Generally speaking, the relationship between two variables X and Y can be expressed as a function with the shape of y = kx (where K is a constant and k≠0), then Y is called the proportional function of X..

Proportional function belongs to linear function, but linear function is not necessarily proportional function. Proportional function is a special form of linear function, that is, in the linear function y=kx+b, if b = 0, that is, the so-called "Y-axis intercept" is zero, it is a proportional function. The relationship of the proportional function is expressed as: y=kx(k is the proportional coefficient).

When k > 0 (one or three quadrants), the larger k is, the closer the image is to the Y axis. The function value y increases with the increase of independent variable X.

When k < 0 (24 quadrants), the smaller k is, the closer the image is to the Y axis. When the value of independent variable x increases, the value of y decreases gradually.

[Edit this paragraph] Properties of proportional function

1. domain: r (real number set)

2. Range: R (real number set)

3. Parity check: odd function

4. Monotonicity: When k>0, the image is located in the first and third quadrants, and Y increases (monotonically) with the increase of X; When k < 0, the image is located in the second and fourth quadrants, and y decreases (monotonically decreases) with the increase of x.

5. Periodicity: Not a periodic function.

6. Symmetry axis: straight line, no symmetry axis.

[Edit this paragraph] Solution of proportional resolution function

Let the analytic formula of the proportional function be y=kx(k≠0), and bring the coordinates of known points into the above formula to get k, then the analytic formula of the proportional function can be obtained.

In addition, if you want to find the intersection coordinates of proportional function and other functions, you can combine two known resolving function equations to find their x and y values.

[Edit this paragraph] Image of proportional function

The image of the proportional function is a straight line passing through the coordinate origin (0,0) and the fixed point (x, kx). Its slope is k, and the horizontal and vertical intercepts are 0.

[Edit this paragraph] Practice of the image of proportional function

1. Take a value within the allowable range of x, and calculate the value of y according to the analytical formula.

2. Draw a point according to the values of x and y obtained in the first step.

3. The straight line between the point drawn in the second step and the origin.

[Edit this paragraph] Application of proportional function

The power of positive 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 the image with y = k/x.

① Proportion: two related quantities, one of which changes and the other changes accordingly. 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 the letters X and Y are used to represent two related quantities and K is used to represent their ratio, the (certain) proportional relationship can be used as follows.

(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.

[Edit this paragraph] Definition of inverse proportional function

Generally speaking, if the relationship between two variables X and Y can be expressed by Y = K/X (where K is a constant and k≠0), then Y is said to be an inverse proportional function of X. ..

Because y=k/x is a fraction, the range of the independent variable x is X≠0. And y=k/x is sometimes written as xy=k or y=kx-? .

[Edit this paragraph] Inverse proportional function expression

Y = k/x where x is an independent variable and y is a function of x.

y=k/x=k？ 1/x

xy=k

y=k？ x^- 1

Y=k\x(k is a constant (k≠0, x is not equal to 0).

[Edit this paragraph] Independent variable value range of inverse proportional function

①k≠0； (2) In general, the range of the independent variable X is a real number of x ≠ 0; (3) The range of function y is also all non-zero real numbers.

[Edit this paragraph] Inverse proportional function image

The image of inverse proportional function belongs to hyperbola,

The curve is getting closer to the X axis and the Y axis, but it will not intersect (K≠0).

[Edit this paragraph] Properties of inverse proportional function

1. When k>0, the image is located in the first and third quadrants respectively; When k < 0, the image is located in the second and fourth quadrants respectively.

2. When k>0 is in the same quadrant, Y decreases with the increase of X; When k < 0, y increases with the increase of x in the same quadrant.

K>0, function in x

The domain is x ≠ 0; The range is y≠0.

3. Because in y=k/x(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.

4. In the inverse proportional function image, take any two points P and Q, the intersection points P and Q are parallel lines of the X axis and the Y axis respectively, the rectangular area enclosed with the coordinate axis is S 1, S2 is S 1 = S2 = | k |.

5. The image of inverse proportional function is not only an axisymmetric figure, but also a centrally symmetric figure. It has two symmetry axes y=x y=-x (that is, the bisectors of the first, third and fourth quadrants), and the center of symmetry is the coordinate origin.

6. If the positive proportional function y=mx and the inverse proportional function y=n/x intersect at two points A and B (m the signs of m and n are the same), then the two points A and B are symmetrical about the origin.

7. Let there be an inverse proportional function y=k/x and a linear function y=mx+n on the plane. If they have a common intersection, then b? +4k？ M≥ (not less than) 0.

8. Inverse proportional function y = k/x: asymptote of X axis and Y axis.

[Edit this paragraph] Application example of inverse proportional function

For example, there is a point P(m, n) on the image of 1 inverse proportional function, whose coordinates are two of the unary quadratic equation t2-3t+k=0 about t, and the distance from P to the origin is the root sign 13. Find the analytic expression of inverse proportional function.

Analysis:

To find the inverse proportional resolution function is to find k, so we need to list an equation about k.

Solution: ∫m, n is two of the equation t2-3t+k=0 about t.

∴ m+n=3，mn=k，

PO= root number 13,

∴ m2+n2= 13，

∴(m+n)2-2mn= 13，

∴ 9-2k= 13。

∴ k=-2

When k=-2 and delta = 9+8 > 0,

∴ k=-2 meets the requirements,

Example 2 The straight line intersects the hyperbola located in the second quadrant at two points A and A 1. After passing point A, it is perpendicular to the X axis and Y axis, with vertical feet of B and C respectively, and the area of right-angled ABOC is 6. Find:

(1) Analytical expressions of straight lines and hyperbolas;

(2) The coordinates of point A and A 1 point.

Analysis: AB side and AC side of rectangular ABOC are vertical line segments from point A to X axis and Y axis respectively.

Let the coordinates of point A be (m, n), then AB=|n|, AC=|m|,

According to the area formula of rectangle |m? n|=6。

Example 3: As shown in the figure, there are two points A and C perpendicular to the X axis, and the vertical feet are B and D respectively, connecting OC and OA. Let OC and AB intersect at E, the area of △AOE is S 1, and the area of quadrilateral BDCE is S2. Compare the size of S 1 and S2.

[Edit this paragraph] Mathematical terminology

Pronunciation y

Explain the basic concept of function: Generally speaking, in a change process, there are two variables X and Y, and for each definite value of X, Y has a unique definite value corresponding to it. Then we say that X is an independent variable and Y is a function of X, which is expressed as y = kx+b (where B is an arbitrary constant and K is not equal to 0). 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.

[Edit this paragraph] Basic definition

Variable: the number of changes

Constant: a constant quantity

The independent variable x and the linear function y of x have the following relationship:

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

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 dependent variable, 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.

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

[Edit this paragraph] Related properties

Functional attribute

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, image, intersection and subtraction.

4. When b=0 (y=kx), the image of a linear function 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 is negative reciprocal, two straight lines are vertical; When k and b are the same, the two straight lines coincide.

Image attribute

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

(1) list

(2) tracking points; [Generally, two points are taken and a straight line is determined by two 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, 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, 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 the first, third and fourth 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 the second, third 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 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, but not 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] Expression

Analytical type

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).

[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).

x y

++in the first quadrant

+-In the fourth quadrant

-+in the second quadrant

-In the third 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 to translate n units to the left.

Formula: right minus left plus (for y=kx+b, only change K)

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

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

Formula: increase or decrease (for y=kx+b, only change b)

[Edit this paragraph] Related applications

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).

mathematical problem

First, determine the range of the letter coefficient.

Example 1 If the proportional function is known, 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

Typical example

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.

A school needs to burn some computer CDs. If you burn in a computer company, you need 8 yuan for each CD. If you burn it yourself, you need a 4 yuan for each CD, in addition to renting a burner from 1.20 yuan. Do you want to burn these CDs in the computer company or burn them yourself?

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 3 If the range of x in the linear function y=kx+b is -2≤x≤6, the range of the corresponding function value 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.