In a certain change process, there are two variables X and Y. If it can be written as y=kx+b(k is a linear coefficient k≠0 and B is a constant), then we say that Y is a linear function of X, where X is an independent variable and Y is a dependent variable.
1, the origin of the function
The word "function" used in China's mathematics books is a translated name, which was translated into "function" by Li, a mathematician in Qing Dynasty, when he translated the book Algebra (1852 to 1859).
The word "Xin" and "Han" were commonly used in ancient China, both of which had the meaning of "Han". The definition given by Li is that "every formula contains Heaven, and Heaven is a function of Heaven". In ancient China, the words "heaven, earth, people and things" were used to represent four different unknowns or variables. The meaning of this definition is: "When the variable X is included in a formula, this formula is called a function of X. But in the early mathematical monograph" Nine Chapters Arithmetic "in China, the word" equation "refers to simultaneous linear equations with many unknowns, that is, linear equations.
2. Basic definition
definition
Generally speaking, if the shape is y=kx+b(k≠0, k and b are constants), then y is called a linear function of X. When b=0, y=kx+b means y=kx, that is, a proportional function (independent variables and dependent variables are proportional). So the proportional function is a special linear function.
Also, if the highest degree of the independent variable is 1, then this function is a linear function.
In a certain change process, there are two variables X and Y. If it can be written as y=f(x) (that is, X gets Y through some operation), that is, each X has a unique Y corresponding to it, then we say that Y is a function of X, where X is an independent variable and Y changes with the change of X. When X takes a value, Y has only one value corresponding to X. If there are two or more values corresponding to X, Representation: The commonly used representations of functions include analytical method, image method and list method.
3. Basic nature
When 1. function is proportional, the quotient of x and y must be (x≠0). In the inverse proportional function, the product of x and y is definite.
In y=kx+b(k, b is constant, k≠0), when x increases m, the function value y increases km; On the contrary, when x decreases m, the function value y decreases km.
2. When x=0, b is the ordinate of the intersection of the linear function image and the Y axis, and the coordinate of this point is (0, b). [2]
3. When b=0, the linear function becomes a proportional function. Of course, the proportional function is a special linear function. [2]
4. In two linear function expressions:
When k and b in the expressions of two linear functions are the same, the images of the two linear functions coincide;
When k is the same and b is different in two linear function expressions, the images of the two linear functions are parallel;
When k and b in two linear function expressions are different, the images of two linear functions intersect;
When k is different and b is the same in two linear function expressions, the two linear function images intersect at the same point (0, b) on the y axis;
When k in two linear function expressions is negative reciprocal, two linear function images are perpendicular to each other. [2]
5. the ratio of two linear functions (y 1=ax+b, y2=cx+d), the new function y3=(ax+b)/(cx+d) is an inverse proportional function, and the asymptote is X =-B/A and y = c/a. ..
4. Special positional relationship
When two straight lines are parallel in the plane rectangular coordinate system, the value of k in the resolution function (that is, the coefficient of the first term) is equal;
When two straight lines are perpendicular to each other in a plane rectangular coordinate system, the values of k in the resolution function are reciprocal (that is, the product of two k values is-1).
Note: The straight line parallel to the Y axis has no resolution function, and the analytical formula of the straight line parallel to the X axis is a constant function, so these two straight lines are excluded from the above properties.
5. Frequently asked questions
Linear functions and their images of common problems are important contents of algebra in junior high school, the cornerstone of analytic geometry in senior high school and the key content of senior high school entrance examination. Among them, finding the resolution function once is a common question. Taking some senior high school entrance examination questions as examples, this paper introduces several common types of questions for solving the first-order discrimination function. I hope it will be helpful to everyone's study.
1. Definition example 1. Assuming that the function is a linear function, find its analytical expression. Solution: We can know from the definition of linear function, so the analytic formula of linear function is attention: when solving analytic function with definition, we must ensure that. In this case, it should be guaranteed that
2. Example 2. Given the image passing point (2,-1) of a linear function, find the analytical expression of this function. Solution: The image of a linear function passes through (2,-1), that is, the analytical expression of this linear function is a variant: when the linear function y =- 1 is known, find the analytical expression of this function.
3. If the coordinates of the intersection of the image of a linear function with the X axis and the Y axis are known as (-2,0) and (0,4) respectively, then the analytical formula of this function is _ _ _ _ _ _ _ _ _. Solution: Let a resolution function get the answer from the meaning of the question. The analytical formula of this linear function is
4. Image type 4. If the image of a linear function is known as shown in the figure, the analytical formula of the function is _ _ _ _ _ _ _ _. Solution: Let the primary resolution function be the image passing points of the primary function (1, 0) and (0,2), then the analytical formula of this primary function is
5. Oblique cutting example 5. Given that the straight line is parallel to the straight line and the intercept on the Y axis is 2, the analytical formula of the straight line is _ _ _ _ _ _ _ _ _. Analysis: two straight lines:; : 。 When, when, the straight line is parallel to the straight line. And the intercept of the straight line on the y axis is 2, then the analytical formula of the straight line is
6. Translation Example 6. The analytical formula of the image obtained by translating the straight line down by 2 units is _ _ _ _ _ _ _ _. Analysis: Let the resolution function be, and the intercept of the straight line and the straight line parallel to the straight line on the Y axis be, then the image analysis formula is seven. Practical application example 7. There is 20 liters of oil in a fuel tank, and the oil flows out of the pipeline at a uniform speed, and the flow rate is 0.2 liters/minute. Then the functional relationship between the remaining oil quantity q (liter) in the fuel tank and the outflow time t (minute) is _ _ _ _ _. Solution: From the meaning of the question, that is, the analytical formula of the function is (). Note: To find the functional relationship of practical application problems, it is necessary to write the range of independent variables.
8. Region type 8. Given that the area of a triangle surrounded by a straight line and two coordinate axes is equal to 4, the analytical formula of the straight line is _ _ _ _ _ _ _. Solution: It is easy to find that the intersection of a straight line and the X axis is (,0), so, so, that is, the analytical formula of a straight line is or.
Nine. Symmetric type If the straight line and the straight line are symmetrical about (1)x, the analytical formula of the straight line L is (2)y-axis symmetry, (3) y=x-axis symmetry, (4) straight line symmetry, (5) origin symmetry, and the analytical formula of the straight line L is Example 9. If the straight line is straight, the solution: The analytical formula of the straight line L obtained from (2) is
X. Open paradigm 10. Assuming that the image of the function passes through points A (1, 4) and B (2, 2), please write two functions with different resolutions that meet the above conditions and briefly describe the solution process. Solution: (1) If the function passing through point A and point B is like a straight line, it can be easily obtained from the two-point formula. (2) Because the product of abscissa and ordinate of point A and point B is equal to 4, the function image passing through point A and point B can also be hyperbola, and the analytical formula is
Second, the inverse proportional function 1, the shape function (k is constant and k≠0) is called the inverse proportional function, where k is called the inverse proportional coefficient, x is the independent variable, y is the function of the independent variable x, the value range of x is all real numbers not equal to 0, and y cannot be equal to 0. When k is greater than 0, the image is in 1 and 3 quadrants. When k is less than 0, the image is located in quadrant 2 and quadrant 4. K represents the area of a rectangle formed by the coordinates of x and y.
2. The range of independent variables
① In general, the value range of independent variable X can be any real number not equal to 0;
② The range of function y is also an arbitrary non-zero real number.
3. Analysis formula
Where x is an independent variable, y is a function of x, and its domain is all real numbers that are not equal to 0.
That is {x|x≠0, x∈R}. Here are some common forms:
(k is a constant (k≠0, x is not equal to 0)
4. Overview
The inverse proportional function image belongs to a hyperbola with the origin as the symmetry center, and each curve in 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 (y≠0).
5. Functional attributes
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
intersection
because in
In (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, and can only be infinitely close to X axis and Y axis.
zone
Functional attribute
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
intersection
because in
In (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, and can only be infinitely close to X axis and Y axis.
6. Conceptual understanding
A function with the shape (k is a constant, 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.
Image properties of inverse proportional function: the image of inverse proportional function is hyperbola.
Since the inverse proportional function belongs to odd function, there is f(x)=f(-x), and 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∣.
Note: the inverse proportional function image can only move towards the coordinate axis infinitely, and cannot intersect with the coordinate axis.
7. Key knowledge
Any point on the over-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|.
For hyperbola, if you add or subtract any real number on the denominator (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)