When y=0, aX 2+bx+c = 0, if the equation has two roots, x 1, x 2, we can know from Vieta's theorem.
x 1+x2=-b/a……( 1)
And by changing y = ax 2+bx+c to the vertex,
Y = ax+(b/2a) 2+(4ac-b 2)/4a We can see that the symmetry axis of the function is x =-b/2a...(2).
This is very similar to the formula (1), except for a coefficient relation, 2× (-b/2a) =-b/a = x1+x2 ... (3).
Explain that the sum of the two roots is twice the symmetry axis.
Generally, it can also be expressed in the following form:
1, intersection: y=a(x-x 1)(x-x2)(a≠0) This means that the abscissa of the intersection of the function and the x axis is x 1, x2.
According to formula (3), the symmetry axis of this function is x=(x 1+x2)/2.
For example, the symmetry axis of y=(x-2)(x-4) is x = (4+2)/2 = 3;
2. Vertex: y = a (x-h) 2+k (a, h, k are constants, a≠0).
Through the vertex, we can intuitively see the symmetry axis x=h of the function.
For example: y = 6 (x+3) 2+9...(4)
The symmetry axis must not be understood as x=3, and further deformation of (4) is needed:
Y = 6x-(-3) 2+9, where h=-3, then the symmetry axis is x=-3.
3. general formula: y = ax 2+bx+c (a, b, c are constants, a≠0)
The symmetry axis of the function x=-b/2a can be obtained by formula (2). For the general formula, be sure to write the function in descending order of X, and then confirm what numbers A, B and C refer to respectively (including the symbol before the numerical value, which is particularly important).
For example: y = 3x-5x 2-9
Firstly, according to the descending power of X, Y =-5x 2+3x-9, where a=-5, b=3 and c=-9.
So the symmetry axis x =-b/2a =-3 (-10) = 3/10.
The above 1, 2,3 are common forms of quadratic functions.
Generally speaking, every form of quadratic function can be skillfully used, and it should be no problem to find the symmetry axis of the function.