First, the knowledge network structure
Second, the main points of knowledge
1. In the same plane, there are two kinds of positional relationships between two straight lines: intersecting and parallel, and verticality is a special case of intersection.
2. On the same plane, two disjoint straight lines are called parallel lines. If two straight lines have only one common point, they are said to intersect; If two straight lines have no common point, they are said to be parallel.
3. Among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are
The properties of adjacent complementary angles: complementary adjacent complementary angles. As shown in figure 1, they are complementary angles,
Fill the corner with neighbors. + = 180 ; + = 180 ; + = 180 ;
+ = 180 。
4. Among the four corners formed by the intersection of two straight lines, two sides of one corner are opposite extension lines of two sides of the other corner, so the two corners are opposite. The nature of antipodal angle: antipodal angle is equal. As shown in figure 1, and they are opposite to each other. = ;
= 。
5. If one of the angles formed by the intersection of two straight lines is a right angle or 90, the two straight lines are said to be perpendicular to each other.
One of them is called the perpendicular of the other. As shown in Figure 2, when = 90, ⊥.
Nature of vertical line:
Property 1: There is one and only one straight line perpendicular to the known straight line.
Property 2: Of all the line segments connecting a point outside the straight line and a point on the straight line, the vertical line segment is the shortest.
Property 3: As shown in Figure 2, when a ⊥ b, = = = 90.
Distance from point to straight line: The length from a point outside a straight line to the vertical section of this straight line is called the distance from point to straight line.
6. The basic characteristics of congruent angle, internal dislocation angle and ipsilateral internal angle:
(1) is on the same side of two straight lines (cut lines) and the same side of the third straight line (cut lines), so that
These two angles are called isosceles angles. In Figure 3, * * * has a pair of isosceles angles; And is an isosceles angle;
And are at the same angle; And are at the same angle; And it's the same angle.
(2) Between two straight lines (secant) and on both sides of the third straight line (secant), such two angles are called inscribed angles. In figure 3; * * has a pair of inner corners; Is the inner corner; And is an inner corner.
(3) Between two straight lines (intersecting lines), both are on the same side of the third straight line (intersecting line), and such two angles are called ipsilateral inner angles. In figure 3; * * There are a pair of inner corners on the same side; And are internal angles on the same side; And it is the same inner angle.
7. Parallelism axiom: At a point outside a straight line, only one straight line is parallel to the known straight line.
Inference of the axiom of parallelism: If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.
Properties of parallel lines:
Property 1: Two straight lines are parallel and equal to the complementary angle. As shown in fig. 4, if a∑b,
Then =; = ; = ; = 。
Property 2: Two straight lines are parallel and the internal dislocation angles are equal. As shown in figure 4, if a∨b, then =; = 。
Property 3: Two straight lines are parallel and complementary. As shown in figure 4, if a∨b,+=180;
+ = 180 。
Property 4: Two lines parallel to the same line are parallel to each other. If a∨b and a∨c, then ∨.
8. Determination of parallel lines:
Judgment 1: congruent angles are equal and two straight lines are parallel. As shown in figure 5, if =
Or = or = or =, then a ∨ b.
Decision 2: The internal dislocation angles are equal and the two straight lines are parallel. As shown in figure 5, if = or =, then a ∨ b.
Judgment 3: The internal angles on the same side are complementary and the two straight lines are parallel. As shown in figure 5, if+=180;
+= 180, then a ∨ b.
Decision 4: Two straight lines parallel to the same straight line are parallel to each other. If a∨b and a∨c, then ∨.
9. A statement that judges a thing is called a proposition. A proposition consists of a topic and a conclusion, which can be divided into true proposition and false proposition. If the topic is established, then the conclusion must be established, and such a proposition is called a true proposition; If the topic holds, then the conclusion may not hold. Such a proposition is called a false proposition. Prove the correctness of the true proposition by reasoning. Such a true proposition is called a theorem, which can be used as the basis for further reasoning.
10. Translation: A figure moves a certain distance in a certain direction in a plane. This movement of graphics is called translation transformation, or translation for short.
After translation, the shape and size of the new picture are exactly the same as the original picture. Every point in the new graphic after translation is obtained by moving a point in the original graphic. Such two points are called corresponding points.
Translation properties: ① The connecting lines of corresponding points in the two images before and after translation are parallel and equal; ② The corresponding line segments are equal; ③ The corresponding angles are equal.
Chapter VI Real Numbers
Knowledge point classification-real number
1. Classification by definition: 2. Classification by natural symbols:
Note: 0 is neither positive nor negative.
Knowledge point 2 Related concepts of real numbers
1. Inverse
Algebraic meaning of (1): There are only two numbers with different signs, and we say that one of them is opposite to the other. The antonym of 0 is 0.
(2) Geometric meaning: On both sides of the origin on the number axis, two points with the same distance from the origin represent two opposite numbers, or on the number axis, the points corresponding to two opposite numbers are symmetrical about the origin.
(3) The sum of two opposites is equal to 0.a and B are opposites a+b=0.
2. Absolute value | a | ≥ 0.
3. The reciprocal (1)0 has no reciprocal. (2) Two numbers whose product is 1 are reciprocal. A and b are reciprocal.
4. Square root
(1) If the square of a number is equal to a, it is called the square root of a, a positive number has two square roots, and the two square roots are in opposite directions. 0 has a square root, and the square root itself is 0; Negative numbers have no square root. The square root of a (a ≥ 0) is written as.
(2) The positive square root of a positive number is called the arithmetic square root of a, and the arithmetic square root of A (A ≥ 0) is recorded as.
5. Cubic root
If x3=a, then x is called the cube root of a, and positive numbers have positive cube roots; Negative numbers have negative cubic roots; The cube root of zero is zero.
Knowledge point 3 Real number and axis
Definition of number axis: the straight line defining the origin, positive direction and unit length is called number axis, and the three elements of number axis are indispensable.
Comparison of real numbers of knowledge point four
1. For any two points on the number axis, the point on the right represents a larger number.
2. Positive numbers are all greater than 0, negative numbers are all less than 0, and two positive numbers, the greater the absolute value, the greater the positive number; Two negative numbers; The absolute value is large but small.
3. The relative size of irrational numbers:
The operation of knowledge point five real numbers
1. Add
Add two numbers with the same sign, take the same sign, and add the absolute values; Add different symbols with different absolute values of two numbers, take the symbol with the larger absolute value, and subtract the symbol with the smaller absolute value from the larger absolute value; Two opposite numbers add up to 0; When a number is added to 0, it still gets the number.
2. subtraction: subtracting a number is equal to adding the reciprocal of this number.
multiply
Multiply several non-zero real numbers, and the sign of the product is determined by the number of negative factors. When there are even negative factors, the product is positive. When there are odd negative factors, the product is negative. Multiply several numbers, one factor is 0 and the product is 0.
break up
Dividing by a number is equal to multiplying the reciprocal of this number. Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.
5. Multipliers and prescriptions
The meaning of (1)an is the product a of n, any power of positive number is positive, even power of negative number is positive, and odd power of negative number is negative.
(2) Positive numbers and 0 can be squared, but negative numbers cannot be squared; Positive numbers, negative numbers and 0 can all be turned on.
(3) Zero exponent and negative exponent
Six significant figures of knowledge points and scientific notation
1. Valid numbers:
A divisor, from the first non-zero number on the left to the exact number, is called the significant digits of this divisor.
2. Scientific symbols:
The method of counting a number in the form of (1 ≤ < 10, where n is an integer) is called scientific notation.
Chapter VII Plane Cartesian Coordinate System
First, the knowledge network structure
Second, the main points of knowledge
1. Ordered number pair: A number pair consisting of two numbers A and B in sequence is called an ordered number pair, and it is recorded as (a, b).
2. Plane rectangular coordinate system: On a plane, two mutually perpendicular number axes with a common origin form a plane rectangular coordinate system.
3. Horizontal axis, vertical axis and origin: the horizontal axis is called X axis or horizontal axis; The vertical axis is called Y axis or vertical axis; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.
4. Coordinate: For any point P on the plane, the intersection point P is perpendicular to the X axis and Y axis respectively, and the vertical foot is on the X axis and Y axis respectively. The corresponding numbers A and B are called the abscissa and ordinate of point P, respectively, and denoted as P(a, b).
5. Quadrant: Two coordinate axes divide the plane into four parts, the upper right part is called the first quadrant, and the counterclockwise direction is called the second quadrant, the third quadrant and the fourth quadrant. The point on the coordinate axis is not in any quadrant.
6. Coordinate characteristics of each quadrant ① The first quadrant: abscissa 0, ordinate 0; ② Points in the second quadrant: abscissa 0, ordinate 0; ③ Points in the third quadrant: abscissa 0, ordinate 0; ④ Points in the fourth quadrant: abscissa 0, ordinate 0.
7. Coordinate characteristics of points on the coordinate axis ① Points on the positive semi-axis of the X axis: abscissa 0, ordinate 0; ② Points on the negative semi-axis of X axis: abscissa 0, ordinate 0; ③ Points on the positive semi-axis of Y axis: abscissa 0, ordinate 0; ④ The point on the negative semi-axis of Y axis: side sitting.
Mark 0, ordinate 0; ⑤ coordinate origin: abscissa 0, ordinate 0. (fill in ">", "<" or "=")
8. The distance from point P(a, b) to the X axis is |b|, and the distance from point P (a, b) to the Y axis is |a|.
9. Coordinate characteristics of symmetrical points ① With regard to two points that are symmetrical about X axis, the abscissa is equal and the ordinate is opposite; (2) With respect to the two points of Y axis symmetry, the ordinate is equal and the abscissa is opposite; ③ For two points symmetrical about the origin, the abscissa and ordinate are opposite.
10, the distance from point P (2 2,3) to the X axis is; The distance to the y axis is; The point coordinate of point P (2 2,3) which is axisymmetrical about X axis is (,); The point coordinate of the point P (2 2,3) which is symmetrical about the Y axis is (,).
1 1. If the abscissas of two points are the same, the straight line passing through these two points is parallel to the Y axis and perpendicular to the X axis; If the vertical coordinates of two points are the same, the straight line passing through these two points is parallel to the X axis and perpendicular to the Y axis. If the abscissas of points P (2 2,3) and Q (2 2,6) are the same, pq∑y axis, PQ⊥x axis; If the coordinates of points P (- 1, 2) and Q (4, 2) are the same, then pq∑x axis, PQ⊥y axis.
12, the vertical coordinates of the points on the straight line parallel to the X axis are the same; The abscissas of the points on the straight line parallel to the Y axis are the same; The abscissa of the points on the bisector of the first and third quadrants is the same as the ordinate; The abscissa and ordinate of a point on the bisector of quadrant angle are opposite. If point P (a, b) is on the bisector of the first and third quadrant angles, the abscissa of point P is the same as the ordinate, that is, a = b；; If point P(a, b) is on the bisector of the quadrant angle, the abscissa and ordinate of point p are opposite, that is, a =-b.
13, the method of indicating the position of a point (or object): 1. Establish a plane rectangular coordinate system accurately and appropriately; The second is to correctly write the coordinates of the point where an object or a place is located. The coordinate origin chosen is different, the plane rectangular coordinate system established is also different, and the coordinates of the same point obtained are also different.
14, the translation of graphics can be converted into the translation of points. The law of coordinate translation: ① When translating left and right, the abscissa is added or subtracted, and the ordinate is unchanged; ② When translating up and down, the abscissa is unchanged, and the ordinate is added and subtracted; ③ When adding and subtracting coordinates, it should be carried out according to the law of "left minus right plus, up plus down minus". For example, the coordinates of the point obtained by translating the point p (2 2,3) to the left by 2 units are (,); When the point P (2 2,3) is moved to the right by 2 units, the coordinate is (,); Translate the point P (2 2,3) upward by 2 units to get the coordinates of the point (,); Translate the point P (2 2,3) downward by 2 units to get the coordinates of the point (,); Point P (2 2,3) is translated by 3 units to the left, and then by 5 units upward, and the coordinate is (,); The coordinate of point P (2 2,3) is (,) after it is translated 3 units to the left and 5 units down; The coordinate of point P (2 2,3) is (,) after it is translated 3 units to the right and 5 units to the upward. After the point P (2,3) is first translated by 3 units to the right, and then translated by 5 units downward, the coordinate of the point P (2,3) is (,).
Chapter VIII Binary Linear Equations
First, the knowledge network structure
Second, the main points of knowledge
1. An equation containing unknowns is called an equation, and the value of the unknowns that make the left and right sides of the equation equal is called the solution of the equation.
2. The equation contains two unknowns, and the number of terms containing the unknowns is 1. Such an equation is called binary linear equation, and the general form of binary linear equation is (constant, sum). The value of the unknown quantity that makes the left and right sides of the binary linear equation equal is called the solution of the binary linear equation, and a binary linear equation generally has countless groups of solutions.
3. The equations contain two unknowns, and the number of terms containing the unknowns is 1. Such equations are called binary linear equations. The value of the unknown quantity that makes the left and right sides of each equation equal is called the solution of a binary linear equation group, and a binary linear equation group generally has a solution.
4. The general steps of solving binary linear equations by substitution method: observe whether another unknown is expressed by a formula containing an unknown in the equation, and if so, directly substitute it into another equation; If not, one of the equations is deformed, and the other unknown is expressed by a formula containing one unknown; Then substitute the expressed unknown into another equation, so as to eliminate one unknown and find the value of another unknown, and substitute the obtained unknown into any equation in the original equations to find the value of another unknown.
5. General steps for solving binary linear equation by addition and subtraction: In two equations of (1) equation, if the coefficients of the same unknown are neither equal nor reciprocal, multiply the two sides of the equation by appropriate numbers to make the coefficients of the same unknown equal or reciprocal; (2) Add or subtract the two sides of the two equations respectively to eliminate an unknown number; (3) Solve this one-dimensional linear equation and find an unknown value; (4) Substituting the obtained unknown into any equation in the original equation set to obtain the value of another unknown, thus obtaining the solution of the original equation set.
6. The general steps to solve the ternary linear equations: ① Observe the coefficient characteristics of the unknown quantity in the equations and determine which unknown quantity should be eliminated first; (2) Using substitution method or addition and subtraction method, one equation and the other two equations in the equations are respectively grouped, and the same unknown number is eliminated, so as to obtain a binary linear equation group about the other two unknowns; (3) solving the binary linear equations to obtain two unknown values; (4) Substitute the values of these two unknowns into a simple equation in the original equation group, and find out the value of the third unknown, so as to get the solution of the original ternary linear equation group.
Chapter 9 Inequality and Unequal Groups
First, the knowledge network structure
Second, the main points of knowledge
1, and the formula that represents inequality relations with inequality symbols is called inequality, and the inequality symbols mainly include: >,
2. In an inequality with unknowns, the value of the unknowns that make the inequality valid is called the solution of the inequality, and the set of all solutions of an inequality with unknowns is called the solution set of this inequality. The solution set of inequality can be expressed on the number axis. The process of finding the solution set of inequality is called solving inequality. An inequality that contains an unknown number and the number of terms containing the unknown number is 1 is called a unary linear inequality.
3. The essence of inequality:
① Property 1: The same number (or formula) is added (or subtracted) on both sides of the inequality at the same time, and the direction of the inequality remains unchanged.
Expressed in letters: if, then; If, then;
If, then; If, then.
② Property 2: Both sides of the inequality are multiplied by (or divided by) the same positive number at the same time, and the direction of the inequality remains unchanged.
Expressed in letters: if, then (or); If, then (or);
If, then (or); If, then (or);
③ Property 3: When both sides of the inequality are multiplied (or divided) by the same negative number at the same time, the direction of the inequality changes.
Expressed in letters: if, then (or); If, then (or);
If, then (or); If, then (or);
4. The general steps of solving linear inequality of one variable: ① denominator; (2) the bracket is removed; ③ shifting items; (4) merging similar projects; ⑤ The coefficient is 1. This is similar to solving a linear equation with one variable, and the steps should be chosen flexibly according to the specific situation of linear inequality with one variable.
5. An inequality group contains an unknown number, and the number of terms containing the unknown number is 1. Such a set of inequalities is called a set of unary linear inequalities. The value of the unknown quantity that makes each inequality in the inequality group hold true is called the solution of the inequality group, and the set of all solutions of an inequality group is called the solution set solution of this inequality group (abbreviated as the solution of the inequality group). The solution set of inequality group can be expressed on the number axis. The process of finding the solution set of inequality group is called solving inequality group.
6. The general steps to solve the one-dimensional linear inequality group: ① Find the solution set of each inequality in this inequality group; ② Find the common part of the solution set of these inequalities by using the number axis, and get the solution set of this inequality group. If the solution sets of these inequalities have no common parts, the inequality group has no solution (at this time, the solution set of this inequality group is also called an empty set).
7. After finding the solution set of each inequality, determine the formula of the solution of the inequality group: take the big one, take the small one, take the middle one, and the big one is nowhere to be found.
Chapter 10 Data Collection, Arrangement and Description
Key points of knowledge
1, the general process of data processing: collecting data, collating data, describing data, analyzing and drawing conclusions.
2. In the process of data collection, there are usually two survey methods: comprehensive survey and sampling survey.
3. In addition to text narration, list and underline, you can also use bar charts, line charts, fan charts and histograms to describe data.
4. Sampling survey is referred to as spot check. It only extracts some objects for investigation, and infers the situation of all objects according to the survey data. All the objects to be investigated are called the population, each object constituting the population is called the individual, and some of the extracted individuals constitute the sample of the population, and the number of individuals in the sample is called the capacity of this sample.
5. Steps of drawing frequency histogram: ① Calculate the number difference (the difference between the value and the minimum value); ② Determine the interval and the number of groups; ③ Column frequency distribution table; ④ Draw frequency histogram.