Curriculum standard requirements
Test center curriculum standards require knowledge and skill objectives.
Understand, understand and master flexible applications.
Quadratic Function Understanding the Meaning of Quadratic Function
The image of quadratic function is drawn by tracing points.
The opening direction, vertex coordinates and symmetry axis of the parabola will be determined.
Through the analysis of practical problems, the expression of quadratic function is determined.
Understand the relationship between quadratic function and unary quadratic equation
According to the image of parabola y=ax2+bx+c (a≠0), the symbols of A, B and C will be determined.
Knowledge carding
1. Definition: Generally speaking, if it is a constant, it is called a quadratic function of.
2. Using collocation method, the quadratic function can be transformed into the following form: where.
3. Three elements of parabola: opening direction, symmetry axis and vertex.
The symbol of ① determines the opening direction of parabola: when closed, the opening is upward; When, the opening is downward;
Equal, the opening size and shape of parabola are the same.
② Record a straight line parallel to the axis (or coincident). In particular, the axis is recorded as a straight line.
4. Vertex determines the position of parabola. If the coefficients of several different quadratic functions are the same, the opening direction and size of parabola are exactly the same, but the positions of vertices are different.
5. Solution of Parabolic Vertex and Symmetry Axis
(1) formula:, ∴ The vertex is and the symmetry axis is a straight line.
(2) Matching method: By using formula method, the analytical expression of parabola is transformed into a form, the vertex is (,) and the symmetry axis is a straight line.
(3) Using the symmetry of parabola: Because parabola is an axisymmetric figure with an axis of symmetry, the perpendicular line connecting the axis of symmetry is the axis of symmetry of parabola, and the intersection point between the axis of symmetry and parabola is the vertex. The vertex obtained by matching method can only be foolproof if it is verified by formula or symmetry.
6. The role of parabola
(1) Determine the opening direction and size, which is exactly the same as in.
(2) and * * * both determine the position of the parabola axis of symmetry. Because the parabola axis of symmetry is a straight line.
So: ①, the symmetry axis is the axis; (2) (that is, the symbols are the same), and the symmetry axis is on the left side of the shaft; (3) (that is, the symbols are different), and the axis of symmetry is on the right side of the axis.
(3) The size determines the position where the parabola intersects the axis.
When there is only one intersection (0,) between the parabola and the axis:
(1), parabola passing through the origin; (2), the positive semi-axis intersecting the shaft; ③ The axis intersects with the negative half axis.
The above three points are still valid when the conclusions and conditions are exchanged. If the symmetry axis of a parabola is on the right side of the axis, then.
7. Use the undetermined coefficient method to find the analytical formula of quadratic function.
(1) general formula:. Given the values of three points or three pairs on an image, a general formula is usually selected.
(2) Vertex: the vertex or symmetry axis of the image is known, and the vertex is usually selected.
(3) Intersection point: the coordinates of the intersection point between the image and the axis are known, and the intersection point is usually selected.
12. Intersection point of straight line and parabola
The intersection of (1) axis and parabola is (0,).
(2) There is only one intersection (,) between the straight line parallel to the axis and the parabola.
(3) the intersection of parabola and axis
The abscissa of the two intersections between the image of quadratic function and the axis is two real roots corresponding to the quadratic equation of one variable. The intersection of parabola and axis can be judged by the discriminant of the root of the corresponding quadratic equation:
① A parabola with two intersection points intersects the axis;
(2) There is an intersection (the vertex is on the axis), and the parabola is tangent to the axis;
(3) no intersection, parabola and axis separation.
(4) The intersection of a straight line parallel to the axis and a parabola
As in (3), there may be 0 intersections, 1 intersections and 2 intersections. When there are two intersections, the vertical coordinates of the two intersections are equal. If the ordinate is, the abscissa is two real roots of.
(5) The intersection of the image of a linear function and the image of a quadratic function is determined by the number of solutions of the equation: ① There are two intersections when the equation has two different solutions; ② When the equations have only one set of solutions, there is only one intersection with them; ③ The equations have no solution and no intersection.
(6) Distance between two intersections of parabola and axis: If two intersections of parabola and axis are, because sum is two roots of equation, therefore,
Ability training
1. Quadratic function y =-x2+6x-5, if and when it increases, it decreases.
2. The vertex coordinates of parabola are in the third quadrant, and the value is ().
A. B.C.
3. The symmetry axis of parabola y = x2-2x+3 is a straight line ().
a . x = 2 b . x =-2 c . x =- 1d . x = 1
4. The function value of quadratic function y = x2+2x-7 is 8, so the corresponding value of x is ().
A.3b.5c.-3 and 5d.3 and -5
5. The vertex coordinate of parabola y = x2-x is ().
6. The image of quadratic function, as shown in figure 1-2-40, indicates that the relationship between a, b, c and 0 is ().
A.a>0，b0，c>0
C.a