Generally speaking, if, then it is called the quadratic function of.

[Note] (1) In the quadratic function, X and Y are variables and constants, and the range of the independent variable X is real numbers, and B and C can be arbitrary real numbers, but they cannot be real numbers of 0;

(2) If, at that time, it becomes a linear function; At that time, it was a constant function;

⑶ To judge whether a function is a quadratic function, three conditions must be met: ① The function relation must be an algebraic expression; ② The maximum number of independent variables after simplification must be 2; ③ The coefficient of the simplified quadratic term shall not be 0;

Knowledge point 2: the image of quadratic function

(1) an important condition hidden in quadratic function is.

⑵ The image of quadratic function is an axisymmetrical curve called parabola, and the intersection of the axis of symmetry and parabola is called the vertex of parabola; The vertex of is the coordinate origin and the axis of symmetry is the axis.

(3) If the straight line parallel to the axis intersects the parabola at two points, it can be known from the symmetry that the midline of the line segment is the axis relative to the axis symmetry.

Knowledge point 3: the step of drawing quadratic function image by tracing point method

(1) list, generally centered on 0, selects the value of the independent variable, and generally takes an integer for convenience.

⑵ Draw points, taking each pair of corresponding values of independent variables and functions as the abscissa and ordinate of points respectively, and draw corresponding points in the plane rectangular coordinate system.

⑶ Connection: Connect the points with smooth curves according to the order of independent variables from small to large.

[Note] (1) stippling only draws a part of the whole function image. Because the independent variable can take all real numbers, the image extends infinitely in two directions;

⑵ The more points, the more accurate the image;

⑶ Images must be connected with smooth curves, and no sharp points can be produced;

(4) The quadratic function image is recorded as a parabola.

Knowledge point 4: Properties of quadratic function

The vertex of a parabola is the coordinate origin, and the axis of symmetry is the axis.

⑵ The relationship between image and symbol of function.

① At that time, the top vertex of the parabolic opening was its lowest point, and then it increased with the increase of; At that time, it decreased with the increase of;

② At that time, the lower vertex of parabolic opening was its highest point, and then it decreased with the increase of θ; At that time, it increased with the increase of;

(3) The analytic form of parabola with vertex as coordinate origin and axis of symmetry is.

Knowledge point five: the image of quadratic function

(1) The image of quadratic function is a parabola with parallel symmetry axes (including coincidence).

⑵ Get the image of quadratic function by translating up (or down).

At that time, the quadratic function was translated upward by unit.

At that time, the quadratic function was translated downward in units.

⑶ Shift the image of quadratic function to the left (or right).

At that time, the quadratic function shifted units to the right.

At that time, the quadratic function shifted to the left.

(4) Move the unit to the left (or right), and then move the unit up (or down) to get the image of the quadratic function. The symmetry axis is a straight line, and the vertex coordinates are

5] The general form of quadratic function can be transformed into vertices by matching method:

Therefore, the symmetry axis of parabola is;

Vertex coordinates are.

[Note] The symbol ① determines the opening direction of the parabola: at this time, the opening is upward; At that time, the opening was downward;

Equal, the opening size and shape of parabola are the same.

② If several different quadratic functions are the same, the direction, size and shape of the opening are exactly the same, but the positions of the vertices are different. Vertex determines the position of parabola, and the movement of parabola mainly depends on the movement of vertex, and translation has nothing to do with the order of moving up, down, left and right.

(3) record a straight line parallel to the axis (or coincidence). In particular, the axis is recorded as a straight line.

Knowledge point 6: Drawing method of quadratic function image

(1) tracking method, the steps are as follows:

Convert the quadratic function into the form of;

Determine the opening direction, symmetry axis and vertex coordinates;

On both sides of the axis of symmetry, draw points symmetrically around the vertex.

【 Note 】 If the parabola intersects with the axis, it is best to choose the intersection, especially when drawing parabola, the following five points should be grasped: (1) opening direction; Symmetry axis; (3) Vertex; (4) the intersection with the axis; 5] Intersection point with the axis.

2 translation method, its steps are as follows:

① Transform quadratic function into fixed vertex by collocation method;

(2) the produced image;

③ Translate the image of parabola to make its vertex move to.

Knowledge point 7: Solution of Parabolic Vertex and Symmetry Axis

⑴ Formula method: ∴ Vertex is, and symmetry axis is a straight line.

⑵ Matching method: Using formula method, the analytical expression of parabola is transformed into a form, the vertex is (,) and the symmetry axis is a straight line.

⑶ Using the symmetry of parabola: Because parabola is an axisymmetric figure, the perpendicular line connecting the symmetrical points is the symmetry axis of parabola, and the intersection of symmetry axis and parabola is the vertex.

Note: Vertex obtained by matching method can be verified by formula method or symmetry.

Knowledge point 8: Properties of quadratic function

(1) in the parabola, role.

① Determine the opening direction and opening size, which are exactly the same as in.

② and * * * both determine the position of the parabola symmetry axis. Because the symmetry axis of parabola 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.

At that time, ∴ parabola and axis have only one intersection (0,):

Ⅰ. Parabola passes through the origin; Ⅱ, the positive semi-axis intersecting the axis; ⅲ, intersecting the axis at the negative semi-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.

④ The image features of several special quadratic functions are as follows:

Vertex coordinates of opening direction, symmetry and extreme symmetry axis of analytic function

At this time, the opening is upward, the left side of the symmetry axis decreases with the increase, the right side of the symmetry axis increases with the increase, and the vertex is the lowest point. When the abscissa of the vertex is taken, the ordinate of the vertex is taken as the minimum value of the function. At this time, the opening is downward, the left side of the symmetry axis increases with the increase, the right side of the symmetry axis decreases with the increase, and the vertex is the highest point. When the abscissa of the vertex is taken, the ordinate of the vertex is taken as the maximum value of the function. (Axis) (0,0)

(Axis) (0,)

(,0)

(,)

()

Knowledge point 9: Find the analytic formula of quadratic function by undetermined coefficient method.

⑴ General formula: Given three points or three pairs of values on an image, the general formula is usually selected.

⑵ 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.

Knowledge point 10: the intersection of a straight line and a parabola

The intersection of (1) axis and parabola is (0,).

⑵ 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,, are 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.

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 the two intersections of parabola and axis: If the two intersections of parabola and axis are, and are the two roots of the equation, therefore,

Knowledge point 1 1: the extreme value of quadratic function within the specified range.

Find the maximum value of the quadratic function within the specified range, that is, when the independent variable range of the quadratic function is limited (at this time, the function image is only a part of a parabola), the maximum value of the quadratic function is often determined by observing or calculating the function value at the end of this parabola and comparing it with the function value at the vertex.

If the range of the independent variable is 0, it depends on whether it is within the range of the independent variable.

If it is within this range, then at that time,

If it is not within this range, you need to consider adding or reducing functions within this range:

If the function increases with the increase of, then

At that time,

At that time,

If the function decreases with the increase of, then

At that time,

At that time,