1. Basic definition

2. Related nature

3. Relationship with binary linear equation

4. Extension of general formula

Linear functions, also known as linear functions, can be represented by straight lines on the x and y axes. When the value of one variable in a linear function is determined, the value of another variable can be determined by a linear equation of one variable.

brief introduction

Basic concept of function: in a certain change process, there are two variables X and Y. If each fixed value of X has a unique fixed value, then we say that Y is a function of X, that is, X is an independent variable and Y is a dependent variable. Expressed as y=kx+b(k≠0, k and b are constants). When b=0, y is the proportional function of x, and the proportional function is a special case of linear function. It can be expressed as y=kx(k≠0), the constant k is called the proportional coefficient or slope, and b is called the longitudinal intercept.

Linear function is now a very difficult chapter in the textbook of junior two, and it is the most widely used and knowledgeable mathematics subject.

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Basic definition

The linear function of y about the independent variable x has the following relationship:

1.y=kx+b (k is any constant other than 0, and b is any real number).

When x takes a value, y has one and only one value corresponding to x, and if there are two or more values corresponding to x, it is not a linear function.

X is an independent variable, y is a function value, k is a constant, and y is a linear function of X.

Especially, when b=0, y is the proportional function of x, that is, the image of the proportional function of y=kx (k is constant, but K≠0) passes through the origin.

Definition domain (function value): the range of independent variables, which should make the function meaningful; It should be realistic.

Common representation methods: analytical method, image method and list method.

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Correlation characteristic

Function properties:

The change value of 1.y is directly proportional to the change value of x, and the ratio is k.k. 。

Namely: y=kx+b(k, b is a constant, k≠0),

When x increases m, k (x+m)+b = y+km, km/m = k.

2. When x=0, b is the point of the function on the Y axis, and the coordinate is (0, b).

When b=0 (y=kx), the image of a linear function becomes a proportional function, which is a special linear function.

4. In two linear function expressions:

When k and b in the expression of quadratic function are the same, the images of quadratic function overlap;

When k is the same and b is different in the expression of quadratic function, the images of quadratic function 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 quadratic function expressions, the images of quadratic functions intersect at the same point (0, b) on the Y axis.

If the relationship between two variables X and Y can be expressed as y = kx+b (k both K and B are constants, and K is not equal to 0), then Y is said to be a linear function of X..

Image attribute

1. Practice and graphics: Through the following three steps:

(1) list.

(2) tracking points; [Generally, two points are taken. According to the principle of "two points determine a straight line", it can also be called "two-point method".

The general image of y=kx+b(k≠0) can be drawn as a straight line passing through (0, b) and (-b/k, 0).

The image of the proportional function y=kx(k≠0) is a straight line passing through the coordinate origin, and generally takes two points (0,0) and (1, k).

(3) Connecting lines can make images of linear functions-straight lines. So the image of a function only needs to know two points and connect them into a straight line. Usually, the intersections of the function image with the X axis and the Y axis are -k points B and 0, 0 and B, respectively.

2. Property: any point P(x, y) on the (1) linear function satisfies the equation: y=kx+b(k≠0). (2) The coordinates of the linear function intersecting with the Y axis are always (0, b), and the images of the proportional function intersecting with the X axis at (-b/k, 0) are all at the origin.

3. Function is not a number, it refers to the relationship between two variables in a certain change process.

4. Quadrant where K, B and function images are located:

When y=kx (that is, b is equal to 0 and y is proportional to x):

When k>0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X;

When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.

When y=kx+b:

When k>0, b>0, when the image of this function passes through the first, second and third quadrants;

When k>0, b<0, when the image of this function passes through the first, third and fourth image limits;

When k < 0, b>0, when the image of this function passes through the first, second and fourth pixel limits;

When k < 0, b<0, when the image of this function passes through the second, third and fourth pixel limits;

When b>0, the straight line must pass through the first and second quadrants;

When b<0, the straight line must pass through the third and fourth quadrants.

Particularly, when b=0, the image of the proportional function is represented by a straight line of the origin o (0 0,0).

At this time, when k>0, the straight line only passes through the first and third quadrants, not through the second and fourth quadrants. When k < 0, the straight line only passes through the second and fourth quadrants and does not pass through the first and third quadrants.

4. Special positional relationship:

When two straight lines in the plane rectangular coordinate system are parallel, the k value in the resolution function (that is, the coefficient of the first term) is equal.

When two straight lines are perpendicular to each other in the plane rectangular coordinate system, the value of k in the resolution function is negative reciprocal (that is, the product of two values of k is-1).

③ formula of point inclination y-y 1=k(x-x 1)(k is the slope of the straight line, (x 1, y 1) is the point where the straight line passes) ④ formula of two points (y-y1)/(. Y 1) and (x2, y3)) ⑤ Intercept formula (A and B are intercepts of a straight line on the X axis and Y axis respectively) ⑤ Practical (based on practical problems).

Limitations of analytic expressions

① There are many conditions needed (2 points, because the method of undetermined coefficient needs a binary linear equation set).

(2) and (3) cannot express straight lines without slope (that is, straight lines perpendicular to the X axis; Note that the expression "a straight line without slope is parallel to the Y axis" is inaccurate, because x=0 coincides with the Y axis).

④ There are many parameters and the calculation is too complicated;

⑤ Cannot represent a straight line parallel to the coordinate axis and a straight line passing through the origin.

Concept of inclination angle

The angle between the X axis and the straight line (the angle formed by the straight line and the positive direction of the X axis) is called the inclination angle of the straight line. Let the inclination of the straight line be α, and the slope of the straight line is k=tanα. The range of tilt angle is [0, π].

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Relationship with binary linear equation

1.( 1) An image and a linear function composed of points whose coordinates are the solutions of the binary linear equation system ax+by = c.

Y=-a/bx+c/b images are the same.

(2) Binary linear equations {a1x+b1y = c1,

The solution of a2x+b2y=c2 can be regarded as two linear functions.

Y =-a1/b1x+c1/d1and y=-a2/b2x+c2/d2.

Method summary:

Rewriting two binary linear equations in the equation into linear functions, then making their images, and finding the intersection of the two images, we can know the solution of the equation.

I. Differences and connections

Difference: A binary linear equation has two unknowns, while a linear function only says that the number of unknowns is one, without limiting several variables, so a binary linear equation is only one of the linear functions.

Connection: (1) In the plane rectangular coordinate system, draw points whose coordinates are the solutions of binary linear equations, and these points are all on the images of corresponding linear functions. For example, the equation 2x+y=5 has countless solutions, such as x= 1 and y = 3；; x=2，y = 1； The points (1, 3) (2, 1) whose coordinates are these solutions are all on the image of linear function y =-2x+5. (2) The coordinates of any point on the linear function image are suitable for the corresponding binary linear equation. Such as the linear function y =-x+2 (.

Therefore, the image composed of all points whose coordinates are the solution of the binary linear equation is the same as the image of the corresponding linear function.

Second, the relationship between the intersection of two graphs of this function and the solution of the equation.

In the same plane rectangular coordinate system, the intersection coordinates of two linear function images are the solutions of the corresponding binary linear equations. On the contrary, the point whose coordinates are the solution of binary linear equations must be the intersection of the corresponding images of two linear functions.

Third, the relationship between the corresponding function images when the equations have no solution.

When the binary linear equations have no solution, the images of the corresponding two linear functions in the plane rectangular coordinate system do not intersect, that is, the images of the two linear functions are parallel. Conversely, when two linear function images are parallel, the corresponding binary linear equations have no solution. If the binary linear equations 3x-y=5 and 3x-y=- 1 have no solution, then the linear function y = 3x-5 is parallel to the image of y=3x+ 1, and vice versa.

Fourthly, solve the binary linear equations by graphic method.

There are generally several steps to solve the binary linear equations by graphic method: (1) rewriting the corresponding binary linear equations into the analytical expression of linear functions; (2) Make the images of these two linear functions in the same plane rectangular coordinate system; (3) Find the coordinates of the image intersection point and get the solution of the binary linear equations.

5. Determine the bending resolution function with binary linear equations.

In practical application, the undetermined coefficient method is often used to construct binary linear equations, so as to determine the analytical formula of linear function.

For example, an airline stipulates that passengers can carry a certain amount of baggage for free, but if the baggage exceeds this amount, they need to buy a baggage ticket. The baggage fee Y (yuan) is a linear function of the baggage weight X (kg). It is understood that Wang Fang brought 30 kilograms of luggage and spent 50 yuan to buy a baggage ticket. Li Gang brought 40 Jin of luggage and bought a baggage ticket of 100 yuan. Then, how many kilograms of luggage can passengers carry for free at most?

Answer: according to the meaning of the question, the analytical formula of linear function can be set to y = kx+b, and then the binary linear equations 50=30k+b, 100=40k+b can be obtained. The solution is k=5 and b=- 100, that is, the analytical formula of linear function is y = 5x- 100. When x=20, y=0. So passengers can carry up to 20 kilograms of luggage for free.

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Commonly used formula

1. Find the k value of the function image: (y 1-y2)/(x 1-x2).

2. Find the midpoint of the line segment parallel to the X axis: |x 1-x2|/2.

3. Find the midpoint of the line segment parallel to the Y axis: |y 1-y2|/2.

4. Find the length of any line segment: √ (x 1-x2) 2+(y 1-y2) 2 (note: the sum of squares of (x1-x2) and (y1-y2) under the root sign).

5. Use a linear function to find the intersection coordinates of two images: solve two functions.

Two linear functions y 1 = k1x+y1= y2 = k2x+B2 make y 1x+b 1 = k2x+b2 replace the solution value of x=x0 back to y1=

6. Find the midpoint coordinates of a line segment connected by any two points: [(x 1+x2)/2, (y 1+y2)/2].

7. Find the first resolution function of any two points: (x-x1)/(x1-x2) = (y-y1)/(y1-y2) (where the denominator is 0 and the numerator is 0).

x y

+,+(positive, positive) is in the first quadrant.

-,+(negative, positive) is in the second quadrant.

-,-(negative, negative) in the third quadrant

+and-(plus or minus) are in the fourth quadrant.

8. If two straight lines y1= k1x+b1∑ y2 = k2x+b2, then k 1=k2, b 1≠b2.

9. If two straight lines y1= k1x+b1⊥ y2 = K2x+B2, then k 1×k2=- 1.

10.

Y=k(x-n)+b is to translate n units to the right.

Y=k(x+n)+b is the translation formula of a linear function that translates n units to the left: right minus left plus (for y=kx+b, only b is changed).

Y=kx+b+n is to translate up by n units.

Y=kx+b-n is a downward translation of n units.

Formula: addition and subtraction (for y=kx+b, only change B) related applications.

1 1.Y line = intersection of kx+b and x axis: (-b/k, 0)

Intersection with y axis: (0, b)

Application in life

1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.

2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t, and the original water quantity s in the pool is set. G = S- feet.

3. When the original length b of the spring (the length when the weight is not hung) is constant, the length y of the spring after the weight is hung is a linear function of the weight x, that is, y=kx+b(k is an arbitrary positive number).