Chapter XII Overview of Plane Rectangular Coordinate System

Coordinate characteristics of points on the plane

1. Coordinate characteristics of the midpoint P(a, b) in each quadrant:

The first quadrant: a>0, b>0; The second quadrant: a

2. Coordinate characteristics of point P(a, b) on the coordinate axis:

On the x axis: a is an arbitrary real number, and b = 0；; On the y axis: b is an arbitrary real number, and a = 0；; Origin of coordinates: a=0, b=0.

(Note: If P(a, b) is on the coordinate axis, then AB = 0；; Conversely, if ab=0, then P(a, b) is on the coordinate axis. )

3. The coordinate characteristics of point P(a, b) on the bisector of two axes: 1. Three quadrants: a = b；; 2. Four quadrants: A =-B.

Coordinate characteristics of symmetrical points

The symmetry point of point P(a, b) about the X axis is (a,-b);

The symmetry point about the Y axis is (-a, b);

The symmetrical point about the origin is (-a, -b)

Distance from point to coordinate axis

The distance from the point P(x, y) to the X axis is ∣y∣, and the distance to the Y axis is ∣x∣.

Variation law of translation coordinates of points

(1) The straight lines of two points with the same abscissa are perpendicular to the X axis and parallel to the Y axis;

(2) The straight lines of two points with the same ordinate are perpendicular to the Y axis and parallel to the X axis.

In the coordinate plane, after the point P(x, y) is translated to the right (or left) by one unit, the corresponding point is (x+a, y) or (x-a, y); Point P(x, y) moves up (or down) by b units, and the corresponding point is (x, y+b) or (x, y-b).

(Note: translation from left to right, horizontal to vertical unchanged, translation to right, horizontal axis increasing, translation to left, horizontal axis decreasing; Translation up and down, vertical change and horizontal unchanged, upward translation, vertical coordinate increase, downward translation, vertical coordinate decrease. The symbol is right plus left minus, up plus down minus).

Chapter 13 Linear Functions

Determine the value range of function independent variables

1. Independent variables appear in the form of algebraic expressions, and the range of independent variables is real numbers;

2. The independent variable appears as a fraction, and the range of the independent variable is a number whose denominator is not 0;

3. Independent variables appear in the form of even roots, and the range of independent variables is to make the number of roots greater than or equal to 0 (that is, the number of roots? 0);

Independent variables appear in the form of odd roots, and the range of values is all real numbers.

4. The independent variable appears in the base of zero power or negative integer power, and the range of the independent variable is a number that makes the base non-zero.

Note: (1) When a resolution function contains several algebraic expressions, the range of independent variables is the common part of the range of independent variables in each algebraic expression;

(2) When the resolution function represents a function with practical significance, the range of independent variables should not only make the resolution function meaningful, but also conform to the practical significance.

The arrangement of knowledge points in the final exam of mathematics in the first volume of the eighth grade is 2 test sites 1: triangle.

The test sites in a triangle are divided into three categories: one is a general triangle, the other is an isosceles triangle and the other is an equilateral triangle.

The calculation of the correlation between the area and perimeter of the triangle and the proof of the congruence correlation of the triangle are often tested in general triangles. The area of a triangle is 1/2 times the base times the height, and the perimeter of the triangle is the sum of three sides. The proof methods of triangle congruence are SSS (three sides correspond to two equilateral triangles), SAS (two sides and their included angles correspond to two equilateral triangles), AAS (two angles and two equilateral triangles correspond to one angle) and ASA (two angles and two triangles correspond to their sandwiched sides correspond to two equilateral triangles).

Isosceles triangle: A triangle with two sides or two equal angles is an isosceles triangle. The bisectors of height, midline and angle on the bottom of isosceles triangle coincide, and this auxiliary line is often constructed for relevant proof in exams.

Equilateral triangle: A triangle with three equilateral sides is an equilateral triangle. All angles of an equilateral triangle are 60 degrees, and all sides are equal in length.

Test site 2: Polygon

The sum of the interior angles of the polygon is: 180(n-2), where n is the variable of the polygon. The range of degrees is often given, and the side length is calculated. The common method is to assume that the number of sides of a polygon is n and the column inequality. Finally, the range of edge number n is obtained, which is an integer. If the sum of the internal angles of a polygon is more than 850 degrees and less than 1000 degrees, find the number of sides of the polygon.

Column inequality: 850

Diagonal number of polygons: n(n-3)/2

Test site 3: Axisymmetric

Axisymmetric images are often related to congruence, mainly the topic of combining numbers and shapes. It will be mentioned in the simulation questions later, you only need to know that the figures that can completely overlap a line are axisymmetric, such as isosceles triangles and squares.

Test site 4: algebraic expression

The test center of the compulsory examination of algebraic expression is algebra-related evaluation, which usually trains students. As long as the exam is solved according to the four operations carefully, it can be simplified first and then solved by numerical value.

Test site 5: Factorization

Factorization is one of the required contents. Let's summarize the steps of factorization: first, look at whether there is a common factor in the formula. If there is a common factor, you must extract it. Then, see if we can use the square difference formula or the complete square formula. If not, you can consider using cross multiplication to decompose. See the factorization problem-solving skills mentioned in the previous course for specific decomposition skills.

Test site 6: scores

The score test site is relatively simple. First, the calculation of fractions is the same as that of algebraic expressions. Secondly, the fractional equation is used to solve the application problem. After solving the application problem, it must be substituted into the original fractional equation for verification. To determine whether the denominator is 0, that is, the solution of the equation is over. Need to add: for the solution of the original fractional equation, verify that x is equal to a certain value. Matters needing attention in solving problems will be explained in detail in the final exam.