1、(20 10? Xianning) It is known that the coordinates of the two intersections between the image of quadratic function y=x2+bx﹣c and the X axis are (m, 0) and (﹣3m, 0) (m ≠ 0) respectively.

(1) proves that 4c = 3b2；;

(2) If the symmetry axis of the function image is a straight line x= 1, find the minimum value of the quadratic function.

2、(20 10? Shiyan) The known equation mx2 ~ (3m ~ 1) x+2m ~ 2 = 0.

(1) Prove that no matter what real number M takes, the equation always has real number roots;

(2) If the distance between the image of the quadratic function y = mx2 ~ (3m ~ 1) x+2m ~ 2 and the two intersections of the X axis is 2, find the analytical expression of parabola;

(3) Draw the function image in (2) in the rectangular coordinate system xoy, and answer the question by combining the image: When the straight line y=x+b and the function image in (2) have only two intersections, find the value range of b. 。

3、(20 10? Nanjing) The known point A( 1, 1) is on the image of quadratic function y = x2-2ax+b.

(1) represents b with an algebraic expression containing a;

(2) If the image of the quadratic function has only one intersection with the X axis, find the vertex coordinates of the image of the quadratic function.

4、(20 10? Loudi) It is known that the image of quadratic function y=ax2+bx+c intersects with X axis at points A and B, and intersects with Y axis at point C, where the coordinate of point A is (-2,0), point B is on the positive semi-axis of X axis, point C is on the positive semi-axis of Y axis, and the length (OC < OB) of line segments OB and OC is the equation x2.

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

(2) Find the analytic expression of this quadratic function.

5、(20 10? Shantou) The image of the known quadratic function y=﹣x2+bx+c is shown in the figure, and its intersection coordinate with the X axis is (﹣ 1, 0), and its intersection coordinate with the Y axis is (0,3).

(1) Find the values of b and c, and write the analytical formula of this quadratic function;

(2) According to the image, write the range of the independent variable X whose function value y is positive.

6、(2009? Zhaoqing) It is known that one root of the unary quadratic equation x2+px+q+ 1=0 is 2.

(1) Find the relationship between q and p;

(2) Verification: the parabola y=x2+px+q has two intersections with the X axis;

(3) Let the vertex of the parabola y=x2+px+q be m and intersect with the X axis at two points A (x 1, 0) and B (x2, 0), and find the analytical expression of the parabola that minimizes the △AMB area.

7、(2009? Xinjiang) (1) quadratic function y = x2-4x+3 is transformed into the form of y = (x-h) 2+k by collocation method.

(2) Draw an image of y = x2-4x+3 in rectangular coordinate system.

(3) if A(x 1, y 1) and B(x2, y2) are two points on the image of function y = x2-4x+3, x 1 < x2 < 1, please compare y 1 and y2.

(4) Represent the root of equation x2-4x+3 = 2 on the image of function y = x2-4x+3.

8、(2009? Qiandongnan prefecture) known quadratic function y = x2+ax+a-2.

(1) Verification: No matter what A is, this function image always has two intersections with the X axis;

(2) Let a < 0, and when the distance between the image of this function and the two intersections of the X axis is, the analytical 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.

9、(2009? Ningxia) As shown in the figure, the parabola y =-x2+x+2 intersects with the X axis at points A and B, and intersects with the Y axis at point C. 。

(1) Find the coordinates of points A, B and C;

(2) Prove that △ABC is a right triangle;

(3) There are P points on the parabola besides point C, so is △ABP a right triangle? If yes, find the coordinates of point P; If not, please explain why.

10、(2009? Meizhou) As shown in the figure, it is known that the two intersections of the parabola Y =-x2+X+ and the X axis are A and B, and they intersect with the Y axis at point C. 。

(1) Find the coordinates of points A, B and C;

(2) Verification: △ABC is a right triangle;

(3) If the point M in the coordinate plane makes a quadrilateral with the points M and A, B and C as vertices, find the coordinates of the point M. (Write the coordinates of the point directly without writing the solution process. )

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

(1) Explore the number of images of the quadratic function y intersecting 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) and B(x2, 0), x 12+x22=5, the intersection with the Y axis be C, and its vertex be M, and find the analytical formula of the straight line CM.

12、(2009? Huangshi) already knows the function y = AX2+X+ 1(A is a constant) about x.

(1) If the image of the function just has an intersection with the X axis, find the value of a;

(2) If the image of the function is a parabola, and the vertex is always above the X axis, find the value range of a. 。

13、(2008? Tianjin) known parabola y=3ax2+2bx+c,

(i) If a=b= 1 and c =- 1, find the common point of the coordinates of the parabola and the X axis;

(2) if a=b= 1, and when ﹣ 1 < x < 1, there is only one common point between the parabola and the x axis, find the value range of c;

(iii) if a+b+c=0 and x 1=0, the corresponding y1> 0; > 0; When x2= 1, the corresponding y2 > 0. Try to judge whether parabola and x axis have something in common when 0 < x < 1. If yes, please prove your conclusion; If not, explain why.

14、(2008? Changchun) has known two quadratic functions about X, y 1 and y2, y 1 = a (x-k) 2+2 (k > 0), y1+y2 = x2+6x+12; When x=k, y2 =17; And the symmetry axis of the image of quadratic function y2 is the straight line X =- 1.

(1) Find the value of k;

(2) Find the expression of function y 1, y2;

(3) Do the images of function y 1 and y2 intersect in the same rectangular coordinate system? Please explain the reason.

15、(2008? Beijing) It is known that the univariate quadratic equation mx2 ~ (3m+2) x+2m+2 = 0 (m > 0) about x 。

(1) proves that the equation has two unequal real roots;

(2) Let the two real roots of the equation be x 1 and x2 respectively (where X 1 < X2). If y is a function about m, Y = X2-2x 1, find the analytical expression of this function;

(3) Under the condition of (2), combined with the image of the function, the answer is: when the range of the independent variable m satisfies any condition, y ≤ 2m.

16、(2007? Tianjin) It is known that the unary quadratic equation x2+bx+c=x has two real roots x 1, x2, and it satisfies X 1 > 0, x2-x1>.

(1) test proves that c > 0；;

(2) Prove that B2 > 2 (b+2c);

(3) For the quadratic function y=x2+bx+c, if the independent variable value is x0 and its corresponding function value is y0, then when 0 < x0 < x 1, try to compare y0 with x 1.

17、(2007? Ningxia) in the quadratic function y=ax2+bx+c, the corresponding values of independent variable x and function y are as follows:

x ﹣ 1﹣0 1 2 3

y ﹣2﹣ 1 2 1﹣﹣2

(1) Determine the opening direction of the quadratic function image and write its vertex coordinates.

(2) The two roots of the unary quadratic equation ax2+bx+c=0(a≠0, A, B and C are constants) are x 1, and the value range of x2 is which of the following options _ _ _ _ _ _ _.

①

②

③

④ .

18、(2007? Maoming) It is known that the abscissas of the two intersections of the image with the function y=x2+2x+c and the X axis are x 1, x2, x 12+X22 = C2-2C. Find the values of X2 C and X 1.

19、(2007? The image of quadratic function y=ax2+bx+c(a≠0) is shown in the figure. According to the picture, answer the following questions:

(1) Write two roots of the equation ax2+bx+c=0;

(2) Write the solution set of inequality AX2+BX+C > 0;

(3) Write the value range of the independent variable X, in which Y decreases with the increase of X;

(4) If the equation ax2+bx+c=k has two unequal real roots, find the range of K. 。

20、(2006? Zibo) It is known that the quadratic functions y = x2-MX+ and y = x2-MX+ about x, and an image of these two quadratic functions intersects the x axis at two different points A and B. 。

(1) Try to determine which image of the quadratic function passes through point A and point B;

(2) If the coordinate of point A is (-1, 0), try to find the coordinate of point B;

(3) Under the condition of (2), for the quadratic function passing through point A and point B, when the value of X is taken, the value of Y decreases with the increase of the value of X. 。

2 1、(2006? Liangshan Prefecture) part of the image with parabola Y =-(x- 1) 2+2 is known (as shown in the figure), so when the image intersects with the X axis again, the coordinates of the intersection point are _ _ _ _ _ _ _ _.

22、(2006? Laiwu) It is known that the quadratic function of x y = ﹣ x2+(m+2) x ﹣ m. 。

(1) It is proved that no matter whether m is any real number, the vertex p of the image of quadratic function is always above the X axis;

(2) Let the quadratic function image intersect with the Y axis at point A, and the parallel lines passing through point A as the X axis intersect with the image at another point B. If the vertex P is in the first quadrant and the value of m is what, △PAB is an equilateral triangle.

23、(2006? Jingzhou) The function of Y about X is known: Y = (k ﹣ 2) x2 ﹣ 2 (k ﹣1) x+k+1satisfies k ≤ 3.

(1) Verification: This function image always intersects with the X axis;

(2) When the equation about z has increased roots, find the coordinates of the intersection point between the function image and the X axis.

24、(2006? Chongzuo) It is known that the image of quadratic function y = mx2-mx+n intersects with X axis at two points, A (x 1, 0), B (x2, 0), X 1 < x2, and the negative semi-axis of the intersecting Y axis is at point C, AB=5, AC⊥BC, so

25、(2005? Jiaxing) known function y = x2-4x+ 1.

(1) Find the minimum value of the function;

(2) Draw the image of the function in the given coordinate system;

(3) Let the intersection of the function image and the X axis be A (X 1 0) and B (X2, 0), and find the value of x 12+x22.

26、(2005? Xinjiang) If the quadratic function y=ax2+2x+c, the highest point of the image is

M(x0, y0), the quadratic function image passes through this point.

P( 1,), if x is x0 n (n = 1, 2, 3 ...), the corresponding function value is y0﹣ n2. ..

(1) Find the analytic expression of quadratic function and draw a picture;

(2) If the intersection of the quadratic function image and the X axis is A and B, find the area of △PAB.

27、(2005? Urumqi) It is known that the image of quadratic function y=x2+bx+c passes through point M(0, ﹣3) and intersects with X axis at points A (x 1 0) and B (x2, 0), and x 12+x22 = 10.

28、(2005? Suzhou) known quadratic function y = 2x2-MX-m2.

(1) Verification: For any real number m, the quadratic function image always has a common point with the X axis;

(2) If the quadratic function image and the X axis have two common points A and B, and the coordinate of point A is (1, 0), find the coordinate of point B. 。

29、(2005? Beijing) It is known that the equation (a+2) x2-2ax+a = 0 about x has two unequal real roots x 1 and x2, and the two intersections of the parabola Y = x2-(2a+ 1) x+2a-5 and the x axis are located at points respectively.

(1) the range of the real number a;

(2) When |x 1|+|x2|=, find the value of a.

30、(20 10? Foshan) (1) Please draw an approximate image of the quadratic function y = x2-2x in the coordinate system;

(2) According to the relationship between the root of the equation and the function image, approximately express (draw) the root of the equation x2-2x = 1 on the graph;

(3) Observe the image and directly write the root of the equation x2-2x = 1 (accurate to 0. 1).