(1) Find the coordinates of point A and point B. 。

(2) If point P is any point on the X axis, it is proved that:

(3) At the maximum, find the coordinates of point P. 。

2. As shown in the figure, the parabola intersects with the X axis at point A and point B, the Y axis intersects with point C, the quadrilateral OBHC is rectangular, and the extension line of CH intersects with the parabola at point D (5,2), connecting BC and AD.

(1) Find the coordinates of point C and the parabolic analytical formula;

(2) Rotate △BCH 90 degrees clockwise around point B, and then fold it in half along the X axis.

△ BEF (point C corresponds to point E), judge whether point E falls on a parabola and explain the reason;

(3) Let the straight line passing through point E intersect with AB at point P and CD at point Q, and ask whether there is point P, so that the area of straight line PQ is divided into two parts: 1∶3. If it exists, find out the coordinates of point P; If it does not exist, please explain why.

3. It is known that a rectangular piece of paper, with a length of 4 and a width of 3, is constructed with the straight line where the length is located as the axis and the coordinate origin.

Vertical rectangular coordinate system; The point is a moving point on the edge (not coincident with the point), so it will be folded along the edge.

Get, and then select appropriate points on the edge to fold the edge, get, and make.

Straight lines coincide.

(1) If the point falls on the edge, as shown in Figure ①, find the coordinates of the point and find the parabola function of the three points;

(2) If the point falls on a rectangular piece of paper, as shown in Figure ②, what is the maximum value?

(3) In the case of (1), is there a right triangle with three points on the right side on the parabola? If it does not exist, explain the reasons; If it exists, find the coordinates of the point.

4. The straight line and the coordinate axis intersect at two points, and their lengths are two points () of the equation. The moving point starts from the point and moves along the route at the speed of/kloc-0 per unit length per second →→→→→ and stops when it reaches the point.

(1) Write the coordinates of two points directly;

(2) The movement time of the set point is (seconds), the area is, and the functional relationship between and (the range of independent variables does not need to be written);

(3) Write the coordinates of the points directly. Is there a point on the coordinate axis at this time that makes the quadrilateral with the vertices of,, and a trapezoid? If it exists, please write down the coordinates of this point directly; If it does not exist, please explain why.

5. as shown in figure 14, the parabola intersects the x axis at two points A(x 1 0) and B(x2, 0), and X 1 > X2 intersects the y axis at point C(0, 4), where x 1 and x2 is the equation x2-2x.

(1) Find the analytical expression of this parabola;

(2) Point P is the moving point on the line segment AB, and the passing point P is PE‖AC, which intersects with BC at point E and connects CP. When the area of △CPE is the largest, find the coordinates of point P;

(3) Inquiry: If point Q is a point on the parabola symmetry axis, is there such a point Q that △QBC becomes an isosceles triangle? If yes, please directly write the coordinates q of all qualified points; If it does not exist, please explain why.

6. (Chifeng City, 2009) 25, (14) As shown in the figure, the vertex coordinates of Rt△ABC are A(0,), B(- 1/2,), C( 1 0), ∠ ABC respectively.

The intersection of BC and Y axis is d, and the coordinate of point D is (0,). A parabola with point D as its vertex and axis Y as its axis of symmetry passes through point B.

(1) Find the analytical expression of this parabola.

(2) Fold △ABC along AC to get point B 1 corresponding to point B for verification.

The quadrilateral AOC B 1 is rectangular, and the ice judgment point B 1 is on the parabola of (1).

(3) Extend the intersection parabola of BA to point E, and take a point P on the line segment BE to make it pass through point P..

The perpendicular of the x axis intersects a parabola at point F. Is there such a point p that the quadrilateral PADF

Is it a parallelogram? If it exists, find the coordinates of point P; If it does not exist, explain why.

7. It is known that the functional relationship between the wholesale unit price and the wholesale quantity of a certain fruit is shown in the figure (1).

(1) Please explain the practical significance of the two function images in the figure.

(2) Write down the wholesale fund amount of this fruit between W (yuan) and M (kg).

Functional relationship; Draw a function image in the following coordinate system; Indicate the amount.

Within a certain range, this kind of fruit can be wholesale in large quantities with the same funds.

(3) After investigation, a letter between the highest daily sales volume and the retail price of this kind of fruit sold by a dealer.

As shown in Figure (2), the dealer plans to sell more than 60 kilograms of this fruit every day.

And the retail price of the day remains unchanged, please help the dealer design the purchase and sale plan.

Maximize the profits made on that day.

8. It is known that the quadratic equation with one variable has real roots and is a positive integer.

( 1);

(2) When the equation has two non-zero integer roots, the image of the quadratic function is shifted down by 8 units to find the analytical expression of the shifted image;

(3) Under the condition of (2), the image of the translated quadratic function is folded along the axis, and the rest of the image remains unchanged to obtain a new image. Please use this new image to answer: when the straight line is straight,

When there are two common points with this image, the value range of.

9. Take point C as CE⊥CD and point AD as E in the middle, and rotate line segment EC counterclockwise around point E to get line segment EF (as shown in figure 1).

(1) Drawing in the drawing1;

① when p is any point on the ray CD (P 1 does not coincide with c), the line EP 1 rotates counterclockwise around point e to get the line segment EC 1. Judge and prove the positional relationship between the straight line FC 1 and the straight line CD;

② When P2 is any point on the extension line of line segment DC, connect EP2 and rotate counterclockwise around point E to get line segment EC2. Judge the positional relationship between the straight line C 1C2 and the straight line CD, draw a picture and write your conclusion directly.

(2) If AD=6, tanB=, AE= 1, under the condition of ①, let CP 1=, S =, and write the value range of the independent variable.

10, as shown in the figure, in the plane rectangular coordinate system, the coordinates of three mechanized wars are as follows.

Extend AC to point D, make CD=, pass point D, and make the extension line of DE‖AB to BC of point E.

(1) Find the coordinates of point D;

(2) Make the symmetrical point F of point C about the straight line DE, and connect DF and EF respectively. If the straight line passing through point B divides the quadrangle CDFE into two quadrangles with equal perimeter, the analytical formula of this straight line is determined;

(3) Let G be a point on the Y axis, and point P starts from the intersection of the straight line and the Y axis, then reaches point G along the Y axis, and then reaches point A along GA. If the moving speed of point P on the Y axis is twice as fast as that on the straight line GA, try to determine the position of point G, so that the time for point P to reach point A is the shortest. (Requirements: Briefly describe the method of determining the position of G-point, but do not require proof)