1. Axisymmetric graph 1. Fold a graph along a straight line. If the parts on both sides of a straight line can completely overlap, then this figure is called an axisymmetric figure. This straight line is its axis of symmetry. At this time, we also say that this figure is symmetrical about this straight line (axis).
2. Fold the chart along a straight line. If it can completely coincide with another figure, the two figures are said to be symmetrical about this line. This straight line is called the axis of symmetry. The overlapping points after folding are the corresponding points, which are called symmetrical points.
3. The difference and connection between axisymmetric graphics and axisymmetric graphics.
4. The essence of axial symmetry.
① Two figures symmetrical about a straight line are conformal.
(2) If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.
③ The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points.
(4) If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.
Second, the vertical line of the line segment
1. A straight line passing through the midpoint of a line segment and perpendicular to this line segment is called the median line of this line segment, also called the median line.
2. The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.
3. The point where the distance between the two ends of a line segment is equal is on the middle vertical line of the line segment.
Third, use coordinates to express the outline of axial symmetry
1. In the plane rectangular coordinate system, the horizontal coordinates of points symmetrical about the X axis are equal, but the vertical coordinates are opposite. The horizontal coordinates of points symmetrical about the Y axis are opposite to each other, and the vertical coordinates are equal.
2. The perpendicular lines of the three sides of a triangle intersect at a point, and the distance from the point to the three vertices of the triangle is equal.
2. At the end of the period, the second edition of Beijing Normal University summarized the knowledge points of eighth grade mathematics.
Function and its related concepts 1, variables and constants
In a certain change process, the quantity that can take different values is called a variable, and the quantity whose value remains unchanged is called a constant.
Generally speaking, there are two variables, X and Y, in a certain change process. If for each value of X, Y has a definite value corresponding to it, then X is an independent variable and Y is a function of X. ..
2. Resolution function
The mathematical formula used to express functional relationship is called resolution function or functional relationship.
The whole set of values of independent variables that make a function meaningful is called the range of independent variables.
3. Three representations of functions and their advantages and disadvantages.
(1) analysis method
The functional relationship between two variables can sometimes be expressed by an equation containing these two variables and the symbols of digital operations. This representation is called analytical method.
(2) List method
A series of values of the independent variable x and the corresponding values of the function y are listed in a table to represent the functional relationship. This representation is called list method.
(3) Image method
The method of expressing functional relations with images is called image method.
4. General steps of drawing images with resolution function.
(1) List: List gives some corresponding values of independent variables and functions.
(2) Point tracking: Take each pair of corresponding values in the table as coordinates, and track the corresponding points on the coordinate plane.
3. At the end of the period, Beijing Normal University Third Edition mathematics eighth grade knowledge points summary
(1) Using the formula method, we know that algebraic multiplication and factorization are inverse deformations. If the multiplication formula is reversed, the polynomial is decomposed into factors. So there are:
a2—b2=(a+b)(a—b)
a2+2ab+b2=(a+b)2
a2—2ab+b2=(a—b)2
If the multiplication formula is reversed, it can be used to factorize some polynomials. This factorization method is called formula method.
(2) Variance formula
formula for the difference of square
Equation (1): A2-B2 = (A+B) (A-B)
(2) Language: the square difference of two numbers is equal to the product of the sum of these two numbers and the difference of these two numbers. This formula is the square difference formula.
(3) Factorization
1, factorization, if there is a common factor, first extract the common factor, and then further decompose.
2. Factorization must be carried out until each polynomial factor can no longer be decomposed.
(4) Complete square formula
(1) Reversing the multiplication formula (a+b)2=a2+2ab+b2 and (a-b) 2 = a2-2ab+b2, we can get:
a2+2ab+b2=(a+b)2
a2—2ab+b2=(a—b)2
That is to say, the sum of squares of two numbers, plus (or minus) twice the product of these two numbers, is equal to the square of the sum (or difference) of these two numbers.
Equations a2+2ab+b2 and A2-2ab+B2 are called completely flat modes.
The above two formulas are called complete square formulas.
(2) the form and characteristics of completely flat mode
① Number of projects: three projects.
② Two terms are the sum of squares of two numbers, and the signs of these two terms are the same.
A term is twice the product of these two numbers.
(3) When there is a common factor in the polynomial, the common factor should be put forward first, and then decomposed by the formula.
(4) A and B in the complete square formula can represent monomials or polynomials. Here as long as the polynomial as a whole.
(5) Factorization must be decomposed until every polynomial factor can no longer be decomposed.
4. At the end of the period, the fourth edition of Beijing Normal University summarized the knowledge points of eighth grade mathematics.
The linear function (1) is a proportional function: it generally has the shape of y=kx (k is a constant, k >；; 0) is called the proportional function, where k is called the proportional coefficient;
(2) Image features of proportional function: some straight lines passing through the origin;
(3) Image attributes:
(1) when k >; 0, the image of function y=kx rises from left to right through the first and third quadrants, that is, y also increases with the increase of x; ② when k
(4) To find the analytic expression of the proportional function: only one non-origin point is known;
(5) Drawing the image of the proportional function: passing through the origin and point (1, k); (or another non-origin)
(6) linear function: the general shape is y=kx+b(k and b are constants, k? 0), called a linear function;
(7) Proportional function is a special linear function; (because when b=0, y=kx+b is y=kx).
(8) Linear function image features: partial straight lines;
(9) Nature:
(1) y=kx and y=kx+b have the same inclination angle, and y=kx+b can be regarded as y=kx translation |b| unit length; (When b>0, translate upward; When b<0, translate downward)
2 when k >; 0, the straight line y=kx+b rises from left to right, that is, y increases with the increase of x;
③ When k < 0, the straight line y=kx+b decreases from left to right, that is, Y decreases with the increase of X;
4 when b>0, the intersection of the straight line y=kx+b and the positive semi-axis of the y axis is (0, b);
⑤ When b
(10) Find the analytical formula of linear function: that is, find the values of k and b;
(1 1) Draw the image of the function once: two points are known;
On Equations (Groups) and Inequalities from the Perspective of Functions
(1) Solving a linear equation with one variable can be transformed into: when the value of a linear function is 0, find the value of the corresponding independent variable; From the image, this is equivalent to knowing the straight line y=kx+b and determining the value of the abscissa of its intersection with the x axis;
(2) Solving the linear inequality of one variable can be regarded as: when the linear function value is greater than (less than) 0, find the value range corresponding to the independent variable;
(3) Each bivariate linear equation corresponds to a univariate linear function, so it also corresponds to a straight line;
5. At the end of the period, the eighth grade mathematics knowledge points of Beijing Normal University Edition were summarized.
Quadrilateral parallelogram definition: A quadrilateral with two groups of opposite sides parallel to each other 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 sets 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.
Definition of rectangle: a parallelogram with a right angle.
The nature of the rectangle: all four corners of the rectangle are right angles; The diagonals of a rectangle are equally divided. AC=BD
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.
Definition of diamond: parallelogram with equal adjacent sides.
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.