I. Basic introduction
Generally speaking, we call a function in the form of y=ax2+bx+c (where A, B and C are constants and a≠0) as a quadratic function, where A is called a quadratic term coefficient, B is a linear term coefficient and C is a constant term. X is the independent variable and y is the dependent variable. The maximum number of independent variables to the right of the equal sign is 2.
main feature
"Variable" is different from "unknown", so it cannot be said that "quadratic function means that the polynomial function with the highest number of unknowns is quadratic". "Unknown" is just a number (the specific value is unknown, but only one value is taken), and "variable" can take any value within a certain range. The concept of "unknown" is applied in the equation (both functional equation and differential equation are unknown functions, but both unknown and unknown functions generally represent a number or function-special circumstances may occur), but the letters in the function represent variables and their meanings have always been different. From the definition of function, we can also see the difference between them, just as function is not equal to function relationship.
The situation of quadratic function image intersecting with x axis
When △ = B2-4ac >; 0, the function image has two intersections with the x axis.
When △=b2-4ac=0, the function image has only one intersection with the x axis.
When △ = B2-4ac
Quadratic and quadratic function images
Make an image of quadratic function y = ax 2+bx+c in plane rectangular coordinate system. It can be seen that the image of quadratic function is an endless parabola. If the drawn graph is accurate, then the quadratic function image will be obtained by general translation.
Axial symmetry
Quadratic function image is an axisymmetric figure. The symmetry axis is a straight line x=-b/2a.
The only intersection of the symmetry axis and the quadratic function image is the vertex p of the quadratic function image.
Especially when b=0, the symmetry axis of the quadratic function image is the Y axis (that is, the straight line x=0).
A and B have the same sign, and the symmetry axis is on the left side of the Y axis.
The symbols of A and B are different, and the symmetry axis is on the right side of Y axis.
pinnacle
The quadratic function image has a vertex p, whose coordinate is P(h, k), that is, (-b/2a, (4ac-b2/4a).
When h=0, p is on the y axis; When k=0, p is on the x axis. It can be expressed as vertex y = a (x-h) 2+k.
h=-b/2a，k=(4ac-b2)/4a .
Direction and size of opening
The quadratic coefficient A determines the opening direction and size of the quadratic function image.
When a>0, the parabola opens upwards; When a<0, the parabola opens downward.
The larger the |a|, the smaller the opening of the quadratic function image.
Factor folding determining the position of symmetry axis
Both linear coefficient b and quadratic coefficient a*** determine the position of the axis of symmetry.
When a>0 has the same number as B (namely ab>0), the symmetry axis is on the left of Y axis; Because the axis of symmetry is on the left, the axis of symmetry is less than 0, which is -b/2a.
When a>0, when it is different from B (i.e. AB; 0, so b/2a should be less than 0, so a and b should have different signs.
It can be simply recorded as left and right differences, that is, when the numbers of A and B are the same (that is, AB >；; 0), the symmetry axis is on the left of the y axis; When a and b have different numbers (i.e. AB
In fact, b has its own geometric meaning: the value of the slope k of the resolution function (linear function) of the tangent of the quadratic function image at the intersection of the quadratic function image and the Y axis. It can be obtained by taking the derivative of quadratic function.
Factors determining the intersection point with y axis
The constant term c determines the intersection point between the quadratic function image and the y axis.
Quadratic function image and Y axis intersect at (0, c)
Note: The coordinate of the vertex is (h, k), and it intersects with the Y axis at (0, c).
The number of times it intersects the x axis.
a & lt0; K>0 or a>0; K<0, the quadratic function image has two intersections with the X axis.
When k=0, there are only 1 intersections between the quadratic function image and the X axis.
a & lt0; K<0 or a>0, k>0, quadratic function image does not intersect with X axis.
When a>0, the function gets the minimum value ymin=k at x=h, and at x; k