Chapter 11 Linear Functions
We call the amount of numerical change a variable.
The values of some quantities are always constant, which we call constants.
In the process of a change, if there are two variables X and Y, and for each definite value of X, Y has a unique definite value corresponding to it, then we say that X is an independent variable and Y is a function of X.
If y=b when x=a, then b is called the function value when the value of the independent variable is a.
A function in the form of y=kx(k is a constant and k≠0) is called a proportional function, where k is called a proportional coefficient.
A function in the form of y = kx+b (where k and b are constants and k≠0) is called a linear function. Proportional function is a special linear function.
When k > 0, y increases with the increase of x; When k < 0, y decreases with the increase of x.
Each binary linear equation group corresponds to two linear functions, so it also corresponds to two straight lines. From the perspective of "shape", solving equations is equivalent to determining the coordinates of the intersection of two straight lines.
Chapter 12 Data Description
We call the number of data in different groups the frequency of groups, and the ratio of frequency to total data is frequency.
Common statistical charts: bar chart (composite bar chart), pie chart, line chart and histogram.
Bar chart: describes the amount of data in each group.
Composite bar chart: you can not only see the data, but also compare it.
Fan chart: describes the percentage of each group of frequencies in the total.
Line chart: describes the changing trend of data.
Histogram: it can display the frequency distribution of each group; It is easy to show the frequency difference between groups.
In the frequency distribution table, we call the number of groups the number of groups, and the difference between the two endpoints of each group is called the group distance.
Find the average of the two endpoints in each group, and these averages are called the median in the group.
Chapter 13 congruent triangles
Two graphs that can completely overlap are called congruent graphs.
Two triangles that can completely coincide are called congruent triangles.
The essence of congruent triangles: congruent triangles's corresponding sides are equal; Congruent triangles's corresponding angles are equal.
The condition of congruent triangles congruence: three sides correspond to two triangles congruence. (SSS)
The angle between two sides and them is equivalent to the combination of two triangles. (SAS)
Two angles and their sides correspond to the congruence of two triangles. (ASA)
The opposite sides of two angles and one of them correspond to the congruence of two triangles. (AAS)
The nature of the bisector: the distance from the point on the bisector to both sides of the angle is equal.
The points with equal distance to both sides of the angle are on the bisector of the angle.
Chapter 14 Axisymmetric
A straight line that passes through the midpoint of a line segment and is perpendicular to the line segment is called the midline of the line segment.
The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points.
The point on the vertical line in a line segment is equal to the distance between the two endpoints of the line segment.
The axisymmetric figure obtained from plane figure is called axisymmetric transformation.
The nature of isosceles triangle;
The two base angles of an isosceles triangle are equal. (equilateral and angular)
The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other. (Three lines in one) (Attached: top angle +2 bottom angle = 180)
If the two angles of a triangle are equal, then the opposite sides of the two angles are equal. (Equiangular and Equilateral)
An isosceles triangle with an angle of 60 is an equilateral triangle.
In a right triangle, if an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse.
Chapter 15 Algebraic Expressions
The product of numbers or letters is called a monomial. A single number or letter is also a monomial.
The numerical factor in a single item is called the coefficient of the item.
In a monomial, the sum of the exponents of all the letters is called the degree of the monomial.
The sum of several monomials is called polynomial. Each monomial is called a polynomial term ($ TERM) and those without letters are called constant terms.
The degree of the term with the highest degree in a polynomial is the degree of this polynomial.
Monomial and polynomial are collectively called algebraic expressions.
Items with the same letter and the same letter index are called similar items.
Merging similar terms in polynomials into one term, that is, adding their coefficients as new coefficients, while the letter part remains unchanged, is called merging similar terms.
The addition and subtraction of several algebraic expressions is usually to enclose each algebraic expression in brackets and then connect them with addition and subtraction signs; Then remove the brackets and merge similar projects.
Same radix power multiplication, constant radix, exponential addition.
Power, constant radix, exponential multiplication
The power of the product is equal to multiplying each factor of the product by the power, and then multiplying it by the power.
Multiply a monomial by a monomial, and multiply them by their coefficients and the same letters respectively. For letters contained only in the monomial, they are used as a factor of the product together with its index.
Multiplying a polynomial by a monomial is to multiply each term of a polynomial by a monomial, and then add the products.
Multiply polynomials by multiplying each term of one polynomial by each term of another polynomial, and then add the products.
(x+p)(x+q)=x^2+(p+q)x+pq
Square difference formula: (a+b) (a-b) = a 2-b 2.
Complete square formula: (a+b) 2 = A2+2ab+B2 (a-b) 2 = A2-2ab+B2.
(a+b+c)^2=a^2+2a(b+c)+(b+c)^2
Same base powers divides, the base remains the same, and the exponent is subtracted.
Any number that is not equal to the power of 0 is equal to 1.
Chapter 16 Scores
If A and B represent two algebraic expressions and B contains letters, then the formula A/B is called a fraction.
The numerator of a fraction is multiplied by the denominator or divided by an algebraic expression that is not equal to 0, and the value of the fraction remains the same.
Law of fractional multiplication: fractional multiplication, the product of molecules is the numerator of the product, and the product of denominator is the denominator.
Law of fractional division: a fraction is divided by a fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor.
Fractional power should be numerator and denominator respectively.
A-n = 1/A n (A ≠ 0) That is to say, A-n (A ≠ 0) is the reciprocal of A n.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
Chapter 17 Inverse proportional function
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The image of inverse proportional function belongs to hyperbola.
When k > 0, the two branches of hyperbola are located in the first quadrant and the third quadrant respectively, and the y value of each quadrant decreases with the increase of x value;
When k < 0, the two branches of hyperbola are located in the second quadrant and the fourth quadrant respectively, and the y value of each quadrant increases with the increase of x value.
Chapter 18 Pythagorean Theorem
Pythagorean Theorem: If the lengths of two right-angled sides of a right-angled triangle are A and B respectively and the length of the hypotenuse is C, then A 2+B 2 = C 2.
Inverse Theorem of Pythagorean Theorem: If the lengths of three sides of triangle A, B and C satisfy A 2+B 2 = C 2, then the triangle is a right triangle.
A proposition that is proved to be correct is called a theorem.
We call two propositions with opposite topics and conclusions reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition. (Example: Pythagorean Theorem and Pythagorean Theorem Inverse Theorem)
Chapter 19 Quadrilateral
A quadrilateral with two sets of parallel opposite sides is called a parallelogram.
The nature of parallelogram: the opposite sides of parallelogram are equal; Diagonal angles of parallelogram are equal. Diagonal bisection of parallelogram.
Determination of parallelogram;
1. Two groups of quadrangles with equal opposite sides are parallelograms;
2. The quadrilateral whose diagonal lines bisect each other is a parallelogram;
3. Two groups of quadrangles with equal diagonal are parallelograms;
4. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
The midline of the triangle is parallel to the third side of the triangle and equal to half of the third side.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
The nature of the rectangle: all four corners of the rectangle are right angles; The diagonals of a rectangle are equally divided.
Rectangular judgment theorem;
1. A parallelogram with a right angle is called a rectangle.
2. Parallelograms with equal diagonals are rectangles.
A quadrilateral with three right angles is a rectangle.
The nature of the diamond: all four sides of the diamond are equal; The two diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals.
Judgement theorem of diamonds;
1. A set of parallelograms with equal adjacent sides is a diamond.
2. Parallelograms with diagonal lines perpendicular to each other are diamonds.
A quadrilateral with four equilateral sides is a diamond.
S diamond = 1/2×ab(a and B are two diagonal lines).
The essence of a square: all four sides are equal and all four corners are right angles.
A square is both a rectangle and a diamond.
Square judgment theorem;
1. A rectangle with equal adjacent sides is a square.
Diamonds with right angles are squares.
A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid.
The nature of isosceles trapezoid: the two angles on the same base of isosceles trapezoid are equal; The two diagonals of an isosceles trapezoid are equal.
Judgment theorem of isosceles trapezoid: two trapezoid with equal angles on the same base are isosceles trapezoid.
The center of gravity of the line segment is the midpoint of the line segment.
The center of gravity of a parallelogram is the intersection of its two diagonals.
The point of doubt when three center lines of a triangle meet is the center of gravity of the triangle.
A rectangle with an aspect ratio of (root number 5- 1)/2 (about 0.6 18) is called a golden rectangle.
Chapter 20 Data Analysis
Arrange a set of data in order from small to large (or from large to small). If the number of data is odd, the middle number is median); This set of data. If the number of data is even, the average of the middle two data is the median of this set of data.
The data with the highest frequency in a set of data is the pattern of this set of data.
The difference between the largest data and the smallest data in a set of data is called the range of this set of data.
The greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation and the more stable it is.
Data collection and sorting steps: 1. Collect data. Arrange data 3. Description data 4. Analyze data 5. Write an investigation report.
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