Calculation formulas of trigonometric functions and quadratic functions in junior high schools

Formulas of trigonometric functions and quadratic functions in junior middle schools

Trigonometric function formula

Square relationship:

sin^2（α） cos^2（α）=1

tan^2（α） 1=sec^2（α）

cot^2（α） 1=csc^2（α）

Business relationship:

tanα=sinα/cosα

cotα=cosα/sinα

Reciprocal relationship:

tanα·cotα=1

sinα·cscα=1

cosα·secα=1

Quadratic function formula

Generally, there is the following relationship between the independent variable x and the dependent variable y:

(1) General formula: y=ax2 bx c (a, b, c are constants, a≠0), then y is called a quadratic function of x. Vertex coordinates (-b/2a, (4ac-b^2)/4a)

(2) Vertex formula: y=a(x-h)2 k or y=a(x m)^2 k(a, h, k are constants, a≠0)

(3) Intersection formula (with x-axis): y=a(x-x1)(x-x2)

(4) Two radical formulas: y=a(x-x1)(x-x2), where x1 and x2 are the abscissas of the intersection of the parabola and the x-axis, that is, the two radicals of the quadratic equation ax2 bx c=0 root, a≠0

illustrate:

(1) Any quadratic function can be transformed into the vertex formula y=a(x-h)2 k through formula. The vertex coordinate of the parabola is (h, k). When h=0, the parabola y=ax2 k The vertex 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 on the origin

(2) When the parabola y=ax2 bx c has an intersection with the x-axis, that is, when the corresponding quadratic equation ax2 bx c=0 has real roots x1 and x2, according to the decomposition formula of the quadratic trinomial ax2 bx c=a(x-x1)(x-x2), the quadratic function y=ax2 bx c can be converted into two radicals y=a(x-x1)(x-x2)

Junior high school formulas about functions

Quadratic function: y=ax^2 bx c (a, b, c are constants, and a is not equal to 0)

a>0 opening upward

aa,b have the same sign, the axis of symmetry is on the left side of the y-axis, otherwise, it is on the right side of the y-axis

|x1-x2|= b^2-4ac divided by |a|

The intersection point with the y-axis is (0,c)

b^2-4ac>0,ax^2 bx c=0 has two unequal real roots

b^2-4acb^2-4ac=0,ax^2 bx c=0 has two equal real roots

Axis of symmetry x=-b/2a

Vertex (-b/2a,(4ac-b^2)/4a)

Vertex formula y=a(x b/2a)^2 (4ac-b^2)/4a

The function moves d (d>0) units to the left. The analytical formula is y=a(x b/2a d)^2 (4ac-b^2)/4a. Moving to the right means minus

The function moves upward by d(d>0) units. The analytical formula is y=a(x b/2a)^2 (4ac-b^2)/4a d, and downward is minus

When a>0, the opening is upward, the parabola is above the y-axis (the vertex is on the x-axis), and extends upward infinitely; when a

4. When drawing the parabola y=ax2, you should first make a list, then draw the points, and finally connect the lines. When selecting the independent variable x value from the list, 0 is always the center, and an integer value is selected that is convenient for calculation and point drawing. When drawing points, be sure to use a smooth curve to connect them, and pay attention to the changing trend.

Several forms of analytic expressions of quadratic functions

(1) General formula: y=ax2 bx c (a, b, c are constants, a≠0).

(2) Vertex formula: y=a(x-h)2 k(a, h, k are constants, a≠0).

(3) Two radical formulas: y=a(x-x1)(x-x2), where x1 and x2 are the abscissas of the intersection of the parabola and the x-axis, that is, the two radicals of the quadratic equation ax2 bx c=0 root, a≠0.

Explanation: (1) Any quadratic function can be transformed into the vertex formula y=a(x-h)2 k through the formula. The vertex coordinate of the parabola is (h, k). When h=0, the parabola y=ax2 The vertex of 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 on the origin.

(2) When the parabola y=ax2 bx c has an intersection with the x-axis, the corresponding quadratic equation ax2 bx c=0 has real roots x1 and

When x2 exists, according to the decomposition formula of quadratic trinomial ax2 bx c=a(x-x1)(x-x2), the quadratic function y=ax2 bx c can be converted into two radicals y=a(x -x1)(x-x2).

Methods for the vertex, axis of symmetry, and maximum value of a parabola

①Assembling method: Convert the analytical expression into the form of y=a(x-h)2 k, the vertex coordinates (h, k), the axis of symmetry is the straight line x=h, if a>0, y has a minimum value, when When x=h, the minimum value of y=k, if a

②Formula method: directly use the vertex coordinate formula (-, ), the vertex; the axis of symmetry is the straight line x=-, if a>0, y has a minimum value, when x=-, the minimum value of y=, if a

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