There are three analytic expressions of quadratic function.
1. General formula: y=ax? +bx+c
2. Vertex: y=a(x+h)? +k
3. Intersection point: y=a(x-x 1)(x-x2)
The intersection is also called two-point or two-point type.
Where x 1 and x2 are the abscissa of the intersection of parabola and x axis.
Is also the corresponding equation ax? Two roots of +bx+c=0
Important and difficult
1. This section focuses on the understanding and flexible application of the image and properties of the quadratic function y=ax2+bx+c, but the difficulty lies in the properties of the quadratic function y=ax2+bx+c and the transformation of the analytical formula into the form of y=a(x-h)2+k through the formula.
2. Learning this section requires careful observation and induction of the characteristics of images and the relationship between different images. Connect different images and find out their uniqueness.
Generally speaking, if the quadratic coefficient a of several different quadratic functions is the same, the opening direction and opening size (i.e. shape) of parabola are exactly the same, but the positions are different.
Any parabola y=a(x-h)2+k can be obtained by appropriately translating the parabola y=ax2. The specific translation method is shown in the following figure:
Note: the law of the above translation is: "H value is positive and negative, shifting left and right; K value is positive, negative, up and down "is actually related to the translation problem of parabola, so we can't memorize the translation law by rote." It is very simple to determine the translation direction and distance according to the position relationship of their vertices.
Images and properties of 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.
The above is the general formula and key analysis of mathematical quadratic function I summarized for you, which is for reference only and I hope it will be helpful to you.