A summary of knowledge points about inverse proportional functions in junior middle schools

The image of the inverse proportional function is a center-symmetric hyperbola with the origin as the center of symmetry. Each curve in each quadrant of the inverse proportional function image will be infinitely close to the X-axis and Y-axis but will not be consistent with the coordinate axes. intersect. What are the summary of knowledge points about inverse proportional functions in junior middle school? Let’s take a look at the summary of knowledge points about inverse proportional functions in junior high schools. Welcome to check it out!

Summary of knowledge points of inverse proportional function

1. Expression of inverse proportional function

X is the independent variable, Y is the function of X

y=k/x=k·1/x

xy=k

y=k·x^(-1) (ie: y is equal to The negative first power of x, where /n

2. The value range of the independent variable in the functional expression

①k≠0; ②In general, the value range of the independent variable x can be not equal to 0 is any real number; ③The value range of function y is also any non-zero real number.

Analytical formula y=k/x where X is the independent variable, Y is the function of X, and its domain is all real numbers not equal to 0

y=k/x= k·1/x

xy=k

y=k·x^(-1)

y=kx (k is a constant (k≠0 ), x is not equal to 0)

3. Image of the inverse proportional function

The image of the inverse proportional function belongs to a centrally symmetric hyperbola with the origin as the symmetry center.

Each curve in each quadrant of the inverse proportional function image will be infinitely close to the X-axis and Y-axis but will not intersect with the coordinate axes (K≠0).

4. What is the geometric meaning of k in the inverse proportional function? What are its applications?

Through the inverse proportional function y=k/x(k≠0), a point P(x on the image , y), draw the perpendicular line of the two coordinate axes. The two vertical feet, the origin, and point P form a rectangle. The area of ??the rectangle S=absolute value of x_absolute value of y=absolute value of (x_y)=|k| < /p>

To study function problems, we must look into the essential characteristics of functions. In the inverse proportional function, the proportional coefficient k has a very important geometric meaning, that is: through any point P on the graph of the inverse proportional function, draw vertical lines PM and PN along the x-axis and y-axis. If the vertical feet are M and N, then the area of ??the rectangle PMON S=PM·PN=|y|·|x|=|xy|=|k|.

Therefore, if we draw perpendiculars to the x-axis and y-axis at any point on the hyperbola, the area of ??the rectangle enclosed by them and the x-axis and y-axis is constant. Thereby we have the absolute value of k. When solving problems related to inverse proportional functions, if you can flexibly use the geometric meaning of k in the inverse proportional function, it will bring a lot of convenience to problem solving.

Summary of knowledge points about mathematical inverse proportional functions

The graph of y=k/x(k≠0) is called a hyperbola.

When k>0, The hyperbola is in the first and third quadrants (in each quadrant, descending from left to right);

When k<0, the hyperbola is in the second and fourth quadrants (in each quadrant, descending from left to right); rising to the right).

Therefore, its increase and decrease properties are opposite to those of a linear function.

I believe that students can master the above explanation of the knowledge points of inverse proportional functions. I hope the students can learn the knowledge points well.

Summary of junior high school mathematics knowledge points: Plane Cartesian Coordinate System

The following is the content of the plane Cartesian Coordinate System. I hope students can master the following content well.

Plane Cartesian Coordinate System

Plane Cartesian Coordinate System: Draw two mutually perpendicular number axes with coincident origins in the plane to form a plane Cartesian coordinate system.

The horizontal number axis is called the x-axis or horizontal axis, and the vertical number axis is called the y-axis or vertical axis. The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.

Elements of the plane Cartesian coordinate system: ① On the same plane ② Two number axes ③ Perpendicular to each other ④ The origins coincide

Three regulations:

① Positive direction The stipulation that the horizontal axis orientation is the positive direction to the right, and the vertical axis orientation is the positive direction

② Regulations on unit length; Generally speaking, the unit length of the horizontal axis and the vertical axis is the same; in practice, it may be different sometimes, but the same must be the same on the number line.

③The provisions of the quadrants: the upper right is the first quadrant, the upper left is the second quadrant, the lower left is the third quadrant, and the lower right is the fourth quadrant.

I believe that the above students have mastered the knowledge of plane rectangular coordinate system well. I hope that all students can succeed in the exam.

What are the properties of the inverse proportional function

1. When k>0, the images are located in the first and third quadrants respectively. In the same quadrant, y decreases as x increases. Small; when k<0, the images are located in the second and fourth quadrants respectively. In the same quadrant, y increases with the increase of x.

2. When k>0, the functions are both decreasing functions on x<0 and decreasing functions on x>0; when k<0, the functions are increasing functions on x<0. It is also an increasing function on x>0. The definition domain is x≠0; the value domain is y≠0.

3. Because in y=k/x(k≠0), x cannot be 0, and y cannot be 0, so the graph of the inverse proportional function cannot intersect with the x-axis, nor can it Intersects the y-axis.

4. Pick any two points P and Q on the graph of an inverse proportional function. Draw parallel lines through the points P and Q respectively for the x-axis and the y-axis. The area of ??the rectangle enclosed by the coordinate axes is S1. S2 then S1=S2=|K|

5. The graph of the inverse proportional function is both an axis-symmetric graph and a centrally symmetric graph. It has two axes of symmetry y=xy=-x (that is, the first Three, two and four quadrants angle bisectors), the center of symmetry is the origin of the coordinates.

6. If the direct proportional function y=mx and the inverse proportional function y=n/x intersect at two points A and B (m and n have the same sign), then the two points AB are symmetrical about the origin.

7. Suppose there are inverse proportional function y=k/x and linear function y=mx+n in the plane. To make them have a common intersection point, then n^2+4k·m≥( Not less than)0.

8. The asymptote of the inverse proportional function y=k/x: x-axis and y-axis.

9. The inverse proportional function is symmetrical about the direct proportional function y=x, y=-x axis, and symmetrical about the center of the origin.

10. Inversely proportional point m moves toward x and y respectively The vertical line intersects q and w, then the area of ??the rectangle mwqo (o is the origin) is |k|

11. Inverse proportional functions with equal k values ??overlap, and inverse proportional functions with unequal k values ??never intersect. .

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

13. The graph of the inverse proportional function is a centrally symmetrical figure, and the center of symmetry is the origin

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