Induction of Mathematics Knowledge Points in the Second Term of Grade Three
Part 1: inverse proportional function
A function in the form of y = k/x (where k is a constant, k≠0, x≠0, y≠0) is called an inverse proportional function.
The range of the independent variable x is all real numbers that are not equal to 0.
Inverse proportional function image properties:
The image of the inverse proportional function is a hyperbola.
Since the inverse proportional function belongs to odd function, let f(-x)=-f(x), and the image is symmetrical about the origin.
In addition, from the analytical formula of inverse proportional function, it can be concluded that any point on the image of inverse proportional function is perpendicular to two coordinate axes, and the rectangular area surrounded by this point, two vertical feet and the origin is a constant, which is ∣k∣.
When K>0, the inverse proportional function image passes through one or three quadrants, which is a decreasing function (that is, Y decreases with the increase of X).
When k < 0, the inverse proportional function image passes through two or four quadrants, which is increasing function (that is, y increases with the increase of x).
Because the independent variable and dependent variable of the inverse proportional function cannot be zero, the image can only approach the coordinate axis infinitely, and cannot intersect the coordinate axis.
1. Any point on the inverse proportional function image is a vertical line segment of two coordinate axes, and the area of the rectangle surrounded by these two vertical line segments and coordinate axes is |k|.
2. For hyperbola y=k/x, if you add or subtract any real number on the denominator (that is, y = k/x (x m) m is a constant), it is equivalent to translating the hyperbola image to the left or right by one unit. (When adding a number, move to the left, and when subtracting a number, move to the right)
Chapter 2: Quadratic function.
Knowledge point 1. rectangular coordinates/ cartesian coordinates
1, plane rectangular coordinate system
Draw two mutually perpendicular number axes with a common origin on the plane to form a plane rectangular coordinate system.
Among them, the horizontal axis is called X axis or horizontal axis, and the right direction is the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation is positive; The intersection o of the two axes (that is, the common * * *) is called the origin of the rectangular coordinate system; The plane on which the rectangular coordinate system is established is called the coordinate plane.
In order to describe the position of a point in the coordinate plane conveniently, the coordinate plane is divided into four parts, namely the first quadrant, the second quadrant, the third quadrant and the fourth quadrant.
Note: The points on the X and Y axes do not belong to any quadrant.
2, the concept of point coordinates
The coordinates of points are represented by (a, b), and the order is abscissa before, ordinate after, and there is a ","in the middle. The positions of horizontal and vertical coordinates cannot be reversed. The coordinates of points on the plane are ordered real number pairs. At that time, (a, b) and (b, a) were the coordinates of two different points.
Knowledge point 2. Coordinate characteristics of different position points
1, the coordinate characteristics of the midpoint of each quadrant.
Point P(x, y) is in the first quadrant.
Point P(x, y) is in the second quadrant.
Point P(x, y) is in the third quadrant.
Point P(x, y) is in the fourth quadrant.
2. Features of points on the coordinate axis
The point P(x, y) is on the X axis, and X is an arbitrary real number.
The point P(x, y) is on the y axis, and y is an arbitrary real number.
Point P(x, y) is on both X and Y axes, and both X and Y are zero, that is, the coordinate of point p is (0,0).
3. Characteristics of the coordinates of points on the bisector of two coordinate axes.
Point P(x, y) is equal to x on the bisector of the first and third quadrants.
Points P(x, y) are opposite to each other on the bisector of the second and fourth quadrants.
4. Coordinate characteristics of points on a straight line parallel to the coordinate axis
The ordinate of each point on the straight line parallel to the X axis is the same.
The abscissa of each point on the straight line parallel to the Y axis is the same.
5. Coordinate characteristics of points symmetrical about X-axis, Y-axis or apogee.
The abscissa of point P and point P' is equal to the axis of X, and the ordinate is opposite.
The vertical coordinates of point P and point P' are symmetrical about the Y axis, and the horizontal coordinates are opposite to each other.
Point p and point p' are symmetrical about the origin, and the abscissa and ordinate are opposite.
6. Distance from point to coordinate axis and origin
Distance from point P(x, y) to coordinate axis and origin:
(1) The distance from the point P(x, y) to the X axis is equal to
(2) The distance from the point P(x, y) to the Y axis is equal to
(3) The distance from point P(x, y) to the origin is equal to
Sorting out the knowledge points of junior high school mathematics
Axisymmetric knowledge point
1. If a graph is folded along a straight line and the parts on both sides of the straight line can overlap each other, then the graph is called an axisymmetric graph; This straight line is called the axis of symmetry.
2. The symmetry axis of an axisymmetric figure is the perpendicular bisector of a line segment connected by any pair of corresponding points.
3. The distance from the point on the bisector of the angle is equal to both sides of the angle.
4. The distance between any point on the vertical line of the line segment and the two endpoints of the line segment is equal.
5. The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
6. The corresponding line segment and the corresponding angle on the axisymmetric figure are equal.
7. Draw an axisymmetric figure about a straight line: find the key points, draw the corresponding points of the key points, and connect the points in the original order.
8. The coordinates of the point (x, y) about the axis symmetry of X are (x, -y).
The coordinates of the point (x, y) that is symmetric about y are (-x, y).
The coordinates of the point (x, y) that is symmetrical about the origin are (-x, -y).
9. The nature of isosceles triangle: the two base angles of isosceles triangle are equal (equilateral and equiangular).
The bisector of the top angle of an isosceles triangle, the height on the bottom edge and the midline on the bottom edge coincide, which is called the integration of the three lines for short.
10. Determination of isosceles triangle: equilateral and equilateral.
1 1. The three internal angles of an equilateral triangle are equal and equal to 60.
12. Determination of equilateral triangle: A triangle with three equal angles is an isosceles triangle.
An isosceles triangle with an angle of 60 is an equilateral triangle.
A triangle with two angles of 60 is an equilateral triangle.
13. In a right triangle, the right angle side of 30 is equal to half of the hypotenuse.
inequality
1. Grasp the basic properties of inequality and use it flexibly;
Add (or subtract) the same algebraic expression on both sides of inequality (1), and the direction of inequality remains the same, that is, if A >；; B, then a+c > b+c，a-c & gt； b-c .
(2) if both sides of the inequality are multiplied by (or divided by) the same positive number, the direction of the inequality remains unchanged, that is, if a >；; B and c>0, then AC & GT200 BC.
(3) If both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality will change, that is, if a >；; B, and c < 0, ac
2. Comparison size: (A and B represent two real numbers or algebraic expressions respectively)
Generally speaking:
If a>b, then a-b is a positive number; On the other hand, if a-b is positive, then a >；; b；
If a=b, then a-b is equal to 0; On the other hand, if a-b is equal to 0, then a = b；;
If a
Namely: a>b<= = = & gta-b & gt；; 0; a = b & lt= = = & gta-b = 0； aa-b & lt； 0。
3. Solution set of inequality: the value of unknown quantity that can make inequality hold is called the solution of inequality; All the solutions of an inequality constitute the solution set of this inequality; The process of finding the solution set of inequality is called solving inequality.
4. Representation of the inequality solution set on the number axis: When the inequality solution set is represented by the number axis, the boundary and direction should be determined: ① Boundary: there are solid circles with equal signs and hollow circles without equal signs; ② Direction: large on the right and small on the left.
The ninth grade mathematics review materials volume one
Knowledge point 1: the basic concept of unary quadratic equation
1, the constant term of the unary quadratic equation 3x2+5x-2=0 is -2.
2. The coefficient of the primary term of the unary quadratic equation 3x2+4x-2=0 is 4, and the constant term is -2.
3. The quadratic term coefficient of the unary quadratic equation 3x2-5x-7=0 is 3, and the constant term is -7.
4. Transform the equation 3x(x- 1)-2=-4x into the general formula 3x2-x-2=0.
Knowledge point 2: Cartesian coordinate system and the position of points
1. In the rectangular coordinate system, point A (3 3,0) is on the Y axis.
2. In the rectangular coordinate system, the abscissa of any point on the X axis is 0.
3. In rectangular coordinate system, point A (1, 1) is in the first quadrant.
4. In rectangular coordinate system, point A (-2,3) is in the fourth quadrant.
5. In rectangular coordinate system, point A (-2, 1) is in the second quadrant.
Knowledge point 3: Find the function value of the known independent variable.
1. When x=2, the value of function y= is 1.
2. When x=3, the value of function y= is 1.
3. When x=- 1, the value of function y= is 1.
Knowledge point 4: the concept and nature of basic functions
1 and function y=-8x are linear functions.
2. The function y=4x+ 1 is a proportional function.
3. This function is an inverse proportional function.
4. The opening of parabola y=-3(x-2)2-5 is downward.
5. The symmetry axis of parabola y=4(x-3)2- 10 is x=3.
6. The vertex coordinate of parabola is (1, 2).
7. The image of the inverse proportional function is in the first and third quadrants.
Knowledge point 5: mean, median and mode of data
1, data 13, 10,12,8,7, the average value is 10.
2. The pattern of data 3, 4, 2, 4, 4 is 4.
3. The median of data 1, 2,3,4,5 is 3.
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