☆ Summary ☆

I. Key concepts

The classification and concept of 1 figure

Digital series table:

Note: "Classification" principle: 1) Proportionality (no weight, no leakage)

2) There are standards

2. Non-negative number: the collective name of positive real number and zero. (Table: x≥0)

Common non-negative numbers are:

Property: If the sum of several non-negative numbers is 0, then every unburdened number is 0.

3. Reciprocity: ① Definition and characterization

② attribute: a.a ≠1/a (a ≠1); B. 1/a，a≠0； c . 0 < a < 1/a > 1； When a > 1,1/a <1; D. the product is 1.

4. Reciprocal: ① Definition and representation

② Properties: A.a when A.a≠0; The position of a and -a on the number axis; The sum of c is 0 and the quotient is-1.

5. Number axis: ① Definition ("three elements")

② Function: a. Visually compare real numbers; B. clearly reflect the absolute value; C. establish a one-to-one correspondence between points and real numbers.

6. Odd number, even number, prime number and composite number (positive integer-natural number)

Definition and expression:

Odd number: 2n- 1

Even number: 2n(n is a natural number)

7. Absolute value: ① Definition (two kinds):

Algebraic definition:

Geometric definition: the geometric meaning of the absolute value top of the number A is the distance from the point corresponding to the real number A on the number axis to the origin.

② A ≥ 0, and the symbol "│ │" is a sign of "non-negative number"; ③ There is only one absolute value of number A; ④ When dealing with any type of topic, as long as "│ │" appears, the key step is to remove the "│ │" symbol.

Second, the operation of real numbers

1. Arithmetic (addition, subtraction, multiplication, division, power and root)

2. Algorithm (five plus [multiplication] commutative law and associative law; [Multiplication versus Addition]

Distribution law)

3. Operation sequence: a. Advanced operation to low-level operation; B. (Operation at the same level) From "Left"

To the "right" (such as 5 ÷ 5); C (when there are brackets) from "small" to "medium" to "large".

Attachment: Typical examples

1. It is known that the positions of a, b and x on the number axis are as follows. Please verify: │x-a│+│x-b│.

=b-a。

2. a-b=-2 and AB are known.

Chapter II Algebraic Formulas

★ Key points ★ Related concepts, properties and operations of algebraic expressions.

☆ Summary ☆

I. Key concepts

Classification:

1. Algebras and Rational Expressions

Formulas that associate numbers or letters representing numbers with operational symbols are called algebraic expressions. independent

The number or letter of is also an algebraic expression.

Algebraic expressions and fractions are collectively called rational forms.

2. Algebraic expressions and fractions

Algebraic expressions involving addition, subtraction, multiplication, division and multiplication are called rational expressions.

Rational expressions without division or division but without letters are called algebraic expressions.

Rational number formula has division, and there are letters in division, which is called fraction.

3. Monomial and Polynomial

Algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)

The sum of several monomials is called polynomial.

Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation. For example,

=x, =│x│ and so on.

4. Coefficients and indices

Difference and connection: ① from the position; (2) In the sense of representation.

5. Similar projects and their combinations

Conditions: ① The letters are the same; ② The indexes of the same letters are the same.

Basis of merger: law of multiplication and distribution

6. Radical form

The algebraic expression of square root is called radical.

Algebraic expressions that involve square root operations on letters are called irrational expressions.

Note: ① Judging from the appearance; ② Difference: It is a radical, but it is not an irrational number (it is an irrational number).

7. Arithmetic square root

(1) The positive square root of a positive number ([the difference between a ≥ 0-and "square root"]);

⑵ Arithmetic square root and absolute value

① Contact: all are non-negative, =│a│.

② Difference: │a│, where A is all real numbers; Where a is a non-negative number.

8. Similar quadratic root, simplest quadratic root, denominator of rational number.

After being transformed into the simplest quadratic root, the quadratic roots with the same number of roots are called similar quadratic roots.

The following conditions are satisfied: ① the factor of the root sign is an integer and the factor is an algebraic expression; (2) The number of roots does not include exhausted factors or factors.

Crossing out the root sign in the denominator is called denominator rationalization.

9. Index

(1)(- power supply, power supply operation)

(1) when a > 0, > 0; ② when a < 0, > 0 (n is even) and < 0 (n is odd)

(2) Zero index: = 1(a≠0)

Negative integer index: = 1/ (a≠0, p is a positive integer)

Second, the law of operation and the law of nature

1. The law of addition, subtraction, multiplication, division, power and root of fractions.

2. The nature of the score

(1) Basic properties: = (m≠0)

(2) Symbolic law:

⑶ Complex fraction: ① Definition; ② Simplified methods (two kinds)

3. Algebraic expression algorithm (bracket deletion and bracket addition)

4. The essence of power operation: ① =; ② ÷ = ; ③ = ; ④ = ; ⑤

Skills:

5. Multiplication rule: (1) single× single; (2) single × many; 3 more x more.

6. Multiplication formula: (plus or minus)

(a+b)(a-b)= 1

(a b) =

7. Division rules: (1) single-single; (2) Too many orders.

8. Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. find the root formula method.

9. The nature of arithmetic roots: =; ; (a≥0，b≥0)； (a ≥ 0, b > 0) (positive and negative)

10. radical algorithm: (1) addition rule (merging similar quadratic roots); (2) multiplication and division; (3) The denominator is reasonable: a; b； c。

1 1. Scientific notation: (1 ≤ A < 10, n is an integer =

Chapter III Preliminary Statistics

★ main points ★

☆ Summary ☆

I. Key concepts

1. population: all the respondents.

2. Individuals: each respondent in the group.

3. Sample: a part of individuals extracted from the population.

4. Sample size: the number of individuals in the sample.

5. Mode: the data with the highest frequency in a set of data.

6. Median: the number of a set of data in order of size (or the average of two data in the middle).

Second, the calculation method

1. Average sample: (1); (2) If,,,, then (A- constant,,,, is close to a more integer constant A); (3) Weighted average: (4) Average is a characteristic number that describes the trend (concentration position) in data concentration. Sample average is usually used to estimate the overall average. The larger the sample size, the more accurate the estimation.

2. Sample variance: (1); (2) If,,,,, then (a- a relative "integer" constant, close to the average value of,,,); If … is less than "whole", then; ⑶ Sample variance is a characteristic number that describes the degree of data dispersion (fluctuation). When the sample size is large, the sample variance is very close to population variance, which is usually used to estimate population variance.

3. Sample standard deviation:

Chapter IV Linetype

★ Emphasis★ Concepts, judgments and properties of intersecting lines and parallel lines, triangles and quadrangles.

☆ Summary ☆

I. Straight lines, intersecting lines and parallel lines

1. Differences and connections between line segments, rays and straight lines

This paper analyzes the graph, representation, boundary, number of endpoints and basic properties.

2. The midpoint of the line segment and its representation

3. Basic properties of straight lines and line segments (using "basic properties of line segments" to demonstrate that "the sum of two sides of a triangle is greater than the third side")

4. The distance between two points (three distances: point-point; Dotted line; Line-line)

5. Angle (flat angle, rounded corner, right angle, acute angle, obtuse angle)

6. Complementary angle, complementary angle and their expressions

7. The bisector of an angle and its representation

8. Vertical line and its basic properties (use it to prove that "the hypotenuse of a right triangle is greater than the right")

9. Vertex angle and its properties

10. Parallel lines and their judgments and properties (reciprocal) (differences and connections between them)

1 1. Common theorems: ① parallel to two straight lines and parallel to one straight line (transitivity); ② Two straight lines parallel to and perpendicular to a straight line.

12. Definition, proposition and composition of proposition

13. Axioms and theorems

14. Inverse proposition

Second, the triangle

Classification: (1) Classification by edge;

(2) according to the angle.

1. Definition (including internal angle and external angle)

2. The relationship between angles of triangle: (1) the sum and inference of angles and angles: (1) inner angles; ② sum of external angles; (3) the sum of the internal angles of the N-polygon; (4) the sum of the external angles of the N-polygon. ⑵ Edge and edge: The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side. ⑶ Angle and edge: In the same triangle,

3. The main part of the triangle

Discussion: ① Define the intersection of ②×× lines-the property of the× center of triangle.

① High line ② Middle line ③ Angle bisector ④ Middle vertical line ⑤ Middle line.

⑵ General triangle ⑵ Special triangle: right triangle, isosceles triangle and equilateral triangle.

4. Determination and properties of special triangles (right triangle, isosceles triangle, equilateral triangle and isosceles right triangle)

5. congruent triangles

(1) Determine the consistency of general triangles (SAS, ASA, AAS, SSS).

⑵ Determination of congruence of special triangle: ① General method ② Special method.

6. Area of triangle

⑴ General calculation formula ⑴ Properties: The areas of triangles with equal bases and equal heights are equal.

7. Important auxiliary lines

(1) The midpoint and the midpoint form the midline; (2) Double the center line; (3) Add auxiliary parallel lines

8. Proof method

(1) direct proof method: synthesis method and analysis method.

(2) Indirect proof-reduction to absurdity: ① Counterhypothesis ② Reduction to absurdity ③ Conclusion.

(3) Prove that line segments are equal and angles are equal, often by proving triangle congruence.

(4) Prove the folding relationship of line segments: folding method and folding method.

5. Prove the sum-difference relationship of line segments: continuation method and truncation method.

[6] Prove the area relationship: indicate the area.

Third, quadrilateral.

Classification table:

1. General Properties (Angle)

⑴ Sum of internal angles: 360.

(2) Parallelogram connecting the midpoint of each side in turn.

Inference 1: Connect the midpoints of the sides of the quadrilateral in turn with equal diagonal lines to get a diamond.

Inference 2: Connect the midpoints of the sides of the quadrilateral in turn with diagonal lines perpendicular to each other to get a rectangle.

⑶ Sum of external angles: 360.

2. Special quadrilateral

(1) General methods to study them:

(2) parallelogram, rectangle, diamond and square; Definition, properties and judgment of trapezoid and isosceles trapezoid

⑶ Determination steps: quadrilateral → parallelogram → rectangle → square.

┗→ Diamond-=

(4) diagonal tie rod:

3. Symmetric graphics

(1) axis symmetry (definition and properties); (2) Central symmetry (definition and nature)

4. Related Theorems: ① Parallel bisection theorem and its inference 1, 2.

② The midline theorem of triangle and trapezoid.

③ The distance between parallel lines is equal everywhere. (For example, find triangles with equal areas in the figure below)

5. Important auxiliary lines: ① Diagonal lines of quadrangles are often connected; ② Trapezoids are often transformed into triangles by translating a waist, translating a diagonal, making a height, connecting the midpoint between the vertex and the waist and extending the intersection with the bottom.

6. Drawing: Divide the line segments randomly.

Four, application examples (omitted)

The fifth chapter equation (group)

★ Emphasis★ Solution of one-dimensional linear equation, one-dimensional quadratic equation and two-dimensional linear equations; Related application problems of the equation (especially travel and engineering problems)

☆ Summary ☆

I found it somewhere else. I hope I can adopt it.