X's" can be viewed as separate variables, and the whole expression.For example:
2x2 + 5x-4 = 0;
(x + 7) 2 + 3 (x + 7) + 2 = 0.
In the case where x stands as an expression, you need to submit it as a variable, and to find the roots of the equation.After that equate them and find the polynomial 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.
the root equal to -2 and -1, we get the following:
x + 7 = 2 and x + 7 = -1;
x = -9, and x = -8.
roots are the x-coordinate value of the point of intersection of the parabola with the x-axis.In principle, their importance is not so important when the goal is to find a vertex of the parabola.But for plotting roots play an important role.
How to find the top of the parabola
return to the initial equation.To answer the question of how to find the top of the parabola, it is necessary to know the following formula:
xvp = -b / 2a,
hvp- which is the value of x-coordinate of the desired point.
But how to find the top of the parabola without value y-coordinates?The expansion is the value of x in the equation and find the desired variable.For example, we solve the following equation:
x2 + 3x-5 = 0
find the value of x-coordinate of the vertex of the parabola:
hvp = -b / 2a = -3 / 2 * 1;
hvp = -1.5.
find the value of y-coordinate for the vertex of the parabola:
y = 2x2 + 4x-3 = (- 1,5) 2 + 3 * (- 1.5) -5;
y = -7.25.
The result is that the vertex of the parabola is located at coordinates (-1.5, -7.25).
parabola Parabola is connecting the dots having a vertical axis of symmetry.For this reason, its very construction is not difficult.The most difficult - is to make correct calculations of coordinates of points.
should pay particular attention to the coefficients of a quadratic equation.
factor and affect the direction of the parabola.In the case when it has a negative value, the branches are directed downward, and the positive sign - up.
coefficient b indicates how wide the sleeve of a parabola.The higher the value, the greater it will be.
factor to indicate an offset of a parabola on the y-axis relative to the origin.
How to find the top of the parabola, we have already learned, and to find the roots, should be guided by the following formulas:
D = b2-4ac,
where D - is the discriminant, which is necessary for finding the roots of the equation.
x1 = (- b + V-D) / 2a
x2 = (- bV-E) / 2a
obtained values of x will correspond to zero values have sinceThey are the points of intersection with the x-axis.
After this note on the coordinate plane vertex of the parabola and the obtained values.For a more detailed schedule is necessary to find a few more points.To do this, choose any value of x, permissible domain, and substitute it in the equation of the function.The result of the calculation will coordinate of the point on the y-axis.
To simplify the process of plotting, you can draw a vertical line through the vertex of the parabola and perpendicular to the x-axis.This will be the axis of symmetry, by means of which, having a single point, it is possible to designate and second equidistant from the drawn line.