As the most basic elementary function, it is both simple and rich in connotation and extension. We can use it as a material to study the monotonicity, parity, maximum value and other properties of functions, and we can also establish the organic relationship among functions, equations and inequalities. As a parabola, we can discuss the relationship with other plane curves. These vertical and horizontal connections make it possible to solve endless and flexible mathematical problems around quadratic functions. At the same time, the content of quadratic function is closely related to the development of modern mathematics and is an important knowledge base for students to enter colleges and universities for further study. So in this sense, it is not surprising that there are frequent questions about quadratic functions in the college entrance examination.
Quadratic function has two typical characteristics: one is analytical formula, and the other is image characteristics. Pure algebraic reasoning can be carried out from the analytical formula, which embodies a person's basic mathematical literacy; Starting from image characteristics, it is a very important thinking method in middle school mathematics to realize the natural combination of numbers and shapes.
The first is algebraic reasoning, because the analytic formula of quadratic function is simple and easy to deform (general formula, vertex type, zero type, etc.) ), so when solving the problem of quadratic function, we often use its analytical formula and deduce the related properties of quadratic function through pure algebraic reasoning. For example, there are three parameters A, B and C in 1, and the general formula is y=ax2+bx+c (c≠0). The key to solving the problem is to "determine" these three parameters through three independent conditions. 2. Using the relationship between the function and the root of the equation, write the zero formula y=a(x-x 1)(x-x2) of the quadratic function. 3. Following the vertex, symmetry axis and maximum of quadratic function, the discriminant is the resultant force. Secondly, the combination of numbers and shapes. The image of quadratic function y=ax2+bx+c (c≠0) is a parabola, which has many beautiful properties, such as symmetry, monotonicity, concavity and so on. Combining these image features to solve the quadratic function problem can make it simple and intuitive. For example: 1, the image of quadratic function is symmetrical about the straight line x=-, and the special relationship x 1+x2= also reflects a symmetry of quadratic function. 2. The image of quadratic function f(x) is continuous, because the quadratic equation has at most two real roots, so there are m and n, f (m) f (n).
Let's talk about the solution of hospital quadratic equation first. Someone summed up a jingle, aiming at optimizing the steps of solving a quadratic equation with one variable. "A decomposition, two formulas, such as x2 = a square; The first three methods are not easy, and the root formula should be used again; The letter coefficient needs to be discussed, and the classification scheme cannot be forgotten. " When solving specific problems, we must analyze specific problems and never ignore some implied conditions.
Through the understanding of quadratic function, I deeply realize that in the future, quadratic function will be an indispensable test center and right-hand man for both school and work, so learning quadratic function well is also an indispensable and important content of my high school career.