1. (Hangzhou, 2009) It is known that the point p (,) is on the image of the function, so the point p should be in the plane rectangular coordinate system.

A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant

2. (Hangzhou, 2009) There are three statements as follows: ① The idea of coordinates was first established by the French mathematician Descartes; ② In addition to the plane rectangular coordinate system, we can also use the direction and distance to determine the position of the object; ③ All points in the plane rectangular coordinate system belong to four quadrants. One of the mistakes is that

A. only1b. Only 2c. Only 3d ① ② ③.

3. (Taizhou, 2009) The corresponding values of the sum of known quadratic functions are as follows:

…

0 1 3 …

…

1 3 1 …

The following judgment is correct (▲)

A. the parabola opens upwards. The parabola intersects the axis at the negative semi-axis.

C. when = 4, > 0 d. the positive root of the equation is between 3 and 4.

4. (Zhou Nan, 2009) The image of parabola is shown in figure 1. According to the image, the analytical formula of parabola may be ().

Kewang

A, y= x2-x-2b, y= subject network.

C, y= D, y= topic network.

5.(2009 Nanchong) The symmetry axis of parabola is a straight line ()

A.B. C. D。

6. (Putian, 2009) How to transform an image of a quadratic function into an image ()

A. translate 1 unit to the left, and then translate 3 units upward.

B. Translate 1 unit to the right, and then translate 3 units upward.

C. translate 1 unit to the left, and then translate 3 units downward.

D. Translate 1 unit to the right, and then translate 3 units downward.

7. (Li Shui, 2009) Knowing the image of quadratic function Y = AX2+BX+C (A ≠ 0), the following conclusions are given:

①a>0。

② The image of this function is symmetrical about a straight line.

③ When the value of function y is equal to 0.

The number of correct conclusions is a.3b.2c.1d.0.

8. (Suining, 2009) Transform the quadratic function into a form by matching method.

A.B.

C.D.

9. (Jiaxing, 2009) It is known that in the same rectangular coordinate system, the image of function sum may be (▲).

10. (Huzhou, 2009) It is known that every small square in a graph is a small square with a side length of 1, and the vertex of each small square is called a grid point. Please draw a parabola in the picture at will. How many meshes can a parabola pass through at most? ( )

a . 6b . 7c . 8d . 9

1 1.(2009 Guangzhou) The minimum value of the quadratic function is ().

(A)2 (B) 1 (C)- 1 (D)-2

12. (Yantai, 2009) The image of quadratic function is shown in the figure, so the image of linear function and inverse proportional function in the same coordinate system is roughly ().

13. (Huangshi, 2009) The image of the known quadratic function y=ax2+bx+c(a≠0) is shown in Figure 3.

The following conclusions are drawn: ① ABC > 0 22A+B < 0 34A-2B+C < 0 4A+C > 0,

The number of correct conclusions is ()

a，4 B，3 C，2 D， 1

14. (Zhou Nan, 2009) The analytical formula of the image of a quadratic function symmetric about the origin o (0,0) is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. Theme network

15. (Huzhou, 2009) It is known that the symmetry axis of parabola (> 0) is a straight line and passes through a point. Try to compare totals:

_ (fill in ">", "< or" = ")

16.(2009 Jingmen) When the function y = (x-2) (3-x) gets the maximum value, x = _ _ _ _ _

17. (Yiwu, 2009) As shown in the figure, the intersection point A between the parabola and the axis is between the points (-2,0) and (-1 0) (inclusive), and the vertex C is a moving point on the rectangular DEFG (including the boundary and the interior), then

(Fill in ""or "");

The value range of is

18. (Chongqing, 2009) There is a functional relationship between the price (yuan) of a certain brand TV set sold by a TV manufacturer in rural areas last year and the month, and there is a functional relationship between the monthly sales volume (10,000 sets) and the month last year. The sales in two months are as follows:

June 65438+ October May

The sales volume was 39,000 units and 43,000 units respectively.

(1) What month did this brand of TV set have the largest sales amount in the countryside last year? What's the highest?

(2) Due to the impact of the international financial crisis, the price of this brand TV set sold to rural areas in February and June this year was lower than that in February last year, and the monthly sales volume was lower than that in February last year. The state implements the policy of "home appliances going to the countryside", that is, rural families buy new home appliances, and the state gives financial subsidies at 13% of the product price. Affected by this policy, from March to May this year, the average monthly sales volume of this kind of TV set sold by this manufacturer in rural areas increased by 6.5438+0.5 million units compared with February this year, while maintaining the price in February this year. If the state gives a financial subsidy of 9.36 million yuan to the sales of this TV set from March to May this year, what is this value (keep one decimal place)?

(Reference data:,,,)

19. (Ningbo, 2009) As shown in the figure, the parabola intersects with the X axis at point A and point B, and passes through point C (5, 4).

(1) Find the value of a and the coordinates of the vertex p of the parabola.

(2) Please design a translation method to make the vertex of the translation parabola fall in the second quadrant, and write the analytical formula of the translation parabola.

In order to keep the humidity and temperature in the warehouse, the walls around the warehouse are equipped with automatic ventilation facilities, as shown in the figure. The ABCD at the lower part of the facility is rectangular, where AB = 2m and BC =1m; The upper CDG is an equilateral triangle, and the fixed point E is the midpoint of AB. △ EMN is a triangular ventilation window whose shape is controlled by computer (the shadow part is not ventilated), and MN is a telescopic crossbar that can slide up and down along the boundary of the facility and always keep parallel to AB.

(1) When the distance between MN and AB is 0.5m, find the area of △EMN at this time;

(2) Let the distance between MN and AB be meters, and try to express the area s (square meters) of △EMN as a function of X;

(3) Please explore whether the area s (square meter) of △EMN has a maximum value, and if so, find this maximum value; If not, please explain why.

2 1. (The full mark of this question is l2)

(Yibin, 2009) As shown in the figure, in the plane rectangular coordinate system xoy, the lower base OA of the isosceles trapezoid OABC is on the positive semi-axis of the X axis, BC∨OA, OC = ab. Tan ∠ Ba0 =, and the coordinate of point B is (7,4).

(1) Find the coordinates of point A and point C;

(2) Find the analytical formula of parabola passing through points 0, b and c;

(3) Is there a point P on the parabola of the first quadrant (2) that a straight line passing through this point P and parallel to the waist of the isosceles trapezoid divides the trapezoid into two parts with equal areas? If it exists, find the abscissa of point P; If it does not exist, please explain why.

22. (The full mark of this question is 12)

(Luzhou, 2009) As shown in figure 12, it is known that the image of quadratic function intersects with the positive semi-axis of X axis at points A, B,

Intersect with the y axis at point C.

(1) Find the value of c;

(2) If the area of △ABC is 3, find the analytic expression of quadratic function;

(3) Let d be the vertex of the quadratic function image determined in (2). Is there a point p on the straight line AC that minimizes the circumference of △PBD? If it exists, find the coordinates of point P; If it does not exist, please explain why.

23.( 12) (Zhou Nan 2009) Known quadratic function.

(1) Prove that no matter what A is, this function image always has two intersections with the X axis.

(2) Let a < 0. When the distance between the image of this function and the two intersections of the X axis is 0, the analytic expression of this quadratic function is obtained.

(3) If the quadratic function image intersects the X axis at points A and B, is there a point P on the function image, so that the area of △PAB is? If yes, find the coordinates of point P, if no, please explain the reason.

24. (Chengdu, 2009) In the plane rectangular coordinate system xOy, it is known that the parabola intersects with the X axis at points A and B (point A is on the left side of point B), and intersects with the Y axis at point C, and its vertex is m. If the functional expression of the straight line MC is, the intersection with the X axis is n, COS∠BCO= =.

(2) Is there a point P on this parabola that is different from point C, so that a triangle with N, P and C as its vertices is a right triangle with NC as its right side? If it exists, find the coordinates of point P; If it does not exist, please explain the reason;

(3) The intersection point A is perpendicular to the X axis and the intersection line MC is at the Q point. If the parabola moves up and down along its symmetry axis, there is always a common point between the parabola and the line segment NQ. How many unit lengths can the parabola move up at most? How many unit lengths can you translate downward at most?

25. (Putian, 2009) It is known that parabola intersects with Y axis at point C, and intersects with X axis at points A and B, and point A is to the left of point B. The coordinates of point B are (1, 0), and oc = 30b.

(1) Find the analytical formula of parabola;

(2) If point D is the moving point on the parabola below the line segment AC, find the maximum ABCD area of the quadrilateral:

(3) If point E is on the X axis and point P is on the parabola. Is there a parallelogram with A, C, E, P as the vertex and AC as one side? If it exists, find the coordinates of point P; If it does not exist, please explain why.

26. (Jiangsu, 2009) As shown in the figure, the vertex of the image of a known quadratic function is. The image and axis of quadratic function intersect at the origin and another point, and its vertex is on the symmetry axis of function image.

(1) Find the coordinates between points;

(2) When the quadrilateral is a diamond, find the relationship between functions.

27. (Taian, 2009) As shown in the figure, △OAB is an equilateral triangle with a side length of 2 and a straight line passing through point A.

(1) Find the coordinates of point E;

(2) Solving the parabolic analytical formula of three points A, O and E;

28 (Suining, 2009) As shown in the figure, the image of quadratic function passes through point D(0,), the abscissa of vertex C is 4, and the length of line segment AB cut on the X axis of this image is 6.

⑴ Find the analytic formula of quadratic function;

⑵ Find a point P on the parabola symmetry axis to minimize PA+PD, and find the coordinates of the point P;

⑶ Is there a point Q on the parabola that makes △QAB similar to △ABC? If it exists, find the coordinates of point Q; If it does not exist, please explain why.

28. (Huzhou, 2009) It is known that the parabola () intersects the axis at a point, and the vertex is. A straight line intersects an axis, the axis intersects at two points, and the straight line intersects at one point.

(1) Fill-in-the-blank problem: Try to express the coordinates of the point sum with the contained algebraic expressions, and then;

(2) Fold along the axis as shown in the figure. If the corresponding point' just falls on the parabola, then the point' intersects the axis, and the value and area of the quadrilateral are calculated;

(3) Is there a point on the parabola () that makes a quadrilateral with vertices a parallelogram? If it exists, find out the coordinates of the point; If it does not exist, please explain why.

29. (Guangzhou, 2009) As shown in figure 13, the image of quadratic function intersects with X axis at points A and B, and intersects with Y axis at point C (0,-1), and the area of Δ Δ ABC is.

(1) Find the relation of quadratic function;

(2) Take a point M(0, m) on the Y axis as the vertical line of the Y axis in the morning. If the vertical line of ABC has a common point with the circumscribed circle, find the range of m;

(3) Is there a point D on the image of quadratic function, which makes the quadrilateral ABCD a right trapezoid? If it exists, find the coordinates of point D; If it does not exist, please explain why.

30. (Jiangxi, 2009) As shown in the figure, the parabola intersects with the axis at two points (the point is on the left side of the point) and intersects with the axis at one point, and the vertex is.

(1) Write three-point coordinates and parabola symmetry axis directly;

(2) Connection, intersecting with the parabola symmetry axis at a point, which is the moving point on the line segment, the intersection point is the parabola at this point, and the abscissa of this point is;

① Use the included algebraic expression to represent the length of the line segment and find out when the quadrilateral is a parallelogram.

② Let the area be a function of sum.

3 1. (Anshun, 2009) As shown in the figure, it is known that a parabola intersects with two points A (- 1 0) and E (3 3,0), and intersects with the axis at point B (0 0,3).

(1) Find the analytical formula of parabola;

(2) Let the vertex of parabola be d, and find the quadrilateral area AEDB.

(3) Are△ AOB and△△ DBE similar? If similar, please give proof; If not, please explain why.

32. (Luojiang, 2009) In order to welcome the 60th anniversary of the founding of New China, a craft factory in our district designed a 20 yuan-made handicraft for trial sale. After investigation, the relationship between the sales unit price (RMB-piece) and the daily sales volume (piece) of this process is in line with the chart.

(1) When the sales unit price is set to 30 yuan sum, please write directly according to the drawing.

The corresponding daily sales volume is around 40 yuan;

(2)① Try to find the functional relationship between sum;

(2) If the price department stipulates that the highest selling unit price of this handicraft can't exceed 45 yuan/piece, then when the selling unit price is set at what level, when the handicraft factory tries to sell this handicraft, it will get the biggest profit every day? What is the maximum profit? (Profit = total selling price-total cost price).

33. (Hengyang, 2009) It is known that the image passing coordinate origin of quadratic function is (1, -2). Find out the relationship of this quadratic function.

34. (Yantai, 2009) A shopping mall sells refrigerators with a purchase price of 2,000 yuan at a price of 2,400 yuan, with an average of 8 refrigerators sold every day. In order to cooperate with the implementation of the national "home appliances to the countryside" policy, shopping malls decided to take appropriate price reduction measures. According to the survey, every time the price of such refrigerators is reduced by 50 yuan, an average of four refrigerators can be sold every day.

(1) Assuming that the price of each refrigerator is reduced by X yuan, the profit of selling this refrigerator in the mall every day is Y yuan. Please write the function expression between y and x; (It is not required to write the range of independent variables)

(2) If a shopping mall wants to make a profit of 4,800 yuan a day in the sales of this refrigerator and benefit the people at the same time, how much should the price of each refrigerator be reduced?

(3) When the price of each refrigerator is reduced by several yuan, what is the maximum profit of selling this kind of refrigerator in the shopping mall every day? What is the highest profit?

35. (Loudi, 2009) It is known that the quadratic function of X is y=x2-(2m- 1)x+m2+3m+4.

(1) Explore the number of images of the quadratic function y intersecting with the X axis when m satisfies any conditions.

B (X2) Let the intersection of the image of the quadratic function Y and the X axis be A (X 1 0), B(x2, 0), and +=5, the intersection with the Y axis be C, and its vertex be M, and find the analytical expression of the straight line CM.

36. (Zhongshan, 2009) The side length of square ABCD is 4, and M and N are two moving points on BC and CD respectively. When m moves on BC, AM and MN remain vertical.

(1) proof: rt △ ABM ∽ rt △ MCN;

(2) Let BM=x and the area of trapezoidal ABCN be y, and find the functional relationship between y and x; When the m point moves to what position, the area of the quadrilateral ABCN is the largest, and the maximum area is calculated;

(3) When the point M moves to what position, Rt△ABM∽Rt△AMN, and find the value of X at this time.

37.

38. (Jingmen, 2009) A parabola with an upward opening intersects the X axis at two points A (m-2,0) and B (m+2,0). Let the vertex of the parabola be C and AC ⊥ be BC.

(1) If m is a constant, find the analytical formula of parabola;

(2) If m is a constant less than 0, what kind of translation can the parabola in (1) make its vertex at the coordinate origin?

(3) Let the positive semi-axis of the parabola intersecting with the Y axis be at point D, and ask whether there is a real number m, so that △BCD is an isosceles triangle? If it exists, find the value of m; If it does not exist, please explain why.

39.( 13) (Luojiang, 2009) In order to welcome the 60th anniversary of the founding of New China, a craft factory in our district designed a product with the cost of 20 yuan, and put it on the market for trial sale. After investigation, the relationship between the unit sales price (RMB/piece) and the daily sales volume (piece) of this process conforms to the chart.

(1) When the sales unit price is set to 30 yuan sum, please write directly according to the drawing.

The corresponding daily sales volume is around 40 yuan;

(2)① Try to find the functional relationship between sum;

(2) If the price department stipulates that the highest selling unit price of this handicraft can't exceed 45 yuan/piece, then when the selling unit price is set at what level, when the handicraft factory tries to sell this handicraft, it will get the biggest profit every day? What is the maximum profit? (Profit = total selling price-total cost price). (Rizhao, 2009) In order to keep the humidity and temperature in the warehouse, automatic ventilation facilities are installed on the walls around the warehouse, as shown in the figure. The ABCD at the lower part of the facility is rectangular, where AB = 2m and BC =1m; The upper CDG is an equilateral triangle, and the fixed point E is the midpoint of AB. △ EMN is a triangular ventilation window whose shape is controlled by computer (the shadow part is not ventilated), and MN is a telescopic crossbar that can slide up and down along the boundary of the facility and always keep parallel to AB.

(1) When the distance between MN and AB is 0.5m, find the area of △EMN at this time;

(2) Let the distance between MN and AB be meters, and try to express the area s (square meters) of △EMN as a function of X;

(3) Please explore whether the area s (square meter) of △EMN has a maximum value, and if so, find this maximum value; If not, please explain why.

40. (Hangzhou, 2009) It is known that the straight line parallel to the X axis intersects with the function and the image of the function at points A and B respectively, and there is a fixed point P (2 2,0).

(1) If, and tan∠POB=, find the length of line segment AB;

(2) In a parabola that passes through two points A and B and the vertex is on a straight line, it is known that the line segment AB= is on the left side of its symmetry axis, and Y increases with the increase of X, so as to find the parabolic analytical formula that meets the conditions;

(3) The parabola passing through three points A, B and P is known, and the image is obtained after translation, and the distance from point P to straight line AB is found.

4 1. (Yiwu, 2009) As shown in the figure, in the rectangular ABCD, AB=3, AD= 1, point P moves on the line segment AB, let AP=, and the paper is folded, so that point D and point P overlap to get a crease EF (points E and F are the intersection of the crease and the rectangular side), and then the paper is restored.

(1) When, the length of the crease EF is; When point E coincides with point A, the length of the crease EF is;

(2) Please write down the range of values that make the quadrilateral EPFD a diamond, and find out the side length of the diamond at that time;

(3) Order, when point E is in AD and point F is in BC, write the functional relationship of AND. Is it similar when the maximum value is taken? If they are similar, it is a calculated value; If not, please explain why.

Tip: folding draft paper may help you!

42. (Yiwu, 2009) It is known that points A and B are moving points on the axis, and points C and D are points on the function image. When the quadrilateral ABCD (points A, B, C and D are arranged in sequence) is a square, this square is called the companion square of this function image. For example, as shown in the figure, the square ABCD is one of the companion squares of the linear function image.

(1) If a function is linear, find the side lengths of all partner squares of its image;

(2) If a function is an inverse proportional function, its partner is ABCD, and the point D(2, m) (m

(3) If a function is a quadratic function, the partner square of its image is ABCD, and the coordinate of a point in C and D is (3,4). Write the coordinates of the other vertex of the partner on the parabola, and write one of the parabolic analytical expressions that meets the meaning of the question, and judge whether the number of partners in the parabola you write is odd or even. . Just write the answer to this little question directly. )

43. (Chongqing, 2009) (Chongqing was known in 2009: As shown in the figure, in the plane rectangular coordinate system, the edge OA of the right-angled OABC is on the positive semi-axis of the shaft, OC is on the positive semi-axis of the shaft, OA=2, OC=3. The bisector passing through the origin O is ∠AOC passes through AB at point D, connecting DC, passing through D is DE⊥DC, and passing through OA is at point E.

(1) Find the analytical formula of parabola passing through points E, D and C;

(2) After rotating ∠EDC clockwise around point D, one side of the angle intersects with the positive semi-axis of the shaft at point F, and the other side intersects with the line segment OC at point G ... If DF intersects with the parabola in (1) at another point M, is it true that the abscissa of point M is, EF=2GO? If yes, please give proof; If not, please explain the reasons;

(3) For point G in (2), is there a point Q on the parabola located in the first quadrant, so that the △PCG formed by the intersection point P of straight lines GQ and AB and points C and G is an isosceles triangle? If it exists, request the coordinates of point q; If it does not exist, please explain why.

44. (Chongqing in 2009) (Known in Chongqing in 2009: As shown in the figure, in the plane rectangular coordinate system, the edge OA of the right-angled OABC is on the positive semi-axis of the shaft, OC is on the positive semi-axis of the shaft, OA=2, OC=3. The bisector passing through the origin O is ∠AOC passes through AB at point D, connecting DC, passing through D is DE⊥DC, and passing through OA is at point E.

(1) Find the analytical formula of parabola passing through points E, D and C;

(2) After rotating ∠EDC clockwise around point D, one side of the angle intersects with the positive semi-axis of the shaft at point F, and the other side intersects with the line segment OC at point G ... If DF intersects with the parabola in (1) at another point M, is it true that the abscissa of point M is, EF=2GO? If yes, please give proof; If not, please explain the reasons;

(3) For point G in (2), is there a point Q on the parabola located in the first quadrant, so that the △PCG formed by the intersection point P of straight lines GQ and AB and points C and G is an isosceles triangle? If it exists, request the coordinates of point q; If it does not exist, please explain why.

45. (Taizhou, 2009) As shown in the figure, it is known that the straight line intersects the coordinate axis at two points, and the line segment is taken as the edge.

Another intersection of a square, a parabola passing through a point and a straight line is.

(1) Please write the coordinates of the point directly;

(2) Find the analytical formula of parabola;

(3) If the square slides down along the ray at the speed of one unit length per second until the vertex falls on the axis, let the area of the part of the square falling under the axis be, find the functional relationship about the sliding time, and write the value range of the corresponding independent variable;

(4) Under the condition of (3), the parabola and the square move together and stop at the same time, and find the area swept by the parabola arc between two points on the parabola.

46. (Nanchong, 2009) As shown in Figure 9, it is known that both the images of the proportional function and the inverse proportional function pass through points.

(1) Find the analytic expressions of proportional function and inverse proportional function;

(2) After downward translation, the straight line OA intersects with the image of the inverse proportional function, and the value and analytical expression of this linear function are obtained;

(3) The image of the linear function in the problem (2) intersects the axis at points C and D respectively, and the analytic expressions of the quadratic function passing through points A, B and D are obtained;

(4) Under the condition of (3), is there a point E on the image of the quadratic function, so that the area of the quadrangle OECD and the area s of the quadrangle OABD satisfy:? If it exists, find the coordinates of point e; If it does not exist, please explain why.

47. (Shenzhen, 2009) 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.

48. (Li Shui, 2009) The position of rhombic ABCD in rectangular coordinate system is shown in the figure, and the coordinates of C and D are (4,0) and (0,3) respectively. At present, two moving points, P and Q, start from A and C at the same time. Point P moves along AD to the end point D, and point Q moves along the dotted line CBA to the end point A, assuming that the moving time is t seconds.

(1) Fill in the blank: the side length of the diamond-shaped ABCD is ▲, and the area is ▲.

The length of high BE is ▲;

(2) Explore the following questions:

(1) If the speed of point P is 1 unit per second and the speed of point Q is 2 units per second, when point Q is on the line BA, find the functional relationship between the area S of △APQ and t, and find the maximum value of S;

② If the speed of point P is 1 unit per second, then the speed of point Q becomes K per second.

Unit, in the process of movement, there is a corresponding k value at any moment, which makes

△APQ is folded along one side, and the four sides formed by the front and rear triangles are folded.

The shape is a diamond. Please explore the situation when t=4 seconds and find out the value of K.

49. (Full score of this question 13) (Ningde, 2009) As shown in the figure, it is known that the vertex of parabola C 1 is p, which intersects with the X axis at two points A and B (point A is to the left of point B), and the abscissa of point B is 1.

(1) Find the coordinates of point P and the value of a; (4 points)

(2) As shown in figure (1), parabola C2 and parabola C 1 are symmetrical about X, and parabola C2 shifts to the right. The parabola after translation is denoted as C3, and the vertex of C3 is M. When point P and point M are symmetrical about the center of point B, the analytical formula of C3 is found. (4 points)

(3) As shown in Figure (2), point Q is a point on the positive semi-axis of X axis. Parabola C 1 rotates around point q 180 to get parabola C4. The vertex of parabola C4 is n, which intersects with X axis at two points E and F (point E is to the left of point F). When the triangle whose vertices are at point P, point N and point F is a right triangle, find the point.

50. (Jiaxing, 2009) As shown in the figure, curve C is the image of the function in the first quadrant, and parabola is the image of the function. Points () are all integers on curve C.

(1) Find all points;

(2) Take any two points in the middle as straight lines and find the number of all different straight lines;

(3) Take any straight line from all the straight lines in (2) and find the probability that the selected straight line and parabola have a common point.

5 1. (Yiyang, 2009) Reading materials:

As shown in figure 12- 1, draw three straight lines perpendicular to the horizontal line through the three vertices of △ABC. The distance between the two outer straight lines is called △ABC's "horizontal width" (a), and the length of the middle line segment of this straight line is called △ABC's "vertical height (h)". We can get a new method to calculate the area of triangle.

Answer the following questions:

As shown in figure 12-2, the vertex coordinates of parabola are point C (1 4), the X axis is at point A (3,0), and the Y axis is at point B. 。

(1) Find the analytical expressions of parabola and straight line AB;

(2) Point P is the moving point on the parabola (in the first quadrant), connecting PA and PB. When point P moves to vertex C, find the vertical heights CD and △ cab;

(3) whether there is a point P, so that S△PAB= S△CAB, and if there is, find out the coordinates of the point P; If it does not exist, please explain why.

52. (Hengyang, 2009) As shown in figure 12, the straight line and the two coordinate axes intersect at point A and point B respectively, and point M is any point on the line segment AB (except point A and point B). When passing through point M, MC⊥OA is at point C and MD⊥OB is at point D. 。

(1) Do you think the perimeter of the quadrangle OCMD will change when the point M moves on AB? And explain the reasons;

(2) When the point M moves to what position, what is the maximum area of the quadrilateral OCMD? What is the maximum value?

(3) When the quadrangle OCMD is a square, move the quadrangle OCMD along the positive direction of the X axis, set the translation distance as, and the area of the overlapping part of the square OCMD and △AOB as S, try to find the functional relationship of S, and draw the image of the function.

53. (Loudi, 2009) As shown in figure 1 1, there is another right trapezoid DEFH at △ABC, ∠ c = 90, BC=8 and AC=6.

(HDE, ∠ hDE = 90), the bottom de falls on CB, and the waist DH falls on CA, with DE=4, ∠DEF=∠CBA, and AH∶AC=2∶3.

(1) Extend the intersection of HF and AB to G, and find the area of △AHG.

(2) Operation: fix △ABC, and set the right-angled trapezoid DEFH at the rate of 1 per second.

The speed of the unit moves to the right in CB direction until point D and point B..

Stop when overlapping, let the movement time be t seconds, and then take the right-angle ladder after movement.

The shape is DEFH' (as shown in figure 12).

Question 1: Can quadrilateral CDH'h be a square in motion? If it's okay,

Request the value of t at this time; If not, please explain why.

Question2: During the movement, △ABC overlaps with right-angled trapezoid DEFH ''.

The area of the part is y, and find the functional relationship between y and t.

54. (Zhou Nan 2009) The quadratic function is known.

(1) Prove that no matter how real A is, there are always two intersections between this function image and the X axis.

(2) Let a < 0. When the distance between the image of this function and the two intersections of the X axis is 0, the analytic expression of this quadratic function is obtained.

(3) If the quadratic function image intersects the X axis at points A and B, is there a point P on the function image, so that the area of △PAB is? If yes, find the coordinates of point P, if no, please explain the reason.