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HomeComputer TutorialsComputer KnowledgeCalculation formulas of trigonometric functions and quadratic functions in junior high schools

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

Jan 24, 2024 pm 01:27 PM
Junior high school function conversion formula

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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