I. Definition and definition expression
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
Y = ax 2+bx+c (a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Three Expressions of Quadratic and Quadratic Functions
General formula: y = ax 2; +bx+c(a, b, c are constants, a≠0).
Vertex: y = a (x-h) 2; +k[ vertex P(h, k) of parabola].
Intersection point: y=a(x-x 1)(x-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis].
Note: Among the three forms of mutual transformation, there are the following relationships: h =-b/2ak = (4ac-b2; )/4ax 1，x2 =(-b √b^2； -4ac)/2a .
Cubic and quadratic functions and univariate quadratic equations
Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c .
When y=0, the quadratic function is a unary quadratic equation about x (hereinafter referred to as the equation), that is, ax? +bx+c=0 .
At this point, whether the function image intersects with the X axis means whether the equation has real roots. The abscissa of the intersection of the function and the x axis is the root of the equation.
1. Quadratic function y=ax? ，y=a(x-h)？ ，y=a(x-h)？ +k，y=ax？ The images of +bx+c (a≠0 in all kinds) have the same shape but different positions.
When h>0, y=a(x-h)? Can the image be represented by parabola y=ax? Move the H unit parallel to the right to get it.
When h < 0, it is obtained by moving |h| units in parallel to the left.
When h>0, k>0 and parabola y=ax? Move h units in parallel to the right, and then move k units up, and you can get y=a(x-h)? The image of+K.
When h>0, k<0 and parabola y=ax? Move H units to the right in parallel, and then move down by | k units, and you can get y=a(x-h)? The image of+K.
When h < 0, k >; 0, move the parabola to the left by |h| units in parallel, and then move it up by k units to get y=a(x-h)? The image of+K.
When h < 0, k<0, move the parabola to the left by |h| units in parallel, and then move it down by |k| units to get y=a(x-h)? The image of+K.
So, study the parabola y=ax? The image of +bx+c(a≠0) is transformed into y=a(x-h) by the formula. In the form of +k, its vertex coordinates, symmetry axis and approximate position of parabola can be clearly determined, which provides convenience for drawing images.
2. parabola y=ax? Image of +bx+c(a≠0): When a>0, the opening is upward, while when
3. parabola y=ax? +bx+c(a≠0), if a>0, when x≤-b/2a, y decreases with the increase of x; When x≥-b/2a, y increases with the increase of x, if a
4. parabola y=ax? The intersection of the image of +bx+c and the coordinate axis:
(1) The image must intersect with the Y axis, and the coordinate of the intersection point is (0, c);
(2) when △ = b 2-4ac >; 0, the image intersects the x axis at two points A(x? , 0) and B(x? 0), where x 1, x2 is the quadratic equation ax? +bx+c=0(a≠0)。 The distance between these two points AB=|x? -x？ |。
When △ = 0, the image has only one intersection with the X axis; When delta < 0. The image does not intersect with the x axis. When a >; 0, the image falls above the X axis, and when X is an arbitrary real number, there is y >；; 0; When a<0, the image falls below the X axis, and when X is an arbitrary real number, there is Y.
5. parabola y=ax? Maximum value of +bx+c: if a >；; 0(a & lt； 0), then when x=-b/2a, the minimum (large) value of y =(4ac-b? )/4a。
The abscissa of the vertex is the value of the independent variable when the maximum value is obtained, and the ordinate of the vertex is the value of the maximum value.
6. Find the analytic expression of quadratic function by undetermined coefficient method.
(1) When the given condition is that the known image passes through three known points or three pairs of corresponding values of known x and y, the analytical formula can be set to the general form: y=ax? +bx+c(a≠0)。
(2) When the given condition is the vertex coordinate or symmetry axis of the known image, the analytical formula can be set as the vertex: y=a(x-h)? +k(a≠0)。
(3) When the given condition is that the coordinates of two intersections between the image and the X axis are known, the analytical formula can be set as two formulas: y=a(x-x? )(x-x？ )(a≠0)。
7. The knowledge of quadratic function can be easily integrated with other knowledge, resulting in more complex synthesis problems. Therefore, the comprehensive question based on quadratic function knowledge is a hot topic in the senior high school entrance examination, which often appears in the form of big questions.
Fourth, the nature of parabola
1. Parabola is an axisymmetric figure. The symmetry axis is a straight line x=-b/2a.
The only intersection of the symmetry axis and the parabola is the vertex p of the parabola. Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
2. The parabola has a vertex p, and its coordinates are: P(-b/2a, (4ac-b 2)/4a) When -b/2a=0, p is on the Y axis; When δ = b 2-4ac = 0, p is on the x axis.
3. Quadratic coefficient A determines the opening direction and size of parabola.
When a > 0, the parabola opens upward; When a < 0, the parabola opens downward. The larger the |a|, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis;
When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the y axis at (0, c).
6. Number of intersections between parabola and X axis
When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
When δ = b 2-4ac < 0, the parabola has no intersection with the X axis. The value of x is an imaginary number (the reciprocal of the value of x =-b √ b 2-4ac, multiplied by the imaginary number I, and the whole formula is divided by 2a).