I. Real numbers
1. 1 rational number
1. 1. 1 definition of rational number: a general term for integers and fractions.
Classification of rational numbers 1. 1.2:
(1) is divided into integers and fractions. Integers are divided into positive integers, zero and negative integers; Scores are divided into positive scores and negative scores.
(2) Divided into positive rational numbers, zero-sum negative rational numbers. Positive rational numbers are divided into positive integers and positive fractions; Negative rational numbers are divided into negative integers and negative fractions.
1. 1.3 axis
1. 1.3. 1 Definition of number axis: The straight line defining the origin, positive direction and unit length is called number axis.
Three elements of number axis: ① origin ② positive direction ③ unit length.
Every rational number can be represented by a point on the number axis.
The reciprocal of 1. 1.4
Definition of 1. 1.4. 1: Only two numbers with different signs are regarded as reciprocal (note: the reciprocal of 0 is 0.
The meaning of 1. 1.4.2 The two numbers represented by two points with the same distance from the origin are opposite.
1. 1.4.3 Discrimination of Inverse Numbers
(1) If, then, and are opposite numbers.
(2) If the absolute values of two numbers are equal and the signs are opposite, then the two numbers are opposite.
The reciprocal of 1. 1.5
Definition of 1. 1.5. 1 reciprocal: If the product of two numbers is equal to 1, then these two numbers are reciprocal. (If ab= 1, A and B are reciprocal) Note: Zero has no reciprocal.
1. 1.6 absolute value
1. 1.6. 1 definition of absolute value: on the number axis, it indicates the distance from the origin (the absolute value of a is expressed as ∣a∣).
1. 1.6.2 the nature of absolute value: ∣a∣≥0.
Comparison of rational numbers of 1. 1.7
1. 1.7. 1 Positive numbers are greater than 0 and negative numbers are less than 0.
1. 1.7.2 A positive number is greater than a negative number.
1. 1.7.3 are two positive numbers, with the larger absolute value and the smaller absolute value; Two negative numbers, the greater the absolute value, the smaller the absolute value.
1. 1.7.4 subtraction: two rational numbers are subtracted. If it is greater than 0, the minuend is large; If it is equal to 0, the two numbers are equal; If it is less than 0, the decline is large.
1. 1.7.5 as commercial law: divide two rational numbers (divisor or denominator is not 0). If it is greater than 1, the dividend is large; If it is equal to 1, the two numbers are equal; If it is less than 1, the divisor is large.
1. 1.8 rational number addition
1. 1.8. 1 Algorithm: ① Add two numbers with the same symbol, take the same symbol and add the absolute values; (2) Add two numbers with different signs and different absolute values, take the addend symbol with larger absolute value, and subtract the number with smaller absolute value from the number with larger absolute value (the sum of two numbers with opposite numbers equals 0); ③ Any rational number plus 0 is still equal to this number.
1. 1.8.2 additive commutative law is still applicable in rational number addition, that is, a+b = b+a.
1. 1.8.3 rational number addition still applies the additive combination law, that is, a+(b+c) = (a+b)+c.
1. 1.9 rational number subtraction
1. 1.9. 1 algorithm: subtracting a number is equal to adding the reciprocal of this number.
1. 1.9.2 rational number subtraction-transformation → rational number addition
1. 1. 10 multiplication of rational numbers
1.1.1.1Algorithm: ① Multiply two numbers, the same sign is positive, the different sign is negative, and the absolute value of multiplication (formula: positive is positive, negative is positive, positive and negative is negative, negative is negative) ② Multiply any rational number by 0 is still equal to 0③ Multiply more. When there are even negative factors, the product is positive.
1. 1. 10.2 multiplication method of substitution is still applicable in rational number multiplication, that is,
1. 1. 10.3 The multiplicative associative law is still applicable in rational number multiplication, that is,
1. 1. 10.4 multiplication and distribution laws are still applicable in rational number multiplication, that is,
Division of rational numbers1.1.11
1.1.11algorithm: dividing by a number is equal to multiplying the reciprocal of this number (the divisor cannot be 0, otherwise it is meaningless).
1.1.11.2 rational number division-transformation → rational number multiplication
1. 1. 12 power of rational number
1.1.12.1the meaning of rational number power: the operation of finding the product of a factor is called power.
1. 1. 12.2 Representation of the power of rational numbers: multiplication by the same factor is expressed as, where it is called base and exponent, and the result of power is called power, which is read as "power of" or "power of" (when =2, it is read as square and abbreviated as square).
1. 1. 12.3 algorithm: ① any power of a positive number is positive; ② The odd power of a negative number is negative; And the even power of negative numbers is positive; ③ Any power of 30 is equal to 0 (except 0 power); (4) Any number whose power is 0 is equal to1(except the power of 0).
The mixed operation of 1. 1. 13 rational numbers
1.1.13.1Operation sequence: ① Calculate the power first (i.e. three-level operation), then multiply and divide (i.e. two-level operation), and finally add and subtract (i.e. one-level operation); (2) if it is the same level operation, from left to right; (3) if there are brackets, first.
1. 1. 14 scientific notation
1.1.14.1definition of scientific notation: the rational number greater than 10 (where 1≤ ≤ 10) is called scientific notation.
Approximation of 1. 1. 15
1.1.15.1Definition of divisor: Numbers close to positive numbers but not equal to positive numbers are called divisors or approximations of this positive number.
1. 1. 15.2 Approximation method: ① Rounding method ② Ending method ③ Tailing method.
1. 1. 15.3 Definition of significant figures: to which place a divisor is accurate, all figures from the first place from the left to this place (including 0) are called the significant figures of this divisor.
1.2 real number
1.2. 1 square root
1.2. 1. 1 definition of square root: If the square of a number is equal to, this number is called a square root (or quadratic square root), which means we say it is a square root.
The representation of the square root of 1.2. 1.2: If (> 0), the square root of is marked as "positive and negative root sign", where it is marked as "quadratic root sign", and 2 is called the root index, which is called the root sign.
The nature of the square root of 1.2. 1.3: a positive number has two square roots, and the two square roots are in opposite directions; There is only one square root of 0, which is 0; Negative numbers have no square root.
1.2. 1.4 definition of square root: the operation of finding the square root of a number is called square root (square root and square are reciprocal operations).
1.2.2 arithmetic square root
1.2.2. 1 definition of arithmetic square root: a positive number has two square roots, and the positive square root of one of them is called the arithmetic square root of, and it is recorded as the "root sign".
1.2.2.2 The nature of the arithmetic square root: ① It has double non-negativity, namely: ≥0, ≥0② =a( ≥0)③ =∣ ∣, when ≥0, = ∣ =; When ≤0, = ∣ =-
1.2.3 cube root
Definition of the cube root of 1.2.3. 1: If the cube of a number is equal to, then this number is called the cube root of (or the cube root of).
1.2.3.2 Representation of cube root: If, then X is called the cube root of A, the root number, and 3 is called the root index.
Properties of the cube root of 1.2.3.3: ① Positive numbers have a cube root, which is still positive, negative numbers have a cube root, which is still negative, and the cube root of 0 is still 0. ②
1.2.3.4 Definition of publisher: The operation of finding the cube root of a number is called publisher (it is the inverse operation with the cube).
1.2.4 irrational number
1.2.4. 1 Definition of irrational number: Infinitely circulating decimals are called irrational numbers.
1.2.4.2 Matters needing attention in judging irrational numbers: ① Numbers with root signs are not necessarily irrational numbers, and if they are rational numbers, they are not irrational numbers; 2 irrational numbers are not necessarily inexhaustible numbers, such as pi.
1.2.5 real number
1.2.5. 1 Definition of real numbers: a general term for rational numbers and irrational numbers.
1.2.5.2 Properties of real numbers: ① Real numbers correspond to points on the number axis one by one; ② The inverse of real number A is -a, and the reciprocal of real number is (≠0)③∣ ∣≥0, ∣∣ = ∣-∣ ④ Yes.
1.2.5.3 Comparison of two real numbers: ① Positive numbers are greater than 0, negative numbers are less than 0, positive numbers are greater than all negative numbers, and the absolute values of the two negative numbers are larger but smaller. (2) the two numbers on the axis, the number on the right is always greater than the number on the left. Commercial law: division of two real numbers (divisor or denominator is not 0). If it is greater than 1, the dividend is large; If it is equal to 1, the two numbers are equal; If it is less than 1, the divisor is large. ④ Difference method: two rational numbers are subtracted. If it is greater than 0, the minuend is large; If it is equal to 0, the two numbers are equal; If it is less than 0, the decline is large.
1.2.6 quadratic radical
1.2.6. 1 definition of quadratic radical: the formula (≥0) is called quadratic radical.
1.2.6.2 The operational property of the quadratic radical is ① (≥0, ≥0)② (≥0, > 0).
1.2.6.3 simplest quadratic root: a quadratic root that meets the following two conditions is called the simplest quadratic root: ① the factor of the square root is an integer and the factor is an algebraic expression; ② The square root does not contain any factor or factor that can be completely opened.
1.2.6.4 The denominator has a physical and chemical definition: in a formula with a radical sign in the denominator, the process of crossing out the radical sign in the denominator is called the denominator has a physical and chemical definition.
1.2.6.5 mixed operation of quadratic roots: first do power operation, then do multiplication and division, and finally do addition and subtraction; If there are brackets, operate in the order of small, medium and curly brackets.
Second, algebraic expressions.
2. 1 algebraic expression
2. 1. 1 Definition of algebraic expression: An expression formed by connecting numbers or letters with operational symbols is called an algebraic expression.
2. 1.2 Classification of algebraic expressions: Algebra can be divided into rational expressions and irrational expressions, rational expressions can be divided into algebraic expressions and fractional expressions, and algebraic expressions can be divided into monomials and polynomials.
2. 1.3 definition of column algebra: column algebra is a formula containing numbers, letters and operation symbols to represent the words related to quantity in the problem.
2. 1.4 generation value: replace the letters in the algebraic expression with numerical values, and the calculated result is called algebraic value.
2.2 algebraic expressions
2.2. The concept of1algebraic expression
2.2. 1. 1 monomial: An algebraic expression that only contains the product of numbers and letters is called a monomial (a single number or letter is also a monomial). Among them, the number factor is called the coefficient of a single item, and the sum of the indexes of all letters in a single item is called the number of times of a single item.
2.2. 1.2 Polynomial: The sum of several monomials is called a polynomial. Each monomial in a polynomial is called a polynomial term, and the term without letters is called a constant term.
2.2. 1.3 degree of polynomial: the degree of the highest coefficient term in polynomial is called the degree of polynomial.
2.2. 1.4 descending (ascending) power arrangement: arrange a polynomial from big (small) to small (big) according to the exponent of a letter.
2.2. 1.5 Definition of algebraic expressions: a general term for monomials and polynomials.
2.2. 1.6 Definition of similar items: items with the same letters and the same time are called similar items.
2.2. 1.7 merging similar terms: the process of merging similar terms in a polynomial is called merging similar terms.
2.2. 1.8 Similar item merging rule: Add the coefficients of similar items, and the obtained results will be taken as coefficients, with the index of letters unchanged.
Operation of algebraic expressions
2.2.2. 1 Algebraic expression addition and subtraction calculation rules: remove brackets first, and then merge similar items.
2.2.2.2 and algebraic expression multiplication and division algorithm: ① same base powers's multiplication rule: same base powers's multiplication, constant base number and exponential addition, that is, (m, n is a positive integer) ② same base powers's division rule: same base powers's division, constant base number and exponential subtraction, that is, (≠0, a positive integer, >) ③ Power: Power, constant base number and exponential multiplication.
2.2.2.3's law of multiplying a monomial by a monomial: When a monomial is multiplied, its coefficient is multiplied by the same letter. For a letter contained only in a monomial, it is used as a factor of the product together with its index. (When calculating the coefficient, determine the symbol first, and then calculate the absolute value. When the coefficient is-1, just write "-"in front of the result. )
2.2.2.4 Polynomial Multiplies Single Term: Multiply each term of a polynomial by a single term, and then add the products.
2.2.2.5's algorithm of dividing a single item by a single item: Generally speaking, a single item is divided by the coefficient and the same base power as a factor of the quotient, and letters only included in the division formula are taken as a factor of the quotient together with their indices.
2.2.2.6 polynomial divided by monomial: When a general polynomial is divided by monomial, the terms of the polynomial are divided by monomial first, and then the obtained quotients are added.
2.2.2.7 Polynomial Multiplication Polynomial: First multiply each term of one polynomial by each term of another polynomial, and then add the products.
2.2.2.8's square difference formula: the product of the sum of two numbers and the difference between the two numbers is equal to the square difference between the two numbers, that is, (Note: in the formula, the content is relatively extensive, and it can represent numbers, monomials or polynomials).
2.2.2.9's 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: (Note: the contents expressed by A and B in the formula are extensive and can represent numbers or monomials or polynomials).
2.2.2. 10 cubic sum and cubic difference formula: the sum (or difference) of two numbers multiplied by the difference (or sum) between the sum of squares and the product is equal to the cubic sum (or cubic difference) of these two numbers, namely
2.2.2. 1 1 Other multiplication formulas:
①
②
factoring
2.2.3. 1 definition of factorization: the product of a polynomial factorization into several monomials is called polynomial factorization.
Matters needing attention in 2.2.3.2 factorization: factorization should be decomposed until it can no longer be decomposed; Factorization and algebraic expression multiplication are reciprocal operations.
Definition of 2.2.3.3 common factor: The same factor contained in each term of a polynomial is called the common factor of each term of the polynomial.
2.2.3.4 factorization method: ① Extraction of common factor method: If every term of a polynomial has a common factor, you can put this common factor outside brackets and write the polynomial in the form of factor product. This factorization is called extraction of common factors. Namely: ② Using formula method: using multiplication formula in reverse, some polynomials can be decomposed into factors, which is called using formula method (commonly used: summation); ③ Grouping decomposition method: using grouping to decompose factors is called grouping decomposition method; ④ Cross multiplication: the quadratic trinomial is decomposed into.
2.3 score
2.3. 1 fraction concept
2.3. 1. 1 Fraction definition: A and B represent two algebraic expressions. If B contains letters, the formula is called a fraction. Where a is called the numerator of the fraction and b is called the denominator of the fraction.
2.3. 1.2 Definition of rational expressions: algebraic expressions and fractions.
2.3. 1.3 Definition of complex fraction: A fraction contains a fraction in the numerator or denominator, and such a fraction is called a complex fraction.
2.3. 1.4 simplest fraction's definition: When the numerator and denominator of a fraction have no common factor, it is called simplest fraction.
2.3. 1.5 Definition of divisor: According to the basic properties of a fraction, the process of rounding the common factor of the numerator and denominator of a fraction is called divisor.
2.3. 1.6 definition of general score: the process of converting a score with different denominators into a score with the same denominator equal to the original score is called general score.
2.3.2 the basic nature of the score
2.3.2. 1 Basic properties of the fraction: both the numerator and denominator of the fraction are multiplied or divided by an algebraic expression that is not 0 at the same time, and the value of the fraction remains unchanged, that is,
2.3.2.2's sign law of fraction: If you change any two of the numerator, denominator and sign of the fraction itself, the value of the fraction will remain unchanged, that is
Fraction operation
The calculation rules of 2.3.2.3 fraction addition and subtraction: add and subtract with denominator fraction, denominator unchanged, numerator addition and subtraction, that is; The addition and subtraction of fractions with different denominators are divided into fractions with the same denominator, and then calculated according to the addition and subtraction law of fractions with the same denominator, that is.
The multiplication and division algorithm of 2.3.2.4 fraction: the fraction is multiplied by the fraction, the product of the numerator is the numerator of the product, and the product of the denominator is the denominator of the product, that is; Divide the fraction by the fraction, invert the numerator and denominator of the division, and then calculate according to the multiplication rule of the fraction.
The mixed operation of 2.3.2.5 fraction: ① first calculate the power (i.e. three-level operation), then multiply and divide (i.e. two-level operation), and finally add and subtract (i.e. one-level operation); (2) if it is the same level operation, from left to right; (3) if there are brackets, calculate the brackets first, then the brackets, and finally the braces.
III. Equations and Equations
3. 1 equations and equations
3. The basic concept of1.1
3. 1. 1. 1 Definition of equation: An equation with an equal sign is called an equation.
3. Properties of1.1.2 equation: ① Adding or subtracting a number or an algebraic expression on both sides of the equation at the same time, the result is still equation ② multiplying or dividing a non-zero number on both sides of the equation at the same time, and the result is still equation.
3. Definition of1.1.3 equation: An equation with unknown numbers is called an equation.
3. Solution of the equation1.1.4: The value of the unknown that makes both sides of the equation equal is called the solution of the equation, and the solution of the equation with only one unknown is also called the root of the equation.
3. 1. 1.5 Definition of solving equations: The process of solving equations is called solving equations.
3. 1. 1.6 One-dimensional linear equation: An equation with an unknown number, degree 1 and coefficient not equal to 0 is called one-dimensional linear equation, and its standard form is ax+b=0, where x is unknown and has a unique solution, (a≠0).
3. 1. 1.7 Binary linear equation: An integral equation with two unknowns and all its terms are 1 is called a binary linear equation.
3. 1. 1.8 One-dimensional quadratic equation: contains only one unknown, and the highest degree of the unknown is 2. This equation is called unary quadratic equation, and its general form is ax+bx+c=0, where ax is called quadratic term, bx is called linear term and c is called constant term.
3. The solution of1.1.9 unary quadratic equation: ① direct open method ② collocation method ③ root method ④ factorization method.
3.1.1.11discriminant of the root of a quadratic equation: it is called the discriminant of a quadratic equation ax+bx+c=0.
3. 1. 1. 12 The relationship between roots and coefficients of a quadratic equation: Let it be two roots of the equation ax+bx+c=0 (a≠0), then the inverse proposition of the relationship between roots and coefficients also holds.
3. 1. 1. 13 Symbol of the root of a quadratic equation: Let the two roots of a quadratic root ax+bx+c=0(a≠0) be. When ≥0 and > 0, +> 0, they are the same sign; When ≥0, and > 0, +< 0, their negative signs are the same; When < 0, when the two symbols are different +> 0, the absolute value of the positive root is larger, and when +< 0, the absolute value of the negative root is larger.
3. 1. 1. 14 integral equation: both sides of the equation are algebraic expressions about unknown quantities, and such an equation is called an integral equation.
3. 1. 1. 15 Fractional equation: An equation with an unknown number in the denominator is called a fractional equation.
3. 1. 1. 16 Rooting: When the equation is deformed, sometimes roots that are not suitable for the original equation may be generated. This root is called the root of the equation (the root that makes the denominator of the equation zero), so it is necessary to check the root when solving the fractional equation. The method of checking the root is usually to substitute the root of the whole equation into the simplest common denominator, so that the root with the simplest common denominator of 0 is the added root.
3. 1. 1. 17 Binary linear equation: The number of terms containing two unknowns is 1, and this equation is called binary linear equation (note: for unknowns, the algebraic expression of the equation must be algebraic expression).
3. 1. 1. 18 solution of the binary linear equation: the values of a pair of unknowns satisfying the binary linear equation are called the solutions of the binary linear equation.
3. 1. 1. 19 solution of binary linear equation: give a fixed value to one of the unknowns, solve the equation about another unknown, and get the value of this unknown, so as to get a solution of binary linear equation.
3. 1. 1.20 Binary linear equations: When two binary linear equations are merged, they are called binary linear equations.
3. 1. 1.2 1 Solution of binary linear equations: The common * * * solution of binary linear equations is called the solution of binary linear equations.
3. 1. 1.22 solution of binary linear equations: the basic idea of solving binary linear equations is to eliminate an unknown and turn it into a linear equation group. The basic elimination methods are substitution and addition and subtraction. (1) substitution method: the basic idea of substitution method is that the same unknown in an equation should represent the same value, so an unknown in one equation can be replaced by an algebraic expression representing this unknown in another equation, thus reducing an unknown and transforming binary linear equations into linear equations. ② Addition and subtraction: The basic idea of addition and subtraction is to make the absolute value of an unknown coefficient in the two equations equal according to the basic properties of equation 2, and then add and subtract the two equations according to the basic properties of equation 1, so that an unknown can be eliminated and transformed into a linear equation. )
3. 1. 1.23 ternary linear equation: There are three unknowns, and the number of unknowns of each equation is 1. Such an equation is called a ternary linear equation.
3. 1. 1.24 Solution of ternary linear equations: The basic idea of solving ternary linear equations is to eliminate an unknown, convert it into binary linear equations, and then solve it according to the solution of binary linear equations.
3.2 equations (equations) for solving application problems
3.2. 1 Basic concepts
3.2. 1.65438+
3.2. 1.2 Methods of setting unknowns: ① Setting elements directly; (2) indirect setting; ③ Set auxiliary unknowns.
Common application problems
3.2.2. 1 Travel problems: Travel problems can be divided into four types: encounter problems, chasing problems, ring problems and water (wind) flow problems. Basic relationship: distance = speed × time ().
3.2.2.2 Engineering Problems: Basic Relationship: Workload = Working Time × Working Efficiency.
3.2.2.3 number problem: (Understand the concepts of several related nouns such as continuous natural number, continuous integer, continuous odd number and continuous even number, and know several representations of multiple digits).
3.2.2.4's growth rate: basic relationship: ① original output+increased output = actual output ② growth rate = growth number/basic number ③ actual output = original output (1+ growth rate).
3.2.2.5's Profit Problem: Basic Relationship: Profit = Selling Price-Buying Price.
Interest rate in 3.2.2.6: (Understand the concepts of principal, interest, sum of principal and interest, number of periods and interest rate) The basic relationship is: sum of principal and interest = principal+interest, and interest = principal × interest rate× number of periods.
3.2.2.7 Geometry: Common formulas: area and perimeter formulas of rectangle, square, triangle, trapezoid and circle.
3.2.2.8 concentration problem: basic relationship: concentration = solute mass/solution mass × 100%.
Other problems in 3.2.2.9: proportional distribution, chicken and rabbit in the same cage, function application …
Four. Unequal and unequal groups
4. 1 inequality
4. Basic concept of1.1
4. 1. 1. 1 inequality: A formula that represents inequality with an inequality symbol is called inequality.
4. 1. 1.2 Inequalities: Commonly used inequalities are: ① < ② > ③ ≠ ④≤⑤≥
4. 1. 1.3 Inequality: ① Add (or subtract) an algebraic expression on both sides of the inequality, and the inequality direction remains unchanged, that is, if >, then both sides of the inequality are multiplied (or divided) by a positive number at the same time, and the inequality direction remains unchanged; ③ Both sides of inequality are multiplied (or divided) by a negative number at the same time, and the sign of inequality changes.
4. 1. 1.4 solution of inequality: the value of the unknown quantity that makes the inequality hold is called the solution of inequality.
4. 1. 1.5 solution set of inequality: all solutions of an inequality constitute the solution set of this inequality.
4. 1. 1.6 Basic methods for solving inequalities: ① denominator ② brackets ③ moving terms ④ merging similar terms ⑤ coefficient is 1.
4.2 Unequal groups
4.2. 1 Basic concepts
4.2. 1. 1 One-dimensional linear inequality group: An inequality group consisting of several one-dimensional linear inequalities is called one-dimensional linear inequality group.
4.2. 1.2 Solution sets of linear inequalities: The common part of the solution sets of several linear inequalities is called the solution sets of linear inequalities.
4.2. 1.3 Solving inequality groups: The process of solving inequality solution sets is called solving inequalities.
Verb (abbreviation for verb) function
5. 1 plane cartesian coordinate system variables and functions
5. Basic concept of1.1
5. 1. 1. 1 Plane Cartesian coordinate system: In order to represent a point on a plane with a pair of real numbers, draw two mutually perpendicular number axes on the plane to form a plane Cartesian coordinate system. Among them, the horizontal axis is called axis or horizontal axis, and the right side is the positive direction; The vertical axis is called the axis or the longitudinal axis, and the direction is positive. Two number axes intersect at point o, which is called coordinate origin.
5. 1. 1.2 Quadrant: The horizontal and vertical axes divide the plane into four quadrants, in which the upper right corner is the first quadrant, the upper left corner is the second quadrant, the lower left corner is the third quadrant and the lower right corner is the fourth quadrant.
5. 1. 1.3 Point coordinates: write in the order of abscissa first and then ordinate, separated by commas.
5. 1. 1.4 Constants and variables: in a certain change process, the quantity whose value remains unchanged is called a constant, and the quantity that can take different values is called a variable.
5. 1. 1.5 function: In a certain change process, if there are two variables and each fixed value of X has a unique fixed value corresponding to it, it is called a function, where it is a dependent variable and an independent variable.
5. 1. 1.6 Range of independent variables: If the function is expressed by an analytic expression, then the range of independent variables is all the independent variables that make the analytic expression meaningful.
5. 1. 1.7 function value: for a certain value of the independent variable in the range, for example, =, the function has a unique and definite corresponding value, which is called the function value when =, or function value for short.
5. Representation of1.1.8 function: ① Analytical method: using mathematical formula to represent the corresponding relationship between two variables ② List: using list to represent the corresponding relationship between two variables ③ Image method: using image to represent the corresponding relationship between two variables in a plane rectangular coordinate system. (Usually the above three methods are used in combination)
5. 1. 1.9 Resolution function Steps to draw an image: list, trace points and connect lines.
5.2 Proportional function
5.2. 1 Basic concepts
5.2. 1. 1 Definition of proportional function: A function with a shape of (≠0) is called a proportional function.
5.2. 1.2 Image of the proportional function: The image of the proportional function is a straight line passing through the coordinate origin.
5.2. Properties of1.3 proportional function: ① When > 0, it increases with the increase of; ② When < 0, it decreases with the increase of.
5.3 linear function
5.3. 1 Basic concepts
5.3. 1. 1 Definition of linear function: A function with a shape of (,which is a constant) is called a linear function.
5.3. 1.2 Image of linear function: The image of linear function is a straight line parallel to the straight line (≠0).
5.3. Properties of1.3 linear function:
① when > 0, y increases with the increase of x.
When > 0, the image passes through one, two or three quadrants.
When < 0, the image passes through one, three and four quadrants.
When =0, it is a proportional function.
② when < 0, y decreases with the increase of x. ..
When > 0, the image passes through one, two and four quadrants.
When < 0, the image passes through 234 quadrants.
When =0, it is a proportional function.
5.4 Inverse proportional function
5.4. 1 Basic concepts
5.4. 1. 1 Definition of inverse proportional function: A function with a shape is called an inverse proportional function.
5.4. 1.2 image of inverse proportional function: the image of inverse proportional function is hyperbola.
5.4. Properties of1.3 inverse proportional function: ① When > 0, it decreases in the first and third quadrants with the increase of x; ② When it is less than 0, it increases with the increase of the second and fourth quadrants.
5.5 Quadratic function
5.5. 1 Basic concepts
5.5. Definition of1.1quadratic function: A function with the form (,is a constant, ≠0) is called a quadratic function.
5.5. 1.2 The image of quadratic function: it is a parabola whose symmetry axis is parallel to the axis.
5.5. Properties of1.3 quadratic function: ① The vertex coordinate of parabola (≠0) is a straight line; ② When > 0, the function has a minimum value; When < 0, the function has a maximum value when (3) when, the parabola (≠0) has two intersections with the X axis; When < 0, the parabola has no intersection with the X axis; When =0, the parabola intersects the x axis. ④ When > 0, the parabolic opening is upward, when A < 0, the parabolic opening is downward ⑤ When > 0, the intersection point is on the positive semi-axis of Y-axis, when C < 0, the intersection point is on the negative semi-axis of Y-axis, when =0, the intersection point is at the coordinate origin ⑦ When A and B have the same sign, it is < 0, and the parabolic symmetry axis is on the left side of Y-axis, when, and symbols are different, it is >.
5.5. 1.4 Three forms of quadratic resolution function: ① general formula; 2 intersection point; ③ Vertex type.
Six, intersection line and parallel line
6. 1 intersection line
6. Basic concept of1.1
6. 1. 1.65438+
6. 1. 1.2 Properties of antipodal angles: antipodal angles are equal.
6. 1. 1.3 Relationship between the definition and properties of vertex angle: The definition of vertex angle reveals the relationship between two angles, and the properties of vertex angle reveal the quantitative relationship of vertex angle. Only by defining two angles as antipodal angles can we draw the conclusion that the two angles are equal according to the nature of the angles.