Y=kx (k is any non-zero real number)
Or y=kx+b (k is any non-zero real number and b is any 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. That is: y=kx (k is any non-zero real number)
The image of the proportional function passes through the origin.
Domain: the range of independent variables should make the function meaningful; It should be realistic.
[Edit this paragraph] Properties of linear functions
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 not equal to 0, and k and b are constants).
2. When x=0, b is a function on the Y axis, and the coordinate is (0, b).
3.k is the slope of the linear function y=kx+b, and k = tan θ (the angle θ is the included angle between the linear function image and the positive direction of the X axis, θ ≠ 90).
Form. Take it. Elephant. Pay. negative
4. When b=0, the linear function image becomes a proportional function, which is a special linear function.
5. Function image properties: when k is the same and b is not equal, the images are parallel; When k is different and b is equal, the images intersect; When k and b are the same, the two straight lines coincide.
[Edit this paragraph] 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 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, 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 the first, second and fourth quadrants.
When k < 0, b<0, then the image of this function passes through two, three and four 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 the first and third quadrants and does not pass through the second and fourth quadrants. When k < 0, the straight line only passes through the second and fourth quadrants, but not 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).
[Edit this paragraph] Determine the expression of a 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.
[Edit this paragraph] 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, and the original water quantity s in the pool is set. G = S- feet.
[Edit this paragraph] Common formulas
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).
k b
++in the quadrant
+-In the four quadrants
-+in the second quadrant
-In the three 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.
10. moving x to the left means B+X, and moving x to the right means b-X.
1 1. Move y up to X +Y and move y down to x-y..
(There is a rule. The value of item B is equal to k times the unit moved up, minus the original item B.. )
(I don't want anyone to add here)
Move up: (A is the number of moves) Y = K (X+A)+B.
Y=kX+ak+b
Move down: (a is the number of moves) Y=k(X-a)+b
Y=kX-ak+xb
[Edit this paragraph] 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, then when k
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.
Example 2
A school needs to burn some computer CDs. If you burn in a computer company, you need 8 yuan for each CD. If the school does it by itself, in addition to renting 120 yuan burners, each CD needs to be burned in 4 yuan. Are these CDs cheaper to burn in computer companies or in schools?
This question should consider the range of X.
Solution: let the total cost be y yuan and burn x copies.
Computer company: Y 1=8X
School: Y2=4X+ 120
When X=30, Y 1=Y2.
When X & gt30: 00, y1>; Y2
When x
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).