Independent variable x and dependent variable y have the following relationship:

Y=kx+b (k is an arbitrary non-zero real number and b is an arbitrary real number)

It is said that y is a linear function of x at this time.

In particular, when b=0, y is a proportional function of x.

Namely: y=kx (k is any non-zero real number)

Domain: the range of independent variables should make the function meaningful; It should be of practical significance.

Image characteristics and properties of linear functions;

b & gt0b & lt0b = 0

K>0 passes through the first, second and third quadrants, passes through the first, third and fourth quadrants and passes through the first and third quadrants.

The image rises from left to right, and y increases with the increase of x.

K<0 passes through the first, second and fourth quadrants, through the second, third and fourth quadrants, and through the second and fourth quadrants.

The image drops from left to right, and y decreases with the increase of x.

Natural linear function

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

That is: y=kx+b(k≠0) (k is any non-zero real number b, take any real number).

2. When x=0, b is the intercept of the function on the y axis.

3.k is the slope of the linear function y=kx+b, and k=tg angle 1 (angle 1 is the included angle between the linear function image and the positive direction of the x axis).

Form. Take it. Elephant. Pay. negative

Images and properties of linear functions

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

(1) list 【 generally take two points and determine a straight line according to them 】;

(2) tracking points;

(3) The connection can be the image of a function-a straight line. So the image of a function only needs to know two points and connect them into a straight line. (Usually find the intersection of the function image with the X and Y axes)

2. Property: any point P(x, y) on the (1) linear function satisfies the equation: y=kx+b(k≠0). (2) The coordinate of the intersection of the linear function and the Y axis is always (0, b), and the image of the proportional function always intersects the origin of the X axis at (-b/k, 0).

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

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

When y=kx

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, then the image of this function passes through the first, second and third quadrants.

When k>0, b<0, then the image of this function passes through one, three and four quadrants.

When k < 0, b<0, then the image of this function passes through two, three and four quadrants.

When k < 0, b>0, then the image of this function passes through the first, second and fourth quadrants.

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

When b < 0, the straight line must pass through three or four 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 one or three quadrants; When k < 0, the straight line only passes through two or four 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).

Determine the expression of linear function

Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..

(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.

(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, we can list two equations: y 1 = kx 1+b … ① and y2 = kx2+b … ②.

(3) Solve this binary linear equation and get the values of K and B. ..

(4) Finally, the expression of the linear function is obtained.

The Application of Linear Function 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 .. Set the original water quantity in the pool. G = S- feet.

Commonly used formula (incomplete, I hope someone will add it)

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. Find the coordinates of the intersection point of the quadratic function image: Solve the quadratic function.

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).

k b

++in the first, second and third quadrants.

+-in quadrants one, three and four

-+in quadrants one, two and four

-In the second, third and fourth quadrants

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.

App application

The properties of the linear function y=kx+b are: (1) when k >; 0, y increases with the increase of x; (2) When k < 0, y decreases with the increase of x, and the following problems can be solved by using the properties of linear functions.

First, determine the range of the letter coefficient.

Example 1. Given the proportional function, when m = _ _ _ _ _ _ _ _ _ decreases with the increase of x.

Solution: According to the definition and properties of proportional function, M is obtained.

Second, compare the size of x value or y value.

Example 2. Given that points P 1(x 1, y 1) and P2(x2, y2) are two points on the image of linear function y=3x+4, Y 1 >: Y2, then the relationship between x 1 and x2 is ().

A.x 1 & gt； x2 b . x 1 & lt； X2c.x 1 = X2D。 Can't be sure.

Solution: according to the meaning of the question, k = 3>0 and y1>; Y2. according to the property of linear function "when k>0, y increases with the increase of x", x1>; X2. So choose A..

Thirdly, judge the position of the function image.

Example 3. The linear function y=kx+b satisfies kb >；; 0, and y decreases with the increase of x, then the image of this function does not pass ().

A. The first quadrant B. The second quadrant

C. The third quadrant D. The fourth quadrant

Solution: Through kb>0, we know that K and B have the same number. Because y decreases with the increase of x, k

Example 1. A spring, without hanging object 12cm, will extend after hanging the object, and the length of extension is proportional to the mass of the suspended object. If the total length of the spring is 13.5cm after a 3kg object is suspended, find the functional relationship between the total length of the spring and the mass x(kg) of the suspended object. If the maximum total length of the spring is

Analysis: This problem has changed from a qualitative problem in physics to a quantitative problem in mathematics, which is also a practical problem. Its core is that the total length of the spring is the sum of the unloaded length and the loaded extension length, and the range of independent variables can be handled by the maximum total length → maximum extension → maximum mass and practical thinking.

Solution: Set the function as y=kx+ 12 from the meaning of the question.

Then 13.5=3k+ 12, and k=0.5.

The resolution function is y=0.5x+ 12.

From 23=0.5x+ 12: x=22。

The value range of the independent variable x is 0≤x≤22.

The key to baking

The definition, image and nature of the linear function are the C-level knowledge points in the interpretation of the senior high school entrance examination, especially the D-level knowledge points in the interpretation of the senior high school entrance examination. It is often combined with inverse proportional function, quadratic function and equation, equation and inequality, and appears in the senior high school entrance examination questions in the form of multiple-choice questions, fill-in-the-blank questions and analytical questions, accounting for about 8 points. In order to solve this kind of problems, classification discussion, combination of numbers and shapes, equations and inequalities are often used.

Example 2. If the value range of x in the linear function y=kx+b is -2≤x≤6, the corresponding function value range is-1 1≤y≤9. Find the analytical expression of this function.

Solution: (1) If k > 0, the equations can be -2k+b=- 1 1.

6k+b=9

If k=2.5 b=-6, then the functional relationship at this time is y = 2.5x-6.

(2) If k < 0, the equations can be -2k+b=9.

6k+b=- 1 1

If k=-2.5 b=4, then the resolution function at this time is y=-2.5x+4.

The key to baking

This question mainly examines students' understanding of the nature of functions. If K > 0, y will increase with the increase of x; If k < 0, y decreases with the increase of x.

Several types of resolution function.

①ax+by+c=0 [general formula]

②y=kx+b[ oblique]

(k is the slope of the straight line, b is the longitudinal intercept of the straight line, and the proportional function b=0).

③y-y 1=k(x-x 1)[ point inclination]

(k is the slope of the straight line, (x 1, y 1) is the point where the straight line passes)

④ (y-y1)/(y2-y1) = (x-x1)/(x2-x1) [two-point formula]

((x 1, y 1) and (x2, y2) are two points on a straight line)

⑤x/a-y/b=0[ intercept type]

(A and B are the intercepts of a straight line on the X axis and the Y axis, respectively)

Limitations of analytical expressions:

① More requirements (3);

② and ③ cannot express straight lines without slope (straight lines parallel to the X 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 a point.

Inclination angle: The included 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 a, and the slope of the straight line be k=tg(a).

If y=kx(k is constant and k is not equal to 0), then y is called a proportional function of X. 。

Proportional function belongs to linear function and is a special form of linear function.

That is, if the linear function y=kx+b and b=0, it is a proportional function.

Image practice

1. directory

Tracking points

3. Connect (make sure to pass through the origin of the coordinate axis)

Secondly, the image of the proportional function is a straight line passing through the sum of the origin (1, k)[ or (2,2k), (3,3k), etc.].

Others: When k>0, its image (except the origin) is in the first and third quadrants, Y increases with the increase of X.

When k < 0, its image (except the origin) is in the second and fourth quadrants, and y decreases with the increase of x.

Summary: y = kx (k is not equal to 0)

From the point of view of equation, this analytical formula can be determined by giving the coordinates of a point on the proportional function.

If we find the coordinates of the intersection of the proportional function and the linear function, the quadratic function or the inverse proportional function, we will combine the two known equations into a set of equations.

Just find its x and y values.

The power of the proportional function in linear programming problem is infinite.

For example, the slope problem depends on the value of k, and the greater the k, the greater the angle between the function image and the X axis, and vice versa.

Also, Y=Kx is the symmetry axis of Y=K/x image.

1) ratio: two related quantities, one changes and the other changes. If the ratio (that is, quotient) of the two numbers corresponding to these two quantities is certain, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship. ① Represented by letters: If two related quantities are represented by letters X and Y, their ratio is represented by K, (.

(2) the changing law of two related quantities in direct proportion: for direct proportion, y = kx(k & gt；; 0), at this time, y and x expand and contract at the same time, and the ratio remains unchanged. For example, the speed of a car per hour is constant, and the distance traveled is directly proportional to the time spent?

The above manufacturers are certain, so dividend and divisor represent two related quantities, which are in direct proportion. Note: When judging whether two related quantities are directly proportional, we should pay attention to these two related quantities. Although they are also a quantity, they change with the change of another, but the proportion of the two numbers they correspond to is not necessarily, so they cannot be directly proportional. Such as a person's age and weight.

Golden section point

Divide a line segment into two parts so that the ratio of one part to the total length is equal to the ratio of the other part to this part. Its ratio is an irrational number, expressed as a fraction (√5- 1)/2, and the approximation of the first three digits is 0.6 18. Because the shape designed according to this ratio is very beautiful, it is called golden section, also called Chinese-foreign ratio. This demarcation point is called the golden section.