Summary of key knowledge of quadratic function;
I. Definition and definition of expressions
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.
Two. Three Expressions of Quadratic Function
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 these three forms of mutual transformation, there are the following relations:
h =-b/2a k=(4ac-b^2； )/4a x 1，x2 =(-b √b^2； -4ac)/2a
Three. Quadratic function image
Do quadratic function y=x in plane rectangular coordinate system? Images of,
It can be seen that the image of quadratic function is a parabola.
Four. Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry 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 parabola is Y axis (that is, straight line x=0).
2. The parabola has a vertex p, and the coordinates are
p[-b/2a ,(4ac-b^2； )/4a ].
-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.
Verb (abbreviation of verb) quadratic function and unary quadratic equation
Especially the quadratic function (hereinafter referred to as function) y = ax 2; +bx+c，
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
That's ax^2；; +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.
Several forms of quadratic resolution function
(1) general formula: Y = AX2+BX+C (A, b, c are constants, a≠0).
(2) Vertex: y = a (x-h) 2+k (a, h, k are constants, a≠0).
(3) two expressions: y = a (X-X 1) (X-X2), where X 1, X2 is the abscissa of the intersection of parabola and x axis, that is, the two roots of quadratic equation AX2+BX+C = 0, a≠0.
Description: (1) Any quadratic function can be transformed into vertex Y = A (X-H) 2+K by formula, and the vertex coordinate of parabola is (h, k). When H = 0, the vertex of parabola Y = AX2+K is on the Y axis; When k = 0, the vertex of parabola a(x-h)2 is on the X axis; When H = 0 and K = 0, the vertex of parabola Y = AX2 is at the origin.
Answer supplement
If the image passes through the origin and the axis of symmetry is the y axis, let y = ax 2; If the symmetry axis is the y axis, but not the origin, let y = ax 2+k.
Defining and defining expressions
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.
X is an independent variable and y is a function of X.
Three Expressions of Quadratic Function
① general formula: y = ax 2+bx+c (a, b and c are constants, a≠0).
② Vertex [vertex P(h, k) of parabola]: y = a (x-h) 2+k.
③ Intersection point [only applicable to parabolas with intersection points A(x 1 0) and B(x2, 0) with the X axis ]: y = a (x-x 1) (x-x2).
The above three forms can be converted as follows:
Relationship between (1) General Formula and Vertex Type
For the quadratic function y = ax 2+bx+c, its vertex coordinates are (-b/2a, (4ac-b 2)/4a), that is.
h=-b/2a=(x 1+x2)/2
k=(4ac-b^2)/4a
② Relationship between general formula and intersection point
X 1, x2 = [-b √ (b 2-4ac)]/2a (that is, the formula for finding the root of a quadratic equation with one variable).