1, monotonicity

When k>0, the image is located in the first and third quadrants respectively. In each quadrant, from left to right, y decreases with the increase of X;

When k < 0, the image is located in the second and fourth quadrants respectively. In each quadrant, from left to right, y increases with the increase of x.

K>0, function in x

2. Area

Take any two points in the inverse proportional function image, the intersection points are parallel lines of the X axis and the Y axis respectively, and the rectangular area enclosed with the coordinate axis is |k|.

A point on the inverse proportional function is perpendicular to the X-axis and the Y-axis respectively, and intersects with the Y-axis and the X-axis respectively, so the area of QOWM is |k|, then the diagonal line connecting the rectangle is connected with OM, and the area of RT△OMQ is =? |k| .

3. Image expression

The asymptotes of the inverse proportional function image that do not intersect with the X axis and the Y axis are: X axis and Y axis.

The inverse proportional function images with equal k value overlap, but the inverse proportional function images with different k values never intersect.

The larger the |k|, the farther the image of the inverse proportional function is from the coordinate axis.

4. Symmetry

The inverse proportional function image is a central symmetric figure, and the symmetric center is the origin; The image of inverse proportional function is also an axisymmetric figure, and its symmetry axis is y=x or y =-x; The points on the inverse proportional function image are symmetrical about the coordinate origin.

The image is symmetrical about the origin. If the positive proportional function y=mx and the inverse proportional function intersect at point A and point B (the signs of m and n are the same), then point A and point B are symmetrical about the origin.

The inverse proportional function is symmetric about the positive proportional function y = x and symmetric about the center of the origin.

Extended data:

1, concept understanding

The range of the independent variable x is all real numbers that are not equal to 0.

Image properties of inverse proportional function: the image of inverse proportional function is hyperbola.

Because the inverse proportional function belongs to odd function and has a symmetrical center, 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∣.

2. Drawing steps of inverse proportional function image:

1. list: independent variables should be selected with the origin as the center, and three pairs (or more than three pairs) of mutually opposite numbers should be taken on both sides of the origin. When you fill in the value of y, you only need to calculate the function value on one side, and the function value on the other side is the corresponding inverse number.

2. Point tracking: After the point on one side is tracked, the point on the other side can be tracked symmetrically according to the center.

3. Connection: Connect all points from left to right and extend them. When connecting, connect with smooth curves according to the order of independent variables from small to large to avoid drawing into broken lines. Note that the two branches of hyperbolic money are disconnected, and the extended part tends to approach the coordinate axis gradually, but it will never intersect with the coordinate axis.

Baidu Encyclopedia-Inverse Proportional Function