As shown in the figure, it is known that the parabola y=-x2+bx+c intersects the X axis and the Y axis at points A (- 1, 0) and B (0 0,3) respectively, and its vertex is d. 。

(1) Find the analytical formula of parabola;

(2) If the other intersection of the parabola and the X axis is E, find the area of the quadrilateral ABDE;

(3) Are △ AOB and △BDE similar? If similar, please prove it; If not, please explain why.

(Note: the vertex coordinates of parabola y=ax2+bx+c(a≠0) are)

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2.(08 Quzhou, Zhejiang) The position of the known right-angled trapezoidal paper OABC in the plane rectangular coordinate system is shown in the figure. The coordinates of the four vertices are O (0 0,0), A (10/0,0), B (8 8,0), C (0 0,0), and the point T is on the line segment OA (not coincident with the line segment endpoint). Fold this paper for emphasis.

(1) Find the number of times ∠OAB, and find the functional relationship between S and T when point A' is on line AB;

(2) When the figure of the overlapping part of the paper is quadrilateral, find the value range of t;

(3) Is there a maximum value for S? If it exists, find this maximum value and find the value of t at this time; If it does not exist, please explain why.

3.(08 Wenzhou, Zhejiang) As shown in the figure, the middle,,, and are the midpoint of the edge, and the point starts from the point and moves in the direction, making an intersection and making an intersection.

When a point coincides with a point, the point stops moving.

(1) Find the length of the distance from the point to the point;

(2) Find the functional relationship about (the range of independent variables is not required);

(3) Is there a point that makes it an isosceles triangle? If it exists, request all the values that meet the requirements; If it does not exist, please explain why.

4.(08 Rizhao City, Shandong Province) in △ABC, ∠ A = 90, AB = 4, AC = 3, where M is the moving point on AB (not coincident with A and B), passing through M is MN∨BC, AC is at N point, and Mn is the diameter, then ⊙O is in.

The area s of (1)△MNP is expressed by an algebraic expression with x;

(2) When what is the value of x, ⊙O is tangent to the straight line BC?

(3) In the process of moving point M, remember that the overlapping area of △MNP and trapezoidal BCNM is y, try to find the functional expression of y about x, and find the value of x and the maximum value of y?

5. (Zhejiang Jinhua, 2007) As shown in figure 1, hyperbola y =(k >；; 0) The straight line y = k ′ x intersects at point A and point B, and point A is in the first quadrant. Try to solve the following problems: (1) If the coordinate of point A is (4,2), the coordinate of point B is; If the abscissa of point A is m, the coordinates of point B can be expressed as:

(2) As shown in Figure 2, make a straight line L after crossing the origin O, and cross the hyperbola y =(k>；; 0) at p and q, point p is in the first quadrant. ① indicates that the quadrilateral APBQ must be a parallelogram; ② the abscissas of points a and p are m and n, respectively. Can quadrilateral APBQ be a rectangle? Will it be a square? If possible, directly write out the conditions that mn should meet; If not, please explain why.

6. (Jinhua, Zhejiang, 2008) As shown in Figure 1, in the plane rectangular coordinate system, it is known that AOB is an equilateral triangle, the coordinates of point A are (0,4), point B is in the first quadrant, and point P is the moving point on the X axis. Connect AP, and rotate AOP counterclockwise around point A to make the edges AO and AB coincide, thus obtaining Abd. (2) When point P moves to point (0), find the length of DP at this time and the coordinates of point D; (3) Whether there is a point P, so that the areas of Δ δOPD are equal, and if there is, the coordinates of the point P meeting the requirements are requested; If it does not exist, please explain why.

7. (Yiwu, Zhejiang, 2008) As shown in figure 1, the quadrilateral ABCD is a square, and G is the moving point on the edge of CD (the G point does not coincide with C and D). Take CG as one side and make a square CEFG outside the square ABCD to connect BG and de. We explore the length relationship between line segment BG and line segment DE and the position relationship of straight line in the following figure:

(1)① guess the length relationship between line segment BG and line segment DE and the position relationship of straight line as shown in figure1;

② Rotate the square CEFG in figure 1 clockwise (or counterclockwise) at any angle around point C, and get the situation as shown in figures 2 and 3. Please observe and measure whether the conclusion drawn from diagram 1 is still valid, and choose Figure 2 to prove your judgment.

(2) The square in the original problem is changed into a rectangle (as shown in Figure 4-6), AB=a, BC=b, CE=ka, CG=kb (a b, k 0). Which conclusions are valid and which are not? If so, take Figure 5 as an example to briefly explain the reasons.

(3) In Figure 5 of the problem (2), connect,, and a=3, b=2, k=, and evaluate.

8. (Yiwu, Zhejiang, 2008) As shown in figure 1, the vertices A and C of the right-angled trapezoidal OABC are on the positive and negative semi-axes of the Y axis respectively. After passing through point B and point C, the straight line translates, and the translated straight line intersects the axis of point D and the axis of point E. 。

(1) translate the straight line to the right, let the translation distance CD be (t 0), and the area swept by the straight line (the shaded part in the figure) be. The correlation function image is shown in Figure 2. OM is a line segment, MN is a part of a parabola, NQ is a ray, and the abscissa of n points is 4.

① Find the length of the trapezoid upper bottom AB and the area of the right-angled trapezoid OABC;

(2) When, find the resolution function of S;

(2) Under the condition of the problem (1), when the straight line moves to the left or right (including overlapping with the straight line BC), is there a point P on the straight line AB, which makes it an isosceles right triangle? If it exists, please directly write the coordinates of all points p that meet the conditions; If it does not exist, please explain why.

9. (Yantai, Shandong Province, 2008) As shown in the figure, the side length of rhombic ABCD is 2, BD=2, E and F are two moving points on the sides of AD and CD respectively, AE+CF=2.

(1) Verification: △ BDE △ BCF;

(2) Judge the shape of △BEF and explain the reason;

(3) Let the area of △BEF be S and find the range of S. 。

10. (Yantai, Shandong, 2008) As shown in the figure, the parabola intersects at point A and point B and at point M. After the parabola is moved to the right by 2 units, it intersects at point C and point D. 。

(1) Find the function expression corresponding to parabola;

(2) Whether the parabola or the part above the axis has a point n, so that the quadrilateral with a, c, m and n as its vertices is a parallelogram. If yes, find out the coordinates of point n; If it does not exist, please explain the reason;

(3) If point P is a moving point on a parabola (point P does not coincide with point A and point B), then whether the symmetric point Q of point P about the origin is on a parabola, please explain the reasons.

1 1.2008 Xijiang Ningbo) In May 2008 1 day, the longest sea-crossing bridge in the world-Hangzhou Bay Sea-crossing Bridge was completed and opened to traffic. After opening to traffic, the distance from Sunan A to Ningbo Port is shortened120km. It is understood that the driving time will be shortened from 3: 20 to 2: 00 when the speed of transport vehicles remains unchanged.

(1) Find the distance from A to Ningbo Port via Hangzhou Bay Bridge.

(2) If the freight cost includes transportation cost and time cost, it is known that the freight cost of a car from A to Ningbo Port is 1.8 yuan per kilometer and the time cost is 28 yuan per hour, then what is the freight cost of this car from A to Ningbo Port via Hangzhou Bay Bridge?

(3) Party A is going to open a outbound route in Ningbo, that is, the goods will be transported from Party A to Ningbo Port via Hangzhou Bay Bridge, and then from Ningbo Port to Party B. If a batch of goods (no more than 10 cars) are transported from Party A to Party B according to the outbound route, the freight is 8,320 yuan, of which the transportation cost of each car transported from Party A to Ningbo Port via Hangzhou Bay Bridge is the same as that in (2). The sea freight from Ningbo Port to B is charged according to a batch of goods not exceeding 10: one car in 800 yuan, and when the goods increase by 1 car, the sea freight per car is reduced by 20 yuan. How many cars are there in this shipment?

12. (Ningbo, Xijiang, 2008) As shown in Figure 1, a piece of standard paper is repeatedly folded to obtain "two-fold" paper, "four-fold" paper, "eight-fold" paper and "16-fold" paper ... As we all know, the short side length of standard paper is

(1) as shown in fig. 2, the "16 folio" paper obtained by folding the standard paper is folded as follows:

Step 1, align and fold the short side and the long side of the rectangle, put it on a point on the table, and flatten it to get a crease;

Step 2, align the long side with the crease, and the points coincide with each other, thus flattening the crease.

The value of and the length of are respectively.

(2) Are the aspect ratios of "two-open" paper, "four-open" paper and "eight-open" paper equal? If they are equal, write this ratio directly; If they are not equal, please calculate their ratios separately.

(3) As shown in Figure 3, a ""pattern consists of eight small squares with the same size, and its four vertices are respectively on the edge of "16" paper.

(4) In the known trapezoid, and the four vertices are all on the edge of the "4-fold" paper, please directly write down the areas of two right-angled trapezoid with different sizes that meet the requirements.

13. (Weihai, Shandong Province, 2008) As shown in the figure, in trapezoidal ABCD, ABCD, AB = 7, CD = 1, AD = BC = 5. Points M and N move on the sides of AD and BC respectively, keeping Mn∑ab, Me ⊥.

(1) Find the area of trapezoidal ABCD;

(2) Find the maximum MEFN area of the quadrilateral.

(3) Try to judge whether the quadrilateral MEFN is square, and if so,

Find the area of square MEFN; If not, please explain why.

14. (Weihai, Shandong, 2008) As shown in the figure, point A (m, m+ 1) and point B (m+3, m- 1) are both on the image of the inverse proportional function.

(1) Find the values of m and k;

(2) If m is a point on the X axis and n is a point on the Y axis,

A quadrilateral whose vertices are points A, B, M and N is a parallelogram.

Find the functional expression of line MN.

(3) Selection problem: in the plane rectangular coordinate system, the coordinates of point P.

Is (5,0), the coordinate of point Q is (0,3), and line PQ is horizontal to the right.

Move it by 4 units, and then move it up by 2 units to get the line segment P 1Q 1.

Then the coordinates of point P 1 are, and the coordinates of point Q 1 are.

15. (Yiyang, Hunan, 2008) We call a closed figure composed of a semicircle and a part of a parabola "egg circle". If a straight line has only one intersection with the "egg circle", then this straight line is called the tangent of the "egg circle".

As shown in figure 12, points A, B, C and D are the intersections of the "egg circle" and the coordinate axis respectively. It is known that the coordinate of point D is (0, -3), AB is the diameter of semicircle, the m coordinate of the center of semicircle is (1, 0), and the radius of semicircle is 2.

(1) Please find out the analytical formula of the parabolic part of the "egg circle" and write down the range of the independent variable;

(2) Can we find the analytical formula of the tangent of the "egg circle" passing through point C? Give it a try;

(3) Use your head and think about it. I believe you can find the analytical formula of the tangent of the egg circle passing through point D.

16. (Shaoxing City, Zhejiang Province, 2008) Put a rectangular piece of paper in a plane rectangular coordinate system, and the moving point starts from this point and moves to the end at the speed of 1 unit per second. When it moves for seconds, the moving point starts from the point and moves to the end point at the same speed. When one of the points reaches the end point, the other point stops moving. The moving time of a point is (seconds.

(1) is represented by the algebraic expression contained;

(2) When, as shown in figure 1, the edge is folded, and the point just falls on the edge, so as to find the coordinates of the point;

(4) Get the link by folding the edge, as shown in Figure 2. Q: Can sum be parallel? and

Can it be vertical? If yes, find the corresponding value; If not, explain why.

17. (Twelve cities in Liaoning Province in 2008) As shown in Figure 16, in the plane rectangular coordinate system, a straight line intersects with an axis at a point, intersects with an axis at a point, and a parabola passes through three points.

(1) Find the analytical formula and vertex coordinates of a three-point parabola;

(2) Whether there is a point on the parabola, make it into a right triangle, and if there is, write the point coordinates directly; If it does not exist, please explain the reason;

(3) Try to explore whether there is a point on the straight line that minimizes the circumference of the straight line. If yes, find out the coordinates of the point; If it does not exist, please explain why.

18. (Shenyang, 2008) As shown in the figure, in the plane rectangular coordinate system, the side of a rectangle is on the negative semi-axis of the shaft, and the side is on the positive semi-axis of the shaft. After rotating clockwise around the point, a rectangle is obtained. The corresponding point of a point is a point, the corresponding point of a point is a point, the corresponding point of a point is a point, and a parabola passes through a point.

(1) Judge whether the point is on the axis and explain the reason;

(2) Find the function expression of parabola;

(3) Whether there is a point above the axis, so that the area of a parallelogram with this point as the vertex is twice that of a rectangle, and the point is on a parabola. If yes, find the coordinates of points and points; If it does not exist, please explain why.

19. (Bazhong City, Sichuan Province, 2008) As shown in Figure 14, a parabola intersects with an axis at one point, and a straight line intersects with an axis at one point.

(1) Write the analytical formula of the straight line.

(2) the area to be searched.

(3) If a point moves from the direction at a speed of 1 unit length per second on a line segment (which is not coincident with it), at the same time, the point moves from the direction at a speed of 2 unit lengths per second on a ray. Let the movement time be seconds, please write the functional relationship between the areas of and, and find out how long the point has been moving and what is the maximum area.

20. (Chengdu, 2008) As shown in the figure, in the plane rectangular coordinate system xOy, the coordinate of vertex A of delta OAB is (10,0), vertex B is in the first quadrant, and =3, sin∠OAB=.

(1) If point C is the symmetrical point of point B relative to the X axis, find the functional expression of parabola passing through O, C and A;

(2) In (1), is there a point p on the parabola, which makes the quadrilateral with the vertices of p, o, c and a trapezoid? If it exists, find the coordinates of point P; If it does not exist, please explain the reason;

(3) If point O and point A are transformed into point Q( -2k, 0) and point R(5k, 0) (k >; Constant 1), let two points (q and r) be set, the intersection of a parabola with QR's median vertical axis and Y axis is n, its vertex is m, and the area of △QNM is the area of △QNR. The value of:.