Summary of knowledge points of inverse proportional function
1, expression of inverse proportional function
X is an independent variable and y is a function of X.
y=k/x=k 1/x
xy=k
Y = k x (- 1) (that is, y is equal to the negative power of x, where x must be a power).
Y=kx(k is constant and k≠0, x≠0) If y=k/nx, the proportional coefficient is k/n.
2. The range of independent variables in the function.
①k≠0； (2) Generally speaking, the value range of the independent variable x' can be any real number not equal to 0; ③ The range of value of function y is also any non-zero real number.
Analytical formula y=k/x, where x is an independent variable and y is a function of x, and its definition fields are 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. Inverse proportional function image
The image of inverse proportional function belongs to hyperbola with the origin as the symmetry center.
Each curve of each quadrant in the inverse proportional function image will be infinitely close to the X axis and the Y axis, but will not intersect with the coordinate axis (K≠0).
4. What is the geometric meaning of k in inverse proportional function? What are the applications?
Overinverse proportional function y=k/x(k≠0), take a point P(x, y) on the image as the vertical line of two coordinate axes, two vertical feet, the origin and point p form a rectangle, and the area of the rectangle s = x _ y = absolute value of x _ y =|k|.
To study the function problem, we should see through the essential characteristics of the function. In the inverse proportional function, the proportional coefficient k has a very important geometric significance, that is, if any point P in the inverse proportional function image is perpendicular to the X-axis and Y-axis PM and PN, and the vertical feet are M and N, then the area of right-angle PMON is S = PM PN = | Y |||| X | =| XY | = | K |.
Therefore, if any point on the hyperbola is perpendicular to the X-axis and Y-axis, the rectangular area they enclose with the X-axis and Y-axis is constant. Then there is the absolute value of K. When solving the problem of inverse proportional function, if we can flexibly use the geometric meaning of K in inverse proportional function, it will bring a lot of convenience to solve the problem.
Induction of knowledge points of mathematical inverse proportional function
The image of y=k/x(k≠0) is called hyperbola.
When k>0, hyperbola is in the first and third quadrants (each quadrant decreases from left to right);
When k < 0, hyperbola is in quadrant 2 and quadrant 4 (in each quadrant, it rises from left to right).
Therefore, its increase or decrease is contrary to a linear function.
The above explanation of the knowledge points of inverse proportional function, I believe that students have mastered it very well, and I hope students can learn the knowledge points well.
Summary of junior high school mathematics knowledge points: plane rectangular coordinate system
The following is the study of the content of plane rectangular coordinate system. I hope students can master the following content well.
Cartesian coordinates/Cartesian coordinates
Plane Cartesian coordinate system: Draw two mutually perpendicular number axes with coincident origin on the plane to form a plane Cartesian coordinate system.
The horizontal axis is called X axis or horizontal axis, the vertical axis is called Y axis or vertical axis, and the intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.
Elements of a plane rectangular coordinate system: ① On the same plane; ② Two axes of numbers are perpendicular to each other; ④ The origin coincides.
Three rules:
① The specified positive direction: the horizontal axis is right, and the vertical axis is oriented in the positive direction.
(2) the provisions of the unit length; Generally speaking, the unit length of the horizontal axis and the vertical axis is the same; In fact, sometimes it can be different, but it must be on the same axis.
③ Quadrant definition: 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 students have mastered the knowledge of plane rectangular coordinate system, and I hope they can all be admitted.
What are the properties of inverse proportional function?
1. when k>0, the image is located in the first and third quadrants respectively, and in the same quadrant, y decreases with the increase of x; When k < 0, the image is located in the second and fourth quadrants respectively, and in the same quadrant, y increases with the increase of x.
2.k>0, function in x
3. Because in y=k/x(k≠0), X can't be 0 and Y can't be 0, so the image of inverse proportional function can't intersect with X axis or Y axis.
4. In the inverse proportional function image, take any two points P and Q, the intersection points P and Q are parallel lines of the X axis and the Y axis respectively, the rectangular area enclosed with the coordinate axis is S 1, S2 is S 1=S2=|K|.
5. The image of inverse proportional function is not only an axisymmetric figure, but also a centrally symmetric figure. It has two symmetry axes y=xy=-x (that is, the bisector of the first, third and fourth quadrant angles), and the center of symmetry is the coordinate origin.
6. If the positive proportional function y=mx and the inverse proportional function y=n/x intersect at two points A and B (m the signs of m and n are the same), then two points AB are symmetrical about the origin.
7. Let there be an inverse proportional function y=k/x and a linear function y=mx+n on the plane. If they have a common intersection, then n 2+4k m ≥ (not less than) 0.
8. Inverse proportional function y = k/x: asymptote of X axis and Y axis.
9. The inverse proportional function is symmetric about the positive proportional function y=x, y=-x, and symmetric about the origin center.
10. On the inverse scale, point M is perpendicular to X and Y respectively, and intersects with Q and W, then the area of rectangle mwqo(o is the origin) is |k|.
The inverse proportional functions with equal 1 1.k value coincide, and the inverse proportional functions with unequal k values never intersect.
12. The larger the | k |, the farther the image of the inverse proportional function is from the coordinate axis.
13. The inverse proportional function image is a central symmetric figure, and the symmetric center is the origin.
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