Generally speaking, there is the following relationship between independent variable x and dependent variable y:
Y = ax 2+bx+c (a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Three Expressions of Quadratic Function
General formula: 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 1)(x-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]
Note: Among these three forms of mutual transformation, there are the following relations:
h=-b/2ak=(4ac-b^2； )/4ax 1，x2 =(-b √b^2； -4ac)/2a
Quadratic function image
Make an image of quadratic function y=x2 in a plane rectangular coordinate system,
It can be seen that the image of quadratic function is a parabola.
Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line
x=-b/2a .
The only intersection of 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 coordinates are
p[-b/2a,(4ac-b^2； )/4a].
-b/2a=0, p is on the y axis; When δ = b 2-4ac = 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 downward.
The larger the |a|, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic 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 left on the 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 parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
When δ = b 2-4ac < 0, the parabola has no intersection with the X axis.
Quadratic function and unary quadratic equation
Especially the quadratic function (hereinafter referred to as function) y = ax 2; +bx+c，
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
That's ax^2；; ; +bx+c=0
At this point, whether the function image intersects with the X axis means whether the equation has real roots.
The abscissa of the intersection of the function and the x axis is the root of the equation.
When drawing a parabola y = ax2, list first, then trace points, and finally connect lines. When choosing the value of independent variable x in the list, it is often centered on 0, so choose an integer value that is convenient for calculation and tracking. When tracking points, be sure to connect them with smooth curves and pay attention to the changing trend.
Several forms of quadratic resolution function
(1) general formula: Y = AX2+BX+C (A, b, c are constants, a≠0).
(2) Vertex: y = a (x-h) 2+k (a, h, k are constants, a≠0).
(3) two expressions: y = a (X-x 1) (X-x2), where x 1, x2 is the abscissa of the intersection of parabola and x axis, that is, the two roots of quadratic equation AX2+BX+C = 0, a≠0.
Description: (1) Any quadratic function can be transformed into vertex Y = A (X-H) 2+K by formula, and the vertex coordinate of parabola is (h, k). When H = 0, the vertex of parabola Y = AX2+K is on the Y axis; When k = 0, the vertex of parabola a(x-h)2 is on the X axis; When H = 0 and K = 0, the vertex of parabola Y = AX2 is at the origin.
If the image passes through the origin and the axis of symmetry is the y axis, let y = ax 2; If the symmetry axis is the y axis, but not the origin, let y = ax 2+k.
Defining and defining expressions
We call the sine, cosine, tangent and cotangent of acute angle ∠ acute angle function, that is, the function with acute angle as independent variable and this value as function value is called acute angle trigonometric function.
The sine (sin), cosine (cos) and tangent (tan), cotangent (cot), secant (sec) and cotangent (csc) of acute angle A are all called acute trigonometric function of angle A [1].
Sine is equal to the hypotenuse of the opposite side; Sina = account
Cosine (cos) is equal to the ratio of adjacent side to hypotenuse; cosA=b/c
Tangent (tan) is equal to the opposite side of the adjacent side; tanA=a/b
Cotangent is equal to the comparison of adjacent edges; cotA=b/a
The definition method of acute trigonometric function value in junior high school is defined in a right triangle, so the calculation of acute trigonometric function value in junior high school is completed by constructing a right triangle, that is, putting this angle in a right triangle.