The general formula of the first and second functions (three points),
When the three-point coordinates A(x 1, y 1), B(x2, y2) and C(x3, y3) are known, the general formula y=ax2+bx+c (a≠0) can be selected, and the coordinate values are substituted into the analytical formula and then listed into a ternary linear equation set, so that the undetermined coefficient can be obtained.
Example: given the image passing point (1, 0), (-1, 6), (2, 3) of the quadratic function y=ax2+bx+c(a≠0), find the analytical formula of this quadratic function. Solution: Let the analytic expression of quadratic function be y=ax2+bx+c(a≠0).
∫ parabola passes through three points (1, 0), (-1, 6), (2, 3),
0=a+b+c Solution: a=2 6=a-b+c b=-3.
3=4a+2b+c c= 1
The analytical formula is y=2x2-3x+ 1.
Vertices of quadratic and quadratic functions (collocation method),
The quadratic function y=ax2+bx+c(a≠0) is formulated to get y=a(x-h)2+k, which is the vertex coordinate of parabola according to the image. Instructions: If there are vertex coordinates of parabola (or symmetry axis and maximum value of function) in the problem, and it passes through another point, it is easier to solve it with vertex. )
Example: The vertex coordinate of a parabola is (2,3). After passing through a point (4,-1), find the analytic expression of this parabola.
Solution: Let the analytic expression of quadratic function be y=a(x-h)2+k(a≠0).
∫a(x-x 1)(x-x2) The vertex coordinate of the parabola is (2,3), which passes through a point (4, 1).
-1=a(4-2)2+3 to get a=- 1,
The analytical formula is y=-(x-2)2+3, that is, y=-x2+4x- 1.
3. Coordinate formula for the intersection of parabola and X axis: Quadratic trinomial ax2+bx+c can be solved by root formula method, that is, ax2+bx+c=a(x-x 1)(x-x2). Therefore, for the function Y = Ax2+BX+C = A (X-X65438).
Note: If there are two intersection coordinates of parabola and X axis in the topic and pass through another point, then the intersection coordinate formula can be used.
Example: Given that the coordinates of the two intersections of a parabola and the X axis are (1, 0) and (-3, 0) respectively, and then through (-2,-1), find the analytical expression of this parabola.
Solution: Let the analytic expression of quadratic function be y= a(x-x 1)(x-x2) (a ≠ 0).
∫ The coordinates of the intersection of parabola and X axis are (1, 0) and (-3, 0) respectively, and then pass through (-2,-1).
- 1=a (-2- 1 )( -2-3)，a= 1/3，
The analytical formula is y = (x-1) (x+3)/3 = x2/3+x/3-1.
To sum up, when finding the form of quadratic function, we should choose the expression form of quadratic resolution function according to the known conditions.