Please talk about the "undetermined coefficient method" in the most popular language. It would be better if it comes with examples ~

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 increases with the increase of x increase and decrease. The following problems can be solved by using the properties of linear functions.

1. Determine the value range of the letter coefficient

Example 1.

The proportional function is known

, then when k<0____________ _, y decreases as x increases.

Solution: According to the definition and properties of the proportional function, we get

and m<0, that is,

and

, so

p>

2. Compare the size of x value or y value

Example 2.

It is known that points p1 (x1, y1) and p2 (x2, y2) are once Two points on the graph of function y=3x+4, and y1>y2, then the size relationship between x1 and x2 is (

)

a.

x1>x2

b.

x1

c.

x1=x2

d. Unable to determine

Solution: According to the meaning of the question, we know that k=3>0, and y1>y2. According to the property of a linear function "when k>0, y increases with the increase of x", we get x1>x2. Therefore choose a.

3. Determine the position of the function graph

Example 3.

The linear function y=kx+b satisfies kb>0, and y increases with x If it is large and decreases, the graph of this function does not pass through (

)

a.

The first quadrant

b.

Second Quadrant

c.

Third Quadrant

d.

Fourth Quadrant

Solution: From kb>0, we know that k and b have the same sign. Because y decreases as x increases, k<0. So b<0. Therefore, the graph of the linear function y=kx+b passes through the second, third, and fourth quadrants, but not the first quadrant.

Therefore, choose a

.

Typical example questions:

Example 1.

A spring is 12cm long without an object hanging on it. When an object is hung on it It will stretch out, and the stretched length is proportional to the mass of the hanging object. If a 3kg object is hung, the total length of the spring is 13.5cm, find the distance between the total length of the spring y (cm) and the mass of the hanging object x (kg) The functional relational expression of . If the maximum total length of the spring is 23cm, find the value range of the independent variable The core is that the total length of the spring is the sum of the no-load length and the extension length after loading, and the value range of the independent variable can be handled by the maximum total length → maximum extension → maximum mass and practical ideas.

Solution: According to the meaning of the question, assume that the desired function is y=kx+12

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

∴The analytical formula of the desired 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 discs. If it is burned by a computer company, each disc will cost 8 yuan. If the school burns it by itself, in addition to renting a burner for 120 yuan, each disc will also cost 4 yuan. , I asked if these CDs should be burned by a computer company, or would it be cheaper for the school to burn them themselves?

This question must consider the range of x

Solution: Let the total cost be y yuan and burn x pictures

Computer company: y1=8x

School

: y2=4x+120

When x=30, y1=y2

When x>30, y1>y2< /p>

When x<30, y1

The definition, image and properties of a linear function are C-level knowledge points in the high school entrance examination instructions, especially finding the function according to the conditions in the problem Analytical expressions and finding functions using the method of undetermined coefficients are D-level knowledge points in the high school entrance examination instructions. They are often combined with inverse proportional functions, quadratic functions and equations, systems of equations, and inequalities, with multiple choice questions, fill-in-the-blank questions, and solution questions. This type of question appears in the high school entrance examination questions, accounting for about 8 points. Mathematical thinking methods such as classification discussion, combination of numbers and shapes, equations and transformations are commonly used to solve this type of problem.

Example 2. If a linear function y The value range of x in =kx+b is -2≤x≤6, and the corresponding function value range is -11≤y≤9. Find the analytical formula of this function.

Solution: (1) If k>0, you can formulate a system of equations

-2k+b=-11

6k+b=9

6k+b=9

p>

The solution is k=2.5

b=-6

, then the functional relationship at this time is y=2.5x—6

(2) If k<0, you can formulate a system of equations

-2k+b=9

6k+b=-11

Solve to k= -2.5

b=4, then the analytical formula of the function at this time is y=-2.5x+4

This question mainly tests students’ understanding of the properties of functions. If k> 0, then y increases with the increase of x; if k<0, then y decreases with the increase of x.

Several types of analytic expressions of linear functions

①ax+by+c=0[general formula]

②y=kx+b[slope-intercept formula]

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

③y-y1=k(x-x1)[point-slope type]

p>

(k is the slope of the straight line, (x1, y1) is a point that the straight line passes through)

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

((x1, y1) and (x2, y2) are two points on the straight line)

⑤x/a-y/b= 0[Intercept expression]

(a and b are the intercepts of the straight line on the x and y axes respectively)

Limitations of analytical expression expression:

① There are many required conditions (3);

② and ③ cannot express a straight line without slope (a straight line parallel to the x-axis);