1 Method and Example of Finding Mathematical Trajectory in Senior High School
Trajectory contains two problems: all points on the trajectory meet the given conditions, which is called the purity of trajectory (also called inevitability); Any points that are not on the trajectory do not match. There are many methods to solve the trajectory equation, such as literal translation, definition, correlation point method, parameter method, intersection method and so on.
2 Common methods
When finding the trajectory of moving point, sometimes there will be a trajectory problem that needs two moving curves to intersect. In this lamp problem, the coordinates (including parameters) of the intersection point are usually obtained by solving the equation, and then the trajectory equation is obtained by eliminating the parameters (if the parameters of the two equations can be directly eliminated, the trajectory equation can also be obtained by directly eliminating the parameters). This method is usually used with parameter methods. By eliminating the parameters in the two dynamic curve equations, an equation without parameters is obtained, which is the trajectory equation of the intersection of the two dynamic curves. This method of solving trajectory equation is called intersection method.
If the trajectory of the moving point can be determined to meet the definition of the known curve, the equation can be written by using the definition of the curve. This method of solving trajectory equation is called definition method. Undetermined coefficient method: If the motion law of moving point P conforms to the definition of known curve (such as circle, ellipse, hyperbola and parabola), we can first establish the trajectory equation, and then work out the constants in the undetermined equation according to the known conditions, which is also called definition method. Judging what kind of figure the trajectory of the moving point is by the geometric properties of the moving point, and then solving its trajectory equation, this method is called definition method. To find its trajectory by definition method, we should master the definitions of common trajectories such as the midline, circle, ellipse, hyperbola and parabola of line segments, and master some property theorems of plane geometry.
3 steps to solve the problem
Establish an appropriate coordinate system and set the coordinates of the scheduling point m; Write a set of points m; List equation = 0; Simplify the equation to the simplest form; Check.
(1) Establish a system-establish a suitable coordinate system;
② set point-set any point on the trajectory P(x, y);
(3) Formula —— List the relationship that the moving point P satisfies;
④ Substitution-according to the characteristics of conditions, the distance formula and slope formula are selected, converted into equations about X and Y, and simplified;
⑤ Proof —— Prove that the equation is a moving point trajectory equation that meets the requirements.
It should be noted that some trajectory problems contain certain implicit conditions, that is, the coordinate range of points on the curve. From the concept of curve and equation, we must pay attention to its "completeness" and "purity" when solving the curve equation, that is, if the trajectory is a part of the curve, it must meet the value range indicated in the equation or the value range indicated at the same time. "Trajectory" and "Trajectory Equation" are both different and related. When solving the "trajectory", we should first solve the "trajectory equation", then explain the trajectory figure of the equation, and finally "trap" and "divide points". If the trajectory has different conditions, it should be discussed separately to ensure its integrity.
4 learning attention
The key to solving the trajectory equation is to find the motion law of moving point P in complex motion changes, that is, the equal relationship that point P satisfies, so we should learn to seek static in motion and constant in change. Trajectory equation can be expressed by ordinary equation and parametric equation. To judge which curve the trajectory equation represents, it is often necessary to turn the parametric equation into a constant equation.
After solving the trajectory equation, we should pay attention to check whether it conforms to the meaning of the question, whether it increases the solution (that is, the point with some solutions of the equation as coordinates is not on the trajectory), and whether it is lost. (that is, some points on the trajectory cannot be expressed by equations), if there is an increased solution, it will be abandoned, and if there is no solution, it will be supplemented. Test method: study special or extreme situations in sports.
The method of seeking trajectory in high school mathematics and examples of related articles;
1. Learning true questions over the years: the most comprehensive method of solving problems in high school mathematics in history.
2. The skills and ideas of solving math problems in senior high school.
3. Mathematics knowledge points in senior three must solve the trajectory equation.
4. High school math problem sets and answers
5. High school math problem-solving skills final sprint scoring questions
6.2 1 Math problem-solving methods and skills in senior high school.
7. Mathematics thought and logic in senior high school: 1 1 summary of mathematics thought methods and explanation of examples.
8. Eight answer templates for solving math problems in senior high school and methods for solving big questions.
9. Skills and methods of answering questions in high school mathematics and jingle.
10. Senior high school mathematics learning methods and answering skills