The conclusion of translation and axial symmetry in plane rectangular coordinate system is introduced.
The symmetry point of point P(m, n) about the X axis is P 1(m, -n), that is, the abscissa is unchanged and the ordinate is opposite;
The symmetry point of point P(m, n) about the Y axis is P2(-m, n), that is, the ordinate is unchanged and the abscissa is opposite;
The symmetrical point of point P(m, n) about the origin is P3(-m, -n), that is, the abscissa and ordinate are opposite;
Translation of coordination point:
Symbolic characteristics of midpoint coordinates in each quadrant
Coordinate characteristics of the first quadrant: (+,+), the second quadrant: (-,+) and the third quadrant: (-,-).
Example 1. Points P(x, y) and x+y<；, x+y <; 0, then point P is at ()
A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant
Analysis: "xy>0" indicates that X and Y have the same sign, that is, they are both positive or negative. Pass "x+y"
Coordinate characteristics of points on coordinate axis
Coordinate characteristics of points on the X axis: the abscissa is arbitrary, and the ordinate is 0; The coordinate characteristics of points on the Y axis: the abscissa is 0 and the ordinate is an arbitrary number; Origin: Both horizontal and vertical coordinates are 0.
Example 2. Point P(a-2, b+3) is known.
If point P is on the X axis, then b = _ _ _ _ _ _ _ _ _
If point P is on the Y axis, then a = _ _ _ _ _ _ _ _ _ _ _ _；
If point P is the origin, then a = _ _ _ _ _ _ _ _ _ and b = _ _ _ _ _ _ _.
Analysis: Through the above conclusions, when point P is on the X axis, b+3=0, and the solution is: b =-3; When point P is on the Y axis, a-2=0, and the solution is: A = 2；; When point P is the origin, a-2=0 and b+3=0.
Characteristics of points on the angular bisector
The horizontal and vertical coordinates of the points on the bisector of the first quadrant and the third quadrant are the same;
The horizontal and vertical coordinates of the points on the bisector of the second and fourth quadrants are opposite.
If the point P(m, n) is on the bisector of the first and third quadrants, then m=n, that is, the abscissa and the ordinate are equal;
If the point P(m, n) is on the bisector of the second quadrant and the fourth quadrant, then m=-n, that is, the horizontal axis and the vertical axis are in opposite directions.
Example 3. Given that point A (5-2m, 3+m) is on the bisector of the third quadrant, the value of m is _ _ _ _ _ _ _.
Analysis: It is known that point A is on the bisector of the third quadrant angle, then the abscissa and ordinate are equal, that is, 5-2m=3-m, and the solution is: m=2.
Coordinate characteristics of points on a straight line parallel to the coordinate axis
The vertical coordinates of points on a straight line parallel to the X axis (or horizontal axis) are the same;
The abscissas of points on a straight line parallel to the Y axis (or vertical axis) are the same.
Example 4: Given point A (1, 2), axis AB∨y, AB = 3, the coordinate of point B is _ _ _ _ _ _ _ _.
Analysis: According to "AB∨y axis", the abscissas of point A and point B are equal, so point B may be three units above point A or three units below point A, and the coordinates of point A are (1,-1) or (1, 5).
Distance from point to coordinate axis
Distance from point to X axis = absolute value of ordinate, and distance from point to Y axis = absolute value of abscissa. That is, the distance from A(x, y) to the X axis =|y|, and the distance to the Y axis =|x|.
Example 5. If there is a point M in the second quadrant of the plane rectangular coordinate system, the distance from the point M to the X axis is 3, and the distance from the Y axis is 4, then the coordinate of the point M is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Analysis: According to "the distance from m point to X axis is 3", if |y|=3, then y = 3；; According to "the distance to the Y axis is 4", if |x|=4, then x = 4, and point M is in the second quadrant, then the coordinate of point M is (-4,3).
These conclusions need to be kept in mind and applied flexibly, and many conclusions are often used in functions.
Edited on 2020-12/2
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