First, master three forms of quadratic function: 1, general formula y=ax2+bx+c 2, vertex y = a (x-h) 2+k.
3. the intersection y=a(x-x 1)(x-x2)
2. Mastering the above three forms, we can write the properties of the image in these three forms, such as opening direction, symmetry axis, vertex coordinates, maximum value, function variability, etc., and understand the meaning of each letter in the three analytical formulas.
Third, you can write their intersections with the coordinate axis in the above three forms. That is, the corresponding values when x is 0 and y is 0. 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 intersections between parabola and X axis is determined by the discriminant of the root.
Fourth, flexibly use the undetermined coefficient method to find the analytical formula of quadratic function.
1. If three or two points are known, find the analytical formula and set the general formula. 2. Given a vertex and another point, set the vertex. It is known that two intersections of the X axis and another point are intersections.
Fifth, the application of functions. There are two main points about the application of functions. One is to find the maximum value, which needs to find the resolution function first, and then find the maximum value with formula or vertex. However, we should pay attention to special circumstances when solving practical problems. Such as whether the maximum value has practical significance. Whether it is within the numerical range. The second is the combination of function, equation and inequality, which uses the value of known independent variables to get the function value or uses the function value to get the value of known independent variables. The inequality should be solved by combining images.
Understand the translation formula of image "addition and subtraction, left plus right subtraction"
Y = AX2 → Y = A (X-H) 2+K "up plus down minus" stands for k, and "left plus right minus" stands for h. 。
Seven. Properties of parabola Y = AX2+BX+C (A ≠ 0, A, B and C are constants) (1). Parabola is an axisymmetric figure. The symmetry axis is a straight line x = -b/2a, and the only intersection point between the symmetry axis and the parabola is the vertex p of the parabola. Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
(2) The parabola has a vertex p, and the coordinate is P [-b/2a, (4ac-b? )/4a].-When b/2a = 0, p is on the y axis; When δδ= b? When -4ac=0, p is on the x axis.
(3) Quadratic coefficient A determines the opening direction and size of parabola.
When a > 0, the parabola opens upward; When a < 0, the parabola opens downwards; The larger the |a|, the smaller the opening of the parabola.
(4) Both the first-order coefficient b and the second-order coefficient a*** determine the position of the symmetry axis.
When the signs of A and B are the same (that is, AB > 0), the symmetry axis is on the left side of Y axis;
When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.
(5) The constant term c determines the intersection of the parabola and the Y axis. The parabola and the y axis intersect at (0, c).
C > intersects with the positive semi-axis of the y axis; 0; C
4. The sign of parabola y=ax2+bx+c;
Symbol of (1)a+b+c:
When x = 1, a+b+c represents the ordinate of the point on the parabola whose abscissa is 1.
The point is above the X axis a+b+c >; 0
This point is located below the X axis a+b+c B+C.
The point on the x axis is a+b+c = 0.
(2) 2) symbols of a-b+c:
It is determined by the position of the point on the parabola when x =- 1
This point is above the x axis, a–b+c > 0.
This point is located below the X axis A–B+C.
The point is on the x axis, and a–b+c = 0.
5. Number of intersections between parabola and X axis
δ= b？ When -4ac > 0, the parabola has two intersections with the x-axis.
δ= b？ When -4ac=0, the parabola has 1 intersections with the X axis.
δ= b？ When -4ac < 0, the parabola has no intersection with the x axis.
If the intersection of the image with y=ax2+bx+c and the X axis is A(x 1, 0), B(x2, 0); Then AB=|x 1-x2|
6. What are the conditions for parabola y=ax2+bx+c to go up and down the X axis?
(1) What is the condition that the parabola y=ax2+bx+c is above the X axis?
a > 0，b2 - 4ac < 0
Variant: What is the condition that the value of function y=ax2+bx+c(a≠0) is always positive no matter what value X takes?
(2) What is the condition that the parabola y=ax2+bx+c is below the X axis?
a & lt0，B2-4ac & lt； 0
Variant: What is the condition that the value of function y=ax2+bx+c(a≠0) is always negative, no matter what value X takes?