Definition of knowledge point 1 inverse proportional function

Usually, a function with the shape (k is a constant) is called an inverse proportional function, which can be understood from the following aspects:

(1) x is an independent variable and y is an inverse proportional function of x;

(2) The range of independent variable x is all real numbers, and the range of function value is;

⑶ Proportional coefficient is an important part of the definition of inverse proportional function;

(4) The inverse proportional function has three expressions:

①(),

②(),

③ (fixed value) ();

5] Functions () and () are equivalent, so when Y is an inverse proportional function of X, X is also an inverse proportional function of Y. ..

(k is a constant,) is part of the inverse proportional function. When k=0, it is not an inverse proportional function. Because there is only one undetermined coefficient in the inverse proportional function (), as long as there is a set of corresponding values, the value of k can be obtained, and thus the expression of the inverse proportional function can be determined.

Knowledge point 2 Find the analytical formula of inverse proportional function by undetermined coefficient method.

Because there is only one undetermined coefficient in the inverse proportional function (), the value of k can be found as long as a set of corresponding values, thus determining the expression of the inverse proportional function.

Image and drawing of inverse proportional function of knowledge point 3

The image of inverse proportional function is a hyperbola, which has two branches, which are located in the first quadrant, the third quadrant or the second quadrant and the fourth quadrant respectively, and they are symmetrical about the origin. Because of the independent variables and function values in the independent variable function of the inverse proportional function, its image does not intersect with the X axis and the Y axis, that is, the two branches of the hyperbola are infinitely close to the coordinate axis, but never reach the coordinate axis.

The inverse proportion drawing method is divided into three steps: (1) list; (2) tracking points; (3) connection.

Pay attention to the following points when making the inverse proportional function image again:

① The values selected in the list should be selected symmetrically;

② The more values selected in the list, the more accurate the picture will be;

(3) When connecting, smooth curves must be used according to the size of independent variables from left to right (or from right to left) to avoid drawing broken lines;

(4) When drawing an image, draw both its branches, but the image cannot intersect with the coordinate axis.

Knowledge Point 4 Properties of Inverse Proportional Function

☆ Regarding the properties of the inverse proportional function, we mainly study the position of the image and the increase or decrease of the function value, as shown in the following table:

The value range of the symbolic image property ① of the inverse proportional function () is Y. At this time, the two branches of the function image are in the first quadrant and the third quadrant respectively, and in each quadrant, Y decreases with the increase of X. The value range of ① is, and the value range of Y is ②. At this time, the two branches of the function image are in the second and fourth quadrants respectively, and in each quadrant, Y increases with the increase of X. ..

Note: When describing the increase or decrease of the function value, it must be pointed out that "in each quadrant ..." Otherwise, generally speaking, y decreased with the increase of x at that time, which is inconsistent with the facts.

The position of the inverse proportional function image and the increase or decrease of the function are determined by the sign of the inverse proportional function coefficient k, and conversely, the sign of k can be inferred from the position of the inverse proportional function image (hyperbola) and the increase or decrease of the function. If it is in the first and third quadrants, you can know.

☆ Geometric meaning of absolute value of proportional coefficient k in inverse proportional function ().

As shown in the figure, any point P(x, y) passing through hyperbola is perpendicular to the X axis and Y axis respectively, and E and F are vertical feet respectively.

rule

In the inverse proportional function (), the bigger it is, the farther the hyperbola is from the coordinate origin; The smaller the hyperbola, the closer it is to the origin of coordinates.

Hyperbola is a central symmetric figure, and the center of symmetry is the origin of coordinates; Hyperbola is also an axisymmetric figure, and the symmetry axis is straight line y=x and straight line y =-X.