Integers (including: positive integer, 0, negative integer) and fractions (including: finite decimal and infinite cyclic decimal) are rational numbers.
Such as: -3, 0.23 1, 0.737373 ... Infinitely cyclic decimals are called irrational numbers ... such as π,-0.101001... (two 65438
Absolute value: a≥0 丨 a 丨 = a；; A≤0 丨 a 丨 =-a.
For example: 买-买 =; 3. 14-π = π-3. 14.
3. A divisor, starting with a non-zero number on the left and ending with the last number, is called the effective number of this divisor. For example, 0.05972 is accurate to 0.00 1 to get 0.060, and two significant numbers 6,0 are obtained.
4. Write a number in the form of a× 10n (where 1 ≤ A
Such as: -40700 =-4.07× 105, 0.000043 = 4.3× 10-5.
5. Every time the square root decimal point moves by 2 digits, the arithmetic square root decimal point moves in the same direction 1 digit; Every time the decimal point of the square root moves by 3 digits, the decimal point of the cube root moves in the same direction by 1 digit.
For example: known =0.4858, then = 48.58; Given = 1.558, then =0. 1588.
6. Multiplication and division of algebraic expressions:
① Multiplication and division of several monomials, multiplication and division of coefficients, combined with the power of the base.
② Polynomials are multiplied by monomials, and each term of polynomials is multiplied by monomials.
(3) Polynomials are multiplied by polynomials, and each term of one polynomial is multiplied by each term of another polynomial.
④ Polynomial is divided by monomial, and every term of polynomial is divided by this monomial.
7. The essence of power operation:
am×an=am+n。
am \an = am-n。
(am)n=amn。
n=anbn。
()n=n。
A-n = n, especially: () -n = () n. ⑦ A0 = 1 (a ≠ 0).
Such as: a3× a2 = a5, a6 ÷ a2 = a4, (a3) 2 = a6, (3a3) 3 = 27a9, (-3)- 1 =-, 5-2 =, () -2 = () 2 =, (-3).
8. Multiplication formula (which in turn is the formula of factorization):
①(a+b)(a-b)=a2-b2。
②(a b)2=a2 2ab+b2。
③(a+b)(a2-ab+b2)=a3+b3。
④(a-b)(a2+a b+B2)= a3-B3； a2+b2=(a+b)2-2ab，(a-b)2=(a+b)2-4ab。
9. The principle of choosing factorization method is: first, see if the common factor can be mentioned. If there is no common factor, binomial uses square difference formula or cubic sum difference formula, trinomial uses cross multiplication (especially complete square formula), and more than three items use group decomposition method. Note: Factorization should be carried out until each polynomial factor can no longer be decomposed.
10. Fractional operation: The multiplication and division method must first decompose the numerator and denominator into factors, divide them in reverse, and multiply them after division; Addition and subtraction must first factorize the denominator, and then divide it (the denominator cannot be removed). Note: The result should be simplified to the simplest score.
1 1. Quadratic radical:
①()2=a(a≥0)，
②=èaè，
③=×,
④=(a & gt； 0，b≥0)。
Such as ① (3) 2 = 45. ② = 6.3a
12. One-variable quadratic equation: For equation: ax2+bx+c=0:
The formula for finding the root is x=, where = B2-4ac is called the discriminant of the root. When δ >; 0, the equation has two unequal real roots; When δ = 0, the equation has an equal real root; When δ
If the equation has two real roots x 1 and x2, then
X 1+x2 =-, X 1x2 =, and the quadratic trinomial ax2+bx+c can be decomposed into a (X-X 1) (X-x2).
③ The quadratic equation with roots A and B is x2-(a+b) x+ab = 0.
13. Solving fractional equations (denominator or substitution) and irrational number equations (square of both sides or substitution) must be tested. The equation in the form: is solved by substitution method; A system of equations, in the form of: first, an equation is decomposed into two linear equations, and then these two equations are combined with another system of equations to form two equations, and then these two equations are solved by method of substitution.
14. Both sides of the inequality are multiplied or divided by the same negative number. If the inequality is not equal, the direction must be changed.
15. Plane rectangular coordinate system:
① The coordinates of the interior points of each defined image are as shown in the figure.
(2) The point on the horizontal axis (X axis) with the ordinate of 0; A point on the longitudinal axis (Y axis) with the abscissa of 0.
(3) With respect to two points that are transversely symmetrical, the abscissa is the same (the ordinate is the opposite number);
With regard to two points of longitudinal symmetry, the ordinate is the same (the abscissa is the opposite number);
For two points with symmetrical origin, the abscissa and ordinate are opposite.
16. the image of the linear function y=kx+b(k≠0) is a straight line (b is the ordinate of the intersection of the straight line and the y axis). When k >; 0, y increases with the increase of x (straight line rises from left to right); When k < 0, y decreases with the increase of x (the straight line decreases from left to right). Especially when b=0, y=kx is also called proportional function (y is proportional to x), and the image must pass through the origin.
17. The image with inverse proportional function y=(k≠0) is called hyperbola. When k >; 0, hyperbola is in the first and third quadrants (decreasing from left to right); When k < 0, hyperbola is in the second and fourth quadrants (rising from left to right). Therefore, its increase or decrease is contrary to a linear function.
18. The image of quadratic function y=ax2+bx+c(a≠0) is called parabola (c is the ordinate of the intersection of parabola and y axis).
A>0, the opening is upward; A<0, opening down.
Vertex coordinates are (-,) and symmetry axis is straight line x =-.
Special: the vertex coordinate of parabola y = a (x-h) 2+k is (h, k), and the symmetry axis is straight line x = h.
Note: the method of finding the analytical formula
(1) if the three-point coordinates are known, it is set to the general form y = ax2+bx+c;
② Given the vertex coordinates (h, k), let it be vertex y = a (x-h) 2+k;
(3) Given the coordinates (x 1, 0) and (x2, 0) of the two intersections of the parabola and the X axis, let it be the intersection point Y = A (X-X 1) (X-X2).
19. relationship between parabola and x axis: for parabola y = AX2+BX+C
①δ& lt； 0, which does not intersect with x.
② When δ = 0, it has only one intersection with the X axis (tangent to the X axis).
③δ& gt； 0, which has two intersections (x 1, 0) and (x2, 0) with the x axis, where x 1 and x2 are two roots of the equation ax2+bx+c=0.
20. Preliminary statistics: (1) Concept:
All the objects to be studied are called population, in which each object is called an individual. Some individuals extracted from the population are called the sample of the population, and the number of individuals in the sample is called the sample size.
In a set of data, the number that appears the most (sometimes more than one) is called the mode of this set of data.
Arrange a set of data in order of size, and the middle number (or the average of two numbers) is called the median of this set of data.
(2) Formula: If there are n numbers x 1, x2, …, xn, then:
① Average value =(x 1+x2+…+xn).
② Variance S2 = [(x1-) 2+(x2-) 2+…+(xn-) 2. (Used when it is an integer)
③S2 =[(x 12+x22+…+xn2)-n()2]。 Note: Use this formula when there are few digits in each data or the average value is a fraction.
(4) If n numbers X 1, X2, ..., Xn are all subtracted from an appropriate number a, and a new number X 1, X2, ..., Xn, then the variance S2 of the original group number = the variance of the new group number, and the larger the mean value =a+, the greater the variance, the more the fluctuation of this group of data. Typically, sample variance is used.
(3) Frequency: ① Divide a group number into several groups, with the group distance = (maximum-minimum) ÷ number of groups (when finding the number of groups, use the ending.
Method is an integer), at this time, the number of data falling in a group is called the frequency of the group, and the frequency of each group is the sum of the data.
The ratio of numbers is called the frequency of this group. Therefore, the sum of each group of frequencies is equal to 1. In the histogram of frequency distribution, the area of each small rectangle is equal to the frequency of each corresponding group. The sum of the areas of each small rectangle is equal to 1.
2 1. Acute trigonometric function:
① Let ∠A be any acute angle of RT δ, then ∠A's sine: sinA=, ∠A's cosine: cosA=, ∠A's tangent: tanA=, and ∠A's cotangent: cotA=.
Sina = COSB, TGA = CTGB, TGA =1,SIN2A+COS2A = 1.0.
② complementary angle formula: sin (900-a) = COSA, COS (900-a) = Sina, TG (900-a) = CTGA, CTG (900-a) = TGA.
③ trigonometric function values of special angles: sin300 = cos600 =, sin450 = cos450 =, sin600 = cos300 =, sin00 =.
cos900=0，sin900=cos00= 1，tg300=ctg600=，tg450=ctg450= 1，tg600=ctg300=，tg00=ctg900=0。
④ The gradient of the slope i==. Let the inclination angle be α, then i=tgα=.
22. Triangle:
(1) In a triangle, equal sides are equal to angles and equal sides are equal to sides.
(2) The methods to prove the congruence of two triples are SAS, AAS, ASA, SSS and HL.
(3) In RT δ, the median line on the hypotenuse is equal to half of the hypotenuse.
(4) The methods to prove that a triangle is a right triangle are:
First prove that there is an angle equal to 900.
It is proved that the square of the longest side is equal to the sum of the squares of the other two sides.
Prove that the center line of one side is equal to half of this side.
④ The midline of the triangle is parallel and equal to half of the three sides.
⑤ In an isosceles triangle, the bisector of the vertex coincides with the midline and height of the bottom.
23. Quadrilateral:
The sum of inner angles of (1)n polygon is equal to (n-2) 1800, and the sum of outer angles is equal to 3600.
(2) The nature of parallelogram: the opposite sides are parallel and equal; Diagonally equal; Adjacent corners are complementary; Diagonal lines will split each other in two.
(3) The methods to prove that a quadrilateral is a parallelogram are:
① Prove that the two groups are parallel to each other.
② Prove that the two groups of opposite sides are equal.
③ First, prove that a group of opposite sides are parallel and equal.
Prove that two diagonal lines are equally divided.
⑤ Prove that the two diagonal lines are equal.
(4) The diagonals of the rectangle are equal and equally divided; The diagonal of the diamond is divided vertically and the four sides are equal.
(5) The methods to prove that a quadrilateral is a rectangle are:
First, it is proved that it has three right angles.
First, it is proved that it is a parallelogram, and then it is proved that one of its angles is right angle or diagonal is equal.
(6) The methods to prove that a quadrilateral is a diamond are:
First, it is proved that its four sides are equal.
First, it is proved that it is a parallelogram, and then it is proved that it has a set of diagonal lines with equal or perpendicular adjacent sides.
(7) A square is both a rectangle and a diamond, and has all the properties of a rectangle and a diamond.
(8) The center line of the trapezoid is parallel to the two bottoms and equal to half of the sum of the two bottoms.
(9) Axisymmetric figures include: line segments, angles, isosceles triangles, isosceles trapezoid, rectangles, diamonds, squares, regular polygons and circles.
(10) The central symmetric figures are: line segment, parallelogram, rectangle, diamond, square, regular polygon with even number of sides and circle.
24. The methods to prove that two triangles are similar are:
It is proved that the corresponding angles of the two groups are equal.
Prove that the two sides are proportional and the included angle is equal.
Prove that the three sides are proportional.
Prove that the hypotenuse is proportional to the right angle. The nature of similar triangles: the ratio of height, angle bisector, center line and perimeter are all equal to similarity ratio. The area ratio is equal to the square of the similarity ratio.
25. parallel cutting theorem: ① as shown in figure 1, DE∑BC =.
② As shown in Figure 2, if AB∨CD∨EF =, =.
26. Projective Theorem: As shown in Figure 3, in δ ABC, if ∠ACB=900,
Then cd⊥ab:①ac2 = ad ab. ② bc2 = BD ba。 ③ ad2 = da db。
27. Related properties of the circle:
(1) vertical diameter theorem: If a straight line has one of the following five properties,
Any two properties: ① passing through the center of the circle; ② Vertical chord; ③ bisect the chord; (4) bisecting the lower arc of the chord;
⑤ If the chord bisects the optimal arc, then this straight line has three other properties.
Note: When ① and ③ are available, the chord cannot be the diameter.
(2) The arcs sandwiched by two parallel chords are equal.
(3) In the same circle or circle, if one of two central angles, two arcs, two chords and the chord-center distance of two chords is equal, the other three corresponding quantities are equal.
(4) The degree of the central angle is equal to the degree of the arc it faces.
(5) The arc faces a circumferential angle equal to half of its central angle.
(6) The circumferential angle is equal to half of the arc it faces.
(7) The chord tangent angle is equal to half the degree of the arc it clamps.
(8) The circumferential angles of the same arc or equal arc are equal.
(9) In the same circle or in the same circle, the circular arcs with equal circumferential angles are equal.
( 10)。 The circumferential angle of 900 is opposite to the chord diameter.
(1 1) The diagonals of the quadrilateral inscribed in the circle are complementary, and the outer angle is equal to its inner diagonal.
28. The relationship between the position of a straight line and a circle:
(1) If the radius of ⊙O is r and the distance from the center of the circle to the straight line L is d, then:
①d & lt； R straight line l and ⊙O intersect.
②d=r The straight line L is tangent to ⊙ O.
③d & gt； R straight lines l and ⊙O are separated.
(2) Judgment theorem of tangent: The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle. On the contrary, the tangent is perpendicular to the radius of the tangent point.
(3) Tangent length theorem, chord tangent angle theorem, intersecting chord theorem and its derivation, section line theorem and its derivation.
(4) The center of the inscribed circle of a triangle is called the center of the triangle. The center of a triangle is the intersection of three internal bisectors. The center of the circumscribed circle of a triangle is called the outer center of the triangle. The outer center of a triangle is the intersection of the vertical lines of three sides.
(5) 5) Within the radius r of inscribed circle of rt δ δ =, within the radius r of inscribed circle of any polygon =.
(6) The sum of one set of opposite sides of the circumscribed quadrilateral is equal to the sum of the other set of opposite sides.
29. The relationship between circles and circles:
Let the radii of two circles be r and r respectively and the center distance be d, then:
①d & gt； R+r are separated from each other.
②d=R+r circumscribes two circles.
③R-R & lt； Two circles of d & ltR+r(R≥r) intersect.
④ D = R-R inscribed with two circles.
⑤d & lt； Two circles contain.
30. Auxiliary lines commonly used in circles:
Two circles intersect, usually as a chord and a line connecting the heart.
These two circles are tangent, which are often used as common tangent and connecting line.
Given a tangent, the tangent point is usually used as the radius.
Known diameter, often used as the circumferential angle of the diameter.
Solve the problem about chords, and make the center distance of chords.
(6) The midpoint of an arc is often connected with the center of the circle.
3 1. Each vertex bisects the circumference of a regular N polygon with equal sides and angles, each inner angle = degrees, and the center angle = outer angle = degrees.
32. Area formula:
S is positive δ =× (side length) 2.
S parallelogram = base × height.
S diamond = base × height =× (diagonal product)
④S circle =πR2.
⑤C circumference = 2 π r.
⑥ arc length L=.
⑦S sector ==LR.
⑧S cylindrical edge = bottom circumference × height.
⑨S cone edge =× bottom circumference× bus =πrR, 2πr= (as shown above).