Definition of inverse proportional function
Definition: The function y = k/x (where k is a constant and k≠0) is called the inverse proportional function, where k is called the proportional coefficient, x is the independent variable, y is the function of the independent variable x, and the value range of x is all real numbers not equal to 0.
Properties of inverse proportional function
The function y=k/x is called inverse proportional function, where k≠0 and x is the independent variable.
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. The value range of x is: x ≠ 0;
The value range of y is: y≠0.
4. 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 and Y axis. However, with the infinite increase or decrease of x, the function value approaches to zero infinitely, so the image is infinitely close to the x axis.
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=x y=-x (that is, the bisectors of the first, third and fourth quadrants), and the center of symmetry is the coordinate origin.
General form of inverse proportional function
(k is a constant, k≠0), then y is said to be an inverse proportional function of X.
Where x is an independent variable and y is a function. Because X is on the denominator, all real numbers that take x≠0 depend on the range of function Y. Because k≠0 and x≠0, the value of function Y cannot be 0.
Supplementary note: 1. The analytical expression of the inverse proportional function can be written as: (k is a constant, k≠0).
2. The analytic expression of inverse proportional function is needed, and k can be obtained by using undetermined coefficient method.
Characteristics of inverse ratio resolution function
(1) The left side of the equal sign is a function, and the right side of the equal sign is a fraction. The numerator is a non-zero constant (also called proportional coefficient), the denominator contains independent variables, and the exponent is 1.
(2) Proportional coefficient
(3) The values of independent variables are all non-zero real numbers.
(4) The function values are all non-zero real numbers.
Mathematics knowledge points of inverse proportional function in senior one.
A function in the form of y = k/x (where k is a constant and k≠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 the inverse proportional function, it can be concluded that any point on the inverse proportional function image is perpendicular to the two coordinate axes, and the rectangular area surrounded by this point, two vertical feet and the origin is a constant value, which is? k？ .
As shown in the figure, the function images when k is positive and negative (2 and -2) are given above.
When K>0, the inverse proportional function image passes through one or three quadrants, it is a decreasing function.
When k < 0, the inverse proportional function image passes through two or four quadrants, which is increasing function.
The inverse proportional function image can only move towards the coordinate axis infinitely, and cannot intersect with the coordinate axis.
Knowledge points:
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 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)
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 In general, the value range of 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.
5. 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 rectangular 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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