Defined by the formula $a((x)^(2))+bx+c$ $(a\ne 0).$ The numbers $a, b$ and $c$ are the coefficients of a quadratic trinomial, they are usually called: a - the leading one, b - second or average coefficient, c - free term. A function of the form y = ax 2 + bx + c is called a quadratic function.
All of these parabolas have their vertex at the origin; for a > 0 this is the lowest point of the graph (the smallest value of the function), and for a< 0, наоборот, наивысшая точка (наибольшее значение функции). Ось Oy есть ось симметрии каждой из таких парабол.
As can be seen, for a > 0 the parabola is directed upward, for a< 0 - вниз.
There is a simple and convenient graphical method that allows you to construct any number of points of the parabola y = ax 2 without calculations, if a point of the parabola other than the vertex is known. Let the point M(x 0 , y 0) lie on the parabola y = ax 2 (Fig. 2). If we want to construct an additional n points between points O and M, then we divide the segment ON of the abscissa axis into n + 1 equal parts and at the division points we draw perpendiculars to the Ox axis. We divide the segment NM into the same number of equal parts and connect the division points with rays to the origin of coordinates. The required points of the parabola lie at the intersection of perpendiculars and rays with the same numbers (in Fig. 2 the number of division points is 9).
The graph of the function y = ax 2 + bx + c differs from the graph y = ax 2 only in its position and can be obtained simply by moving the curve on the drawing. This follows from the representation of the quadratic trinomial in the form
from which it is easy to conclude that the graph of the function y = ax 2 + bx + c is a parabola y = ax 2, the vertex of which is moved to the point
and its axis of symmetry remained parallel to the Oy axis (Fig. 3). From the resulting expression for a quadratic trinomial, all its basic properties easily follow. The expression D = b 2 − 4ac is called the discriminant of the quadratic trinomial ax 2 + bx + c and the discriminant of the associated quadratic equation ax 2 + bx + c = 0. The sign of the discriminant determines whether the graph of the quadratic trinomial intersects the x-axis or lies on the same side from her. Namely, if D< 0, то парабола не имеет общих точек с осью Ox, при этом: если a >0, then the parabola lies above the Ox axis, and if a< 0, то ниже этой оси (рис. 4). В случае D >0 the graph of a quadratic trinomial intersects the x-axis at two points x 1 and x 2, which are the roots of the quadratic equation ax 2 + bx + c = 0 and are equal, respectively
At D = 0 the parabola touches the Ox axis at the point
The properties of the quadratic trinomial form the basis for solving quadratic inequalities. Let's explain this with an example. Suppose we need to find all solutions to the inequality 3x 2 - 2x - 1< 0. Найдем дискриминант квадратного трехчлена, стоящего в левой части неравенства: D = 16. Так как D >0, then the corresponding quadratic equation 3x 2 − 2x − 1 = 0 has two different roots, they are determined by the formulas given earlier:
x 1 = −1/3 and x 2 = 1.
In the quadratic trinomial under consideration, a = 3 > 0, which means that the branches of its graph are directed upward and the values of the quadratic trinomial are negative only in the interval between the roots. So, all solutions to the inequality satisfy the condition
−1/3 < x < 1.
Various inequalities can be reduced to quadratic inequalities by the same substitutions by which various equations are reduced to quadratic ones.
Definition
Parabola is called the graph of a quadratic function $y = ax^(2) + bx + c$, where $a \neq 0$.
Graph of the function $y = x^2$.
To schematically plot the graph of the function $y = x^2$, we will find several points that satisfy this equality. For convenience, we write down the coordinates of these points in the form of a table:
Graph of the function $y = ax^2$.
If the coefficient $a > 0$, then the graph $y = ax^2$ is obtained from the graph $y = x^2$ either by vertical stretching (for $a > 1$) or compression to the $x$ axis (for $0< a < 1$). Изобразим для примера графики $y = 2x^2$ и $y = \dfrac{x^2}{2}$:
| $y = 2x^2$ | $y = \dfrac(x^2)(2)$ |
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If $a< 0$, то график функции $y = ax^2$ можно получить из графика $y = |a|x^2$, отразив его симметрично относительно оси $x$. Построим графики функций $y = - x^2$, $y = -2x^2$ и $y = - \dfrac{x^2}{2}$:
| $y = - x^2$ | $y = -2x^2$ | $y = - \dfrac(x^2)(2)$ |
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Graph of a quadratic function.
To plot the function $y = ax^2 + bx + c$, you need to isolate a complete square from the quadratic trinomial $ax^2 + bx + c$, that is, represent it in the form $a(x - x_0)^2 + y_0$ . The graph of the function $y = a(x - x_0)^2 + y_0$ is obtained from the corresponding graph $y = ax^2$ by shifting by $x_0$ along the $x$ axis, and by $y_0$ along the $y$ axis. As a result, point $(0;0)$ will move to point $(x_0;y_0)$.
Definition
The top the parabola $y = a(x - x_0)^2 + y_0$ is the point with coordinates $(x_0;y_0)$.
Let's construct a parabola $y = 2x^2 - 4x - 6$. Selecting the complete square, we get $y = 2(x - 1)^2 - 8$.
| Let's plot $y = 2x^2$ | Let's move it to the right by 1 | And down by 8 |
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The result is a parabola with its vertex at point $(1;-8)$.
The graph of the quadratic function $y = ax^2 + bx + c$ intersects the $y$ axis at the point $(0; c)$ and the $x$ axis at the points $(x_(1,2);0)$, where $ x_(1,2)$ are the roots of the quadratic equation $ax^2 + bx + c = 0$ (and if the equation has no roots, then the corresponding parabola does not intersect the $x$ axis).
For example, the parabola $y = 2x^2 - 4x - 6$ intersects the axes at the points $(0; -6)$, $(-1; 0)$ and $(3; 0)$.
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