22. As shown in Figure 9, in the plane rectangular coordinate system, the quadratic function? The vertex of the image is point d,

It intersects with the Y axis at point C, and intersects with the X axis at points A and B. Point A is on the left side of the origin, and the coordinate of point B is (3,0).

OB=OC？ ，tan∠ACO=？ .

(1) Find the expression of this quadratic function.

(2) The straight line passing through points C and D intersects the X axis at point E. Is there such a point F on this parabola, and the quadrilateral with points A, C, E and F as its vertices is a parallelogram? If it exists, request the coordinates of point f; If it does not exist, please explain why.

(3) If the straight line parallel to the X axis intersects the parabola at two points, M and N, and the circle with the diameter of MN is tangent to the X axis, find the length of the radius of the circle.

(4) As shown in figure 10, if point G(2, y) is a point on the parabola and point P is a moving point on the parabola below the straight line AG, when point P moves to what position, what is the maximum area of △APG? Find the coordinates of point P and the maximum area of △APG at this time.

Solution: (1) Method 1: From the known: C(0, -3),

A(- 1，0)？ ......................... 1 min.

Substitute the coordinates of a, b and c? ....................., two points.

Solution:? ..............................., 3 points.

So the expression of this quadratic function is

Method 2: From the known: C(0, -3), a (- 1, 0). .................................................................................................................................

Let this expression be:? ..............................., two points.

Substitute the coordinates of point C.

So the expression of this quadratic function is

(Note: The final result of the expression will not be deducted in any of the three forms)

(2) Method 1: Existence, and the coordinate of point F is (2, -3)? ..............................., 4 points.

Reason: D( 1, -4) is easy to get, so the analytical formula of linear CD is:?

∴ The coordinate of point E is (-3,0) ............................ 4 points.

From the coordinates of a, c, e and f, AE = cf = 2, AE ‖ cf.

A quadrilateral with vertices a, c, e and f is a parallelogram.

∴ There is a point f with coordinates (2, -3)? Five points.

Method 2: D( 1, -4) is easy to obtain, so the analytical formula of linear CD is:?

∴ The coordinate of point E is (-3,0) ............................ 4 points.

A quadrilateral with vertices A, C, E and F is a parallelogram.

∴ The coordinates of point F are (2, -3) or (-2, -3).

Or (-4,3)?

Only (2, -3) satisfies the parabolic expression test.

∴ There is point F, and the coordinates are (2, -3). ...................................................................................................................................................

(3) As shown in the figure, ① When the straight line MN is above the X axis, let the radius of the circle be r (r >; 0), then N(R+ 1, r),

Substitute the expression of parabola and find the solution? ........................ scored six points.

② When the straight line MN is below the X axis, let the radius of the circle be r (r >; 0),

Then N(r+ 1, -r),

Substitute the parabola expression and you get ... 7 points.

∴: What is the radius of this circle? Or? . ? Seven points

(4) When the Y axis intersects with AG at point Q, the parallel line passing through point P,

It is easy to get G(2, -3), and the straight line AG is? .........................., eight.

Let P(x,? ), then Q(x, -x- 1), PQ? .

Nine points

What time? △APG has the largest area.

At this point, what are the coordinates of point P? ,? .......................... 10.

Analysis of Mathematics Final Exam of Shenzhen Senior High School Entrance Examination in 2009

22.(9 o'clock) As shown in the figure, in the rectangular coordinate system, the coordinate of point A is (-2,0), connect OA, and rotate the line segment OA clockwise around the origin o 120 to get the line segment OB.

(1) Find the coordinates of point B;

(2) Find the analytical formula of parabola passing through points A, O and B;

(3) Is there a point C on the axis of symmetry of the parabola in (2) that minimizes the circumference of △BOC? If it exists, find the coordinates of point C; If it does not exist, please explain why.

(4) If point P is the moving point on the parabola in (2) and below the X axis, is the area of △PAB the largest? If yes, calculate the coordinates of point P and the maximum area of delta delta △PAB at this time; If not, please explain why.

Solution: (1)B( 1,? )

(2) Let the analytical formula of parabola be y=ax(x+a) and substitute it into point B (1,). What is the result? ,

So what?

(3) As shown in the figure, the symmetry axis of the parabola is the straight line X =- 1. When point C is located at the intersection of symmetry axis and line segment AB, the circumference of △BOC is the smallest.

Let the straight line AB be y = kx+B, then? ,

So the straight line AB is? ,

When x =- 1 ,

So the coordinates of point C are (-1,? ).

(4) As shown in the figure, parallel lines with P as the Y axis intersect AB at D 。

When x =-? What is the maximum delta δ△PAB area? At this time? .

Solution: (1)⊙P is tangent to the x axis.

The straight line Y =-2x-8 intersects the X axis at a (4 4,0),

Intersect with the y axis at B(0, -8),

∴OA=4,OB=8.

From the meaning of the question, op =-k,

∴PB=PA=8+k.

In Rt△AOP, k2+42=(8+k)2,

∴ k =-3, ∴op =∫p radius,

∴⊙P is tangent to the x axis.

(2) Let ⊙P intersect with the straight line L at two points C and D to connect PC and PD. When the center p is on the line segment OB, do PE⊥CD at point E.

∫△PCD is a regular triangle, ∴DE=? CD=？ ，PD=3，

∴PE=？ .

∠∠AOB =∠PEB = 90，？ ∠ Po =∠PBE,

∴△AOB∽△PEB，

∴,

∴?

∴? ,

∴? ,

∴? .

When the center p is on the extension line of line segment OB, P (0,-? -8),

∴k=-？ -8,

When k=? -8 or k =-? At -8, the triangle whose vertex is the intersection of ⊙P and the straight line L and the center P is a regular triangle.

Solving problems in the final exam of mathematics for senior high school entrance examination

Solution questions occupy a considerable proportion in the senior high school entrance examination, mainly composed of comprehensive questions. As far as the types of questions are concerned, they include calculation questions, proof questions and application questions. Their characteristics and examination functions determine the complexity of thinking and the diversity of problem-solving design. Generally speaking, the problem-solving design depends on the method of solving the problem, whether it is overall consideration or local association. The principles of determining the method are: familiarity principle and concreteness principle; Simplification principle, harmony principle, etc.

(1) To solve the comprehensive and final questions, we should grasp the following links:

1. Examination: This is the beginning and basis for solving the problem. It is necessary to comprehensively investigate all the conditions and answering requirements of the question type, so as to correctly and comprehensively understand the meaning of the question, grasp the characteristics and structure of the question type as a whole, and facilitate the selection of problem-solving methods and the design of problem-solving steps.

We should grasp the "three characteristics" in the thinking of examining questions, that is, to clarify the purpose, improve the accuracy and pay attention to the meaning. The practice of solving problems shows that conditional suggestion can recognize and inspire the means of solving problems, and the conclusion can predict and summarize the direction of solving problems. Only by carefully examining the questions can we get as much information as possible from the questions themselves. Don't be afraid of slowness in this step. In fact, there is "quickness" in "slowness". The direction of solving the problem is clear and the means of solving the problem are reasonable and appropriate.

2. Seeking reasonable ideas and methods to solve problems: Breaking away from the model and striving for innovation are the remarkable characteristics of the mathematics test questions in the senior high school entrance examination in recent years, especially solving problems. Therefore, it is forbidden to apply mechanical mode to seek ideas and methods for solving problems, but to identify the conditions and conclusions of problems from different sides and angles, and to understand the relationship between conditions and conclusions, the relationship between geometric characteristics of figures and the number and structural characteristics of numbers and formulas. Carefully determine the ideas and methods to solve the problem. When thinking is blocked, we should adjust our thinking and methods in time, re-examine the meaning of the question, pay attention to excavating the implied conditions and internal relations, and prevent us from falling into a dead end and giving up easily.

Parabolic figure

Generally speaking, most of these problems are final, and the basic idea of solving them is analysis and synthesis. In addition to using the core knowledge of algebra and geometry flexibly, we should also pay attention to the application of basic mathematical thinking methods such as classification, combination of numbers and shapes, and transformation.

23. As shown in the figure, in the plane rectangular coordinate system, the straight line L: Y =-2x-8 intersects the X axis and the Y axis at two points A and B respectively, and the point P(0, k) is a moving point on the negative semi-axis of the Y axis, with P as the center and 3 as the radius.

(1) Connect PA, if PA=PB, try to judge the position relationship between ⊙P and X axis, and explain the reasons;

(2) When k is what value, is the triangle whose vertex is ⊙P and the intersection of straight line L and center p a regular triangle?