The concept of quadratic function 1 Concept of quadratic function: A function with a general shape of (is a constant,) is called a quadratic function. What needs to be emphasized here is that, similar to the unary quadratic equation, the coefficient of the quadratic term can be zero, and the domain of the quadratic function is all real numbers.
2. The structural characteristics of quadratic function:
(1) The left side of the equal sign is the function, and the right side is the quadratic form of the independent variable, with the highest degree of 2.
It is a constant, quadratic coefficient, linear coefficient and constant term.
There are three general expressions of quadratic function in junior high school mathematics: y = ax 2+bx+c (a, b, c are constants, a≠0).
Vertex: y = a(x-h)2+k[ vertex P(h, k) of parabola].
Intersection point: y=a(x-x? )(x-x？ ) [only when it is related to the x axis A(x? , 0) and B(x? 0) parabola].
Note: Among the three forms of mutual transformation, there is the following relationship: h =-b/2ak = (4ac-b 2)/4ax? ，x？ =(-b √b^2-4ac)/2a。
Properties of quadratic function 1 Property: any point P(x, y) on the (1) linear function satisfies the equation: y = kx+b. (2) The coordinate of the intersection of the linear function and the y axis is always (0, b), and the image of the proportional function always intersects the origin of the x axis at (-b/k, 0).
2. Quadrant where K, B and function images are located:
When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;
When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When b > 0, the straight line must pass through the first and second quadrants;
When b=0, the straight line passes through the origin;
When b < 0, the straight line must pass through three or four quadrants.
Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.
At this time, when k > 0, the straight line only passes through one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.
The image of the quadratic function in grade three mathematics is different from the general formula;
①y=ax2+bx+c and y=ax2-bx+c are symmetrical about Y ..
② The two images Y = AX2+BX+C and y=-ax2-bx-c are symmetrical about X axis.
③y=ax2+bx+c and y=-ax2-bx+c-b2/2a are symmetrical about the vertex.
④y=ax2+bx+c and y=-ax2+bx-c are symmetrical about the center of the origin. (i.e., the graph obtained after rotating 180 degrees around the origin)
For vertices:
① two images, y = a (x -h) 2+k and y=a(x+h)2+k, are symmetrical about y axis, that is, vertices (h, k) and (-h, k) are symmetrical about y axis, and the abscissas are opposite, but the ordinate is the same.
② two images, y = a (x-h) 2+k and y=-a(x-h)2-k, are axisymmetrical about x, that is, vertices (h, k) and (h, -k) are axisymmetrical about x, with the same abscissa and opposite ordinate.
③y=a(x-h)2+k and y=-a(x-h)2+k are symmetrical about the vertices, that is, the vertices (h, k) and (h, k) are the same and the opening directions are opposite.
④y=a(x-h)2+k and y=-a(x+h)2-k are symmetrical about the origin, that is, the vertices (h, k) and (-h, -k) are symmetrical about the origin, and the abscissa and ordinate are opposite. (In fact, 13④ is the case of f (-x), -f(x) and -f (-x) versus f(x)). )