1. definition of quadratic function: generally speaking, a function with the shape of y = ax2+bx+c(a, b and c are constants, and a≠0) is called a quadratic function. The structural characteristics of (1) quadratic function: ① The left side of the equal sign is the function y, the right side is the quadratic form of the independent variable x, and the highest degree of x is 2. ②a, b. C are constant terms. According to the definition of quadratic function, we should grasp the key that the coefficient of quadratic term is not zero. Example 1, among the following functions, the one that is not a quadratic function is () (a); (b) and (c) and (d) 2. A special form of quadratic function: (1) y = ax2 (2) y = ax2+bx (3) y = ax2+c Quadratic function contains three coefficients: A, B and C. Generally, the values of A, B and C can be determined by giving three sets of corresponding values of X and Y. 3. Determine the values of letters according to the definition of quadratic function. Example 2. If the function is quadratic, find the value of the constant m. Example 3, the quadratic function is known, (1) When m is a value, this function is quadratic; (2) When m is a value, the function is linear. 4. Determine the expression of quadratic function in practical problems. Example 4: The circumference of a rectangle is 50cm, one side is xcm, and the area of this rectangle is ycm2. (1) Try to write the resolution function of y and x; (2) Is y a quadratic function of x? Why? Example 5. A shopping mall buys a commodity at the price of 30 yuan. In the process of trial sale, it is found that the daily sales volume m (pieces) of this commodity and the sales price x (yuan) of each commodity satisfy the linear function relationship m =162–3x. Try to write the functional relationship between the daily sales profit y (yuan) of this commodity and the sales price x (yuan) of each commodity, and make a judgment. The general formula of the 1. quadratic function is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. Using collocation method, the general formula of quadratic function can be rewritten as _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. For example, if the vertices of parabola Y = 3 x2-(A- 1)x+A+ 1 and parabola Y = 2 x2+(2 b- 1)x-2b are the same, then their vertex coordinates are _ _ _ _ _ _ _ _ _ _ _ _. Firstly, the general formula of quadratic function is transformed into vertex; Determine the opening direction, vertex and symmetry axis of the image, then determine several points on both sides of the symmetry axis, and then draw points. For example, draw an image of quadratic function, and write down the opening direction, vertex coordinates, symmetry axis and intersection coordinates with Y axis and X axis of this image. 3. Quadratic function y = a(x+m)2+ k(a≠0) Open direction: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Vertex coordinates: _ _ _ _ _ _ _ _ _. Symmetry axis: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _; Increase or decrease: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _。 4. The function of the coefficient of quadratic function: (1) The function of three coefficients in the general formula y = ax2+bx+c(a≠0): A: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. b:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _； c:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _； B2–4ac:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _； (2) The function of three letters in vertex Y = a (x+m) 2+k (a ≠ 0): A: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ m: _ _ _ _ _ _ _ _ _ _ _ _ _ _ k:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _； 1- 1o 1, as shown in the figure, then ① 0; ② 0; ③ 0; ④ 0; ⑤ _______0; ⑥ _______0; All landowners _ _ _ _ 0 cases 2. In the same plane rectangular coordinate system, the images of the linear function y = ax+b and the quadratic function y = ax2+bx may be () OXYOXYOXY.

(C)(A)(B)(D) Example 3. If the image of quadratic function y=ax2+bx+c is shown in the figure, then point A (a, b) is in the first quadrant of (a); (b) the second quadrant; The third quadrant; The fourth quadrant. Example 4. It is known that the quadratic function y =–x2+(m–2) x+3 (m+1): when m ≠–4, the image of the quadratic function must have two intersections with the X axis; 5. Determination of quadratic resolution function: (1) General formula y = ax2+bx+c(a≠0) When three points of the quadratic function image are known, you can generally use the general formula and the known parabola intersection points (1, 3), (–2,–3) and (3). (2) Vertex y = a(x+m)2+ k(a≠0) When the vertex or symmetry axis or the maximum value is known, the vertex of the quadratic function is known as A( 1,–4) and passes through point (4,5), so as to find the analytical formula of this quadratic function; (3) two formulas y = a (x–x 1) (x–x2) (a ≠ 0) where x1and x2 are two solutions of the equation ax2+bx+c = 0. When the two intersections between the image of quadratic function and the X axis are known, two formulas can be used, but the results are generalized to general formulas or vertex examples. It is known that the image of quadratic function passes through three points (–2,0), (1, 0) and (0,2). Then the analytic expression of this quadratic function is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0)(5) If the vertex of the quadratic function is known to be on the X-axis (or has only one intersection with the X-axis), let the quadratic function be y = a (x+m) 2 (a ≠ 0). 6. Intersection point of parabola and other curves: (1). (2) The symmetry axes of quadratic functions are all straight lines parallel (or coincident) with the Y axis. (So "straight line" should not be missed), the straight line parallel to the Y axis has only one intersection with the parabola, but the straight line with only one intersection with the parabola is not necessarily parallel to the Y axis (3). The intersection of the quadratic function and the x axis is determined by the following formula. Distance between two intersections of parabola and X axis (4) A straight line parallel to X axis has an intersection with parabola, and parabola is the vertex and has two intersections. The vertical coordinates of these two intersections are the same. That is, the quadratic function y = ax2+bx+c(a≠0). When x takes x 1 and x2, and the values of the functions are equal, then the symmetry axis of the function is a straight line example of 1, and the quadratic function y=2x2+9x+34 is known. When the independent variable x takes two different values of x 1, in Example 2, a part of the parabola y=ax2+2ax+a2+2 is as shown in the figure, then the coordinate of the intersection point between the parabola on the right side of the Y axis and the X axis is _ _ _ _ _ _ _ _ _ _ _ _ _ _. Example 3. Given that the vertex coordinates of the parabola y = ax2+bx+c (a ≠ 0) are (3,–2), and the distance between the two intersections of the parabola y = ax2+bx+c(a≠0) and the x axis is 4, then the analytical formula of the function is _ _ _ _ _ _ _ _ _ _ _ _. (5) parabola y = ax2+bx+c(a≠0) When a > 0 is combined, the parabola is above the X axis (that is, the function values are all greater than zero); When a