I. Numbers and algebra

1. Sum of numbers

(1) real number

Properties of real numbers:

① The inverse of real number A is -A, and the reciprocal of real number A is (a ≠ 0);

(2) Absolute value of real number A:

(3) The positive number is greater than 0, the negative number is less than 0, and there are two negative real numbers, with the larger absolute value being smaller.

Quadratic radical:

① Operational properties of the square root of product sum quotient:

(a≥0，b≥0)；

(a≥0，b > 0)；

(2) The properties of quadratic roots:

(2) Algebraic expressions and fractions.

(1) the same base power rule: the same base power, the same base, exponential addition, that is, (m, n is a positive integer);

(2) same base powers's division rule: same base powers divides, the base number remains unchanged, and the exponent is subtracted, that is, (a≠0, m, n is a positive integer, m >；; n)；

(3) Power Law: Power is power, the base is constant, and it is multiplied by exponent, that is, (n is a positive integer);

④ Zero index: (a ≠ 0);

⑤ Negative integer index: (a≠0, n is a positive integer);

6 square difference formula: the product of the sum of two numbers and the difference between these two numbers is equal to the square of these two numbers, that is;

⑦ Complete square formula: the square of the sum (or difference) of two numbers is equal to the sum of their squares, plus (or minus) twice their product, that is;

mark

① Basic properties of a fraction: both the numerator and denominator of the fraction are multiplied by (or divided by) the same algebraic expression that is not equal to zero, and the value of the fraction remains unchanged, that is; Where m is an algebraic expression not equal to zero;

(2) the multiplication rule of fractions:

(3) the law of division of fractions:

(4) Fractional power law: (n is a positive integer);

⑤ Fraction addition and subtraction rules with the same denominator:

⑥ Addition and subtraction rules of fractions with different denominators:

2. Equality and inequality

The root formula of (1) quadratic equation (a≠0);

(2) Discriminant formula of the root of a quadratic equation with one variable: discriminant formula of the root of a quadratic equation with one variable (a ≠ 0);

This equation has two unequal real roots;

This equation has two equal real roots;

This equation has no real root;

③ The relationship between the roots and coefficients of a quadratic equation: Let it be two roots of equation (a≠0), then+=, =;

Basic properties of inequality:

① Add (or subtract) the same number or the same algebraic expression on both sides of the inequality, and the direction of the inequality remains unchanged;

② Both sides of inequality are multiplied (or divided) by the same positive number, and the direction of inequality remains unchanged;

③ When both sides of inequality multiply (or divide) the same negative number, the direction of inequality changes;

3. Function

Image of linear function: the image of function y=kx+b(k and b are constants, k≠0) is a straight line passing through point (0, b) and parallel to straight line y=kx;

Properties of linear function: let y=kx+b(k≠0), then k >；; 0, y increases with the increase of x; When k < 0, y decreases with the increase of x;

Image of proportional function: The image of the function is a straight line passing through the origin and point (1, k).

Properties of proportional function: If, then:

(1) when k >; 0, y increases with the increase of x;

② when k

Image of inverse proportional function: function (k≠0) is hyperbola;

Properties of inverse proportional function: let (k≠0), if k >；; 0, then when x>0 or x

The image of quadratic function: the image of the function is a parabola whose symmetry axis is parallel to the Y axis;

① Opening direction: When a>0, the parabolic opening is upward, while when

② Symmetry axis: straight line;

③ Vertex coordinates (;

④ increase or decrease: when a >; 0, if, then y decreases with the increase of x, if, then y increases with the increase of x; When a<0, if, then Y increases with the increase of X, if, then Y decreases with the increase of X;

Second, space and graphics.

1. Understanding of graphics

(1) angle

The nature of angle bisector: the points on the angle bisector are equidistant from both sides of the angle, and the points from the inside to both sides of the angle are on the angle bisector.

(2) Intersecting lines and parallel lines

The complementary angles of the same angle or equal angle are equal, and the complementary angles of the same angle or equal angle are equal;

The nature of antipodal angle: antipodal angle is equal.

Nature of vertical line:

(1) There is one and only one straight line perpendicular to the known straight line;

(2) The vertical line segment is the shortest among all the line segments with a point outside the line connected to each point on the line;

Definition of the midline of the line segment: the line passing through the midpoint of the line segment and perpendicular to the line segment is called the midline of the line segment;

The nature of the vertical line in the line segment: a point on the vertical line in the line segment has the same distance from both ends of the line segment, and the point with the same distance from both ends of the line segment is the vertical line of the line segment;

Definition of parallel lines: two straight lines that do not intersect on the same plane are called parallel lines;

Determination of parallel lines:

(1) Same angle, two straight lines are parallel;

② The internal dislocation angles are equal and the two straight lines are parallel;

③ The internal angles on the same side are complementary, and the two straight lines are parallel;

Characteristics of parallel lines:

(1) Two straight lines are parallel and the same angle is equal;

② Two straight lines are parallel and the internal dislocation angles are equal;

③ Two straight lines are parallel and complementary;

Parallelism axiom: One and only one straight line is parallel to the known straight line through a point outside the straight line.

(3) Triangle

Trilateral relation theorem and inference of triangle: the sum of two sides of triangle is greater than the third side, and the difference between the two sides is smaller than the third side;

Theorem of sum of interior angles of triangle: the sum of three interior angles of triangle is equal to;

Theorem of sum of exterior angles of triangle: one exterior angle of triangle is equal to the sum of two non-adjacent exterior angles;

External angle of triangle and theorem reasoning: one external angle of triangle is greater than any internal angle that is not adjacent to it;

The three bisectors of a triangle intersect at a point (center);

The perpendicular lines of the three sides of a triangle intersect at one point (outer center);

Triangle midline theorem: the line connecting the midpoints of two sides of a triangle is parallel to the third side and equal to half of the third side;

Congruent triangles's judgment:

(1) edge axiom (SAS)

(2) Axiom of Angle (ASA)

③ Angular Edge Theorem (AAS)

④ Edge axiom (SSS)

(5) Axiom of hypotenuse and Right Angle (HL)

The nature of isosceles triangle;

① The two base angles of an isosceles triangle are equal;

② The bisector of the top angle of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide (the three lines are one).

Determination of isosceles triangle;

Two triangles with equal angles are isosceles triangles;

Properties of right triangle:

① The two acute angles of a right triangle are complementary angles;

② The median line on the hypotenuse of the right triangle is equal to half of the hypotenuse;

③ The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse (Pythagorean theorem);

(4) The right side of a right triangle is equal to half of the hypotenuse;

Determination of right triangle;

① A triangle with two complementary angles is a right triangle;

(2) A triangle is a right triangle (the inverse theorem of Pythagorean theorem) if its three-side lengths A, B and C have the following relations.

(4) quadrilateral

Theorem of the sum of internal angles of polygons: the sum of internal angles of n polygons is equal to (n≥3, n is a positive integer);

Properties of parallelogram:

① The opposite sides of the parallelogram are equal;

② The diagonals of parallelograms are equal;

(3) diagonal bisection of parallelogram;

Determination of parallelogram;

① Two groups of quadrangles with equal diagonal angles are parallelograms;

② Two groups of quadrangles with equal opposite sides are parallelograms;

③ Quadrilaterals whose diagonals are bisected are parallelograms;

④ A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

Properties of rectangle: (except all properties of parallelogram)

① All four corners of a rectangle are right angles;

② The diagonals of rectangles are equal;

Determination of rectangle:

① A quadrilateral with three right angles is a rectangle;

② Parallelograms with equal diagonals are rectangles;

Characteristics of rhombus: (Except all the properties of parallelogram,

① The four sides of the diamond are equal;

(2) The diagonals of the rhombus are bisected vertically, and each diagonal bisects a set of diagonals;

Diamond decision:

A quadrilateral with four equilateral sides is a diamond;

Features of the square:

① The four sides of a square are equal;

② All four corners of a square are right angles;

(3) The two diagonals of a square are equal and vertically bisected, and each diagonal bisects a set of diagonals;

The trial in the square:

① A diamond with a right angle is a square;

② A group of rectangles with equal adjacent sides are squares.

Characteristics of isosceles trapezoid;

① The two internal angles on the same base of the isosceles trapezoid are equal.

② The two diagonals of the isosceles trapezoid are equal.

Determination of isosceles trapezoid:

① Two trapeziums with equal internal angles on the same base are isosceles trapeziums;

② Two trapeziums with equal diagonals are isosceles trapeziums.

Mosaic of plane graphics;

Any triangle, quadrilateral or regular hexagon can be inlaid with a plane;

(5) circle

The positional relationship between the point and the circle (let the radius of the circle be R and the distance from the point P to the center O be D);

(1) point p on the circle, then d=r, and vice versa;

(2) point P is in the circle, then D.

③ Point P is outside the circle, then d>r, and vice versa;

The relationship among central angle, chord and arc: in the same circle or equal circle, as long as one group of central angle, chord and arc is equal, the other two groups can be equal;

Determination of circle: three points not on a straight line determine a circle;

Vertical diameter theorem (and the inference of vertical diameter theorem): the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord;

Equal arcs of parallel chords: the arcs of two parallel chords of a circle are equal;

Theorem of central angle: the degree of central angle is equal to the degree of arc it faces;

Theorem and inference of the relationship among central angle, arc, chord and chord center distance: in the same circle or in the same circle, the arc opposite to the equal central angle is equal, and the chord center distance of the opposite chord is equal;

Inference: In the same circle or in the same circle, if one set of quantities in two central angles, two arcs, two chords or the center distance between two chords are equal, the corresponding other set of quantities are equal respectively;

Angle theorem of circle: the angle of a circle is equal to half the angle of the arc it faces;

Inference of the fillet theorem: the fillet corresponding to the diameter is a right angle, on the contrary, the chord corresponding to the fillet is a diameter;

Judgment theorem of tangent: the straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle;

The property theorem of tangent: the tangent of a circle is perpendicular to the radius of the tangent point;

Tangent length theorem: two tangents of a circle are drawn from a point outside the circle, and the line segments from this point to the two tangents are equal, and the line connecting them with the center of the circle bisects the included angle of the two tangents;

Calculation formula of arc length: (r is the radius of the circle, n is the degree of the central angle of the arc, and it is the arc length)

Sector area: or (r is the radius, n is the degree of the central angle of the sector, and it is the arc length of the sector)

Bow area

(6) Ruler drawing (basic drawing, making triangles and circles with basic graphics)

Make a line segment equal to a known line segment and an angle equal to a known angle; Make a bisector with a known angle; Perpendicular bisector as a line segment; A vertical line passing through a point is a known straight line;

(7) Views and predictions

Draw three views (front view, left view and top view) of basic geometry (straight prism, cylinder, cone and sphere);

According to the unfolded diagram of basic geometric figures (except balls), the three-dimensional model is judged and established;

2. Graphics and transformations

Axisymmetry of graphs

The basic properties of axial symmetry: the line segments connected by corresponding points are equally divided by the axis of symmetry;

Isosceles triangle, rectangle, diamond, isosceles trapezoid, regular polygon and circle are axisymmetric figures;

Graphic translation

The basic properties of graphic translation: the connecting lines of corresponding points are parallel and equal;

Graphic rotation

The basic properties of graphic rotation are: the distance between the corresponding point and the rotation center is equal, the distance between the corresponding point and the rotation center is equal, and the angles formed by the connecting line between the corresponding point and the rotation center are equal to each other;

Parallelogram, rectangle, diamond, regular polygon (even number of sides) and circle are central symmetric figures;

Similarity of graphics

The basic nature of proportion: if, then, if, then.

Similar triangles's classification method: ① Two groups of angles are equal and corresponding; (2) The two sides are in direct proportion and the included angle is equal; ③ Three sides are in direct proportion.

The nature of similar triangles: ① the corresponding angles of similar triangles are equal; ② The corresponding sides of similar triangles are proportional; ③ The ratio of similar triangles perimeter is equal to similarity ratio; ④ The area ratio of similar triangles is equal to the square of similarity ratio;

Properties of similar polygons:

① The angles corresponding to similar polygons are equal; ② The corresponding edges of similar polygons are proportional;

③ The area ratio of similar polygons is equal to the square of the similarity ratio;

The relationship between graphic similarity and graphic similarity: two graphic similarities are not necessarily graphic similarities, but they must be similar;

At Rt△ABC, ∠C=, SinA=, cosA=, tanA=,

CotA=

Trigonometric function value of special angle:

Sinα

Coase α

tanα

1

Cotα

1

Three. Probability and Statistics

1. Statistics

Data collection methods and data representation methods (statistical tables and fan charts, broken line charts and bar charts)

(1) population and sample

All the objects to be investigated are called the 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 capacity of the sample.

Data analysis and decision-making (with the help of the learned statistical knowledge, sort out and analyze the collected data, and then make judgments and decisions on the analysis results)

(2) mode and median

Mode: the data with the highest frequency in a group of data;

Median: a set of data is arranged in the order from largest to smallest, and is in the middle position.

(3) Histogram of frequency distribution

Frequency =, the sum of each group of frequencies is equal to the total number, and the sum of each group of frequencies is equal to 1. The area of each small rectangle in the frequency distribution histogram is the frequency of each group.

(4) Two average formulas.

The average of n, ..., is:

(2) If there are times, times and times in the number n, and ++= n, then;

(5) Calculation formula of range, variance and standard deviation:

(1) extremely poor:

The difference between the maximum value and the minimum value of a set of data reflects the range of this set of data. The difference obtained by this method is called extreme range, that is, extreme range = maximum-minimum;

② Variance:

The variance of the data, ..., is,

Then =

③ Standard deviation:

Standard deviation of data, ...,

Then =

The greater the variance of a set of data, the greater the fluctuation of this set of data.

2. Possibility

① If the probability of an event is expressed by p, then 0 ≤ p (a) ≤1;

P (inevitable event) =1; P (impossible event) = 0;

② Understand the meaning of probability in specific situations, and calculate the probability of simple events by enumeration (including list and tree drawing).

③ The frequency of repeated experiments can be regarded as an estimate of the probability of events;

3. The preliminary knowledge and probability of statistics are widely used in social life, and the learned knowledge can be used to solve practical problems.