The quadratic function Y = AX2+BX+C (A ≠ 0, A, B and C are constants) contains two variables X and Y. As long as we determine one of the variables first, we can find the other variable by analytical formula, that is, we can get a set of solutions. And a set of solutions is the coordinates of a point, in fact, the image of quadratic function is a graph composed of countless such points.
Familiar with the images and properties of several special quadratic functions.
1. Observe the shapes and positions of y=ax2, Y = AX2+K, Y = A (X+H) 2 images by tracing points, and get familiar with the basic features of their respective images. On the contrary, according to the characteristics of parabola, we can quickly determine which analytical formula it is.
2. Understand the translation formula of image "addition and subtraction, left plus right subtraction".
Y = AX2 → Y = A (X+H) 2+K "addition and subtraction" is K, and "adding left and subtracting right" is H. 。
In a word, if the coefficients of quadratic terms of two quadratic functions are the same, their parabolas have the same shape, but the translation of parabolas is essentially the translation of vertices because of their different coordinates and positions. If parabolas are in general form, they should be converted into vertices and then translated.
3. Through drawing and image translation, we understand and make it clear that the characteristics of analytical expressions are completely corresponding to the characteristics of images. When solving problems, we should have a picture in mind and see the function to reflect the basic characteristics of its image in our hearts.
4. On the basis of being familiar with the function image, through observing and analyzing the characteristics of parabola, we can understand the properties of quadratic function, such as increase and decrease, extreme value and so on. Distinguish the coefficients A, B, C, △ of quadratic function and the symbols of algebraic expressions composed of coefficients by images.
Third, we should make full use of the function of parabola "vertex".
1. We should be able to find the "vertex" accurately and flexibly. The form is y = a (x+h) 2+k → vertex (-h, k). For other forms of quadratic function, we can turn it into a vertex to find the vertex.
2. Understand the relationship between vertex, symmetry axis and maximum value of function. If the vertex is (-h, k) and the symmetry axis is x =-h, the maximum (minimum) value of y = k；; On the other hand, if the symmetry axis is x=m and the maximum value of y is n, then the vertex is (m, n); Understanding the relationship between them can achieve the effect of analyzing and solving problems.
3. Draw a sketch with vertices. In most cases, we only need to draw a sketch to help us analyze and solve problems. At this time, we can draw the approximate image of parabola according to the vertex and opening direction of parabola.
Understand and master the solution of the intersection of parabola and coordinate axis.
Generally speaking, the coordinates of a point consist of abscissa and ordinate. When we find the intersection of parabola and coordinate axis, we can give priority to one of the coordinates, and then find the other coordinate by analytical formula. If the equation has no real root, it means that the parabola and the X axis do not intersect.
From the process of finding the intersection point above, we can see that the essence of finding the intersection point is to solve the equation, which is related to the discriminant of the root of the equation, and the number of times the parabola intersects the X axis is determined by the discriminant of the root.
Quadratic functions are all parabolic functions (its function trajectory is like the trajectory of a ball, of course this is not important), so we can grasp the quadratic function by grasping its function image.
Pay attention to several points in the function image (standard formula y = ax 2+bx+c, a is not equal to 0):
1, the opening direction is related to the quadratic coefficient a, indicating that the opening is upward, and vice versa.
2. There must be an extreme point, which is also the maximum point. If the opening is upward, it is easy to imagine that this extreme point should be the minimum point, and vice versa. The abscissa of the extreme point is -b/2a. Extreme points are prone to application problems.
3. it doesn't necessarily intersect with the x axis. When the determinant δ = b 2-4ac
Δ = 0, then there is just an intersection point, that is, we say that the X axis is tangent to the function image. The corresponding equation has a unique real number solution. δ& gt； 0, there are two intersections, and the corresponding equation has two real number solutions.
4. Inequality. Knowing the above three points, we can definitely solve the inequality of reference function image.