inverse proportion function

Selected questions of "Inverse Proportional Function" in 2007 senior high school entrance examination

First, multiple-choice questions:

(Chenzhou City, 2007) The value range of the independent variable in the function y= is ()

A.0 B. 2 C. -2 D. =2

(Nanchang, 2007) For the inverse proportional function, the following statement is incorrect ()

A. The point is on its image B, and its image is in the first and third quadrants.

C. when, it increases with the increase of d, and when, it decreases with the increase of.

(Hebei Province, 2007) As shown in the figure, if the image of an inverse proportional function passes through point m (,1), then this inverse proportional function

The expression is ()

A.B.

C.D.

(Huai 'an City, 2007) About the function of image, the following statement is wrong ().

A, passing through the point (1,-1) b, in the second quadrant, y increases with the increase of x.

C is an axisymmetric figure, and the axis of symmetry is the Y axis; D is a central symmetric figure, and the center of symmetry is the origin of coordinates.

(Yueyang City, 2007) In the figure below, the image of the inverse proportional function is roughly (D).

(Lishui, Zhejiang, 2007) If the inverse proportional function is known, the image of this function must be passed.

A.(2， 1) B. (2，- 1) C. (2，4)d .(2)

(Taizhou, 2007) Among the following functions, () decreases with the increase of.

A. () BC

(Jiangxi Province, 2007) For the inverse proportional function, the following statement is incorrect ()

A. The point is on its image B, and its image is in the first and third quadrants.

C. when, it increases with the increase of d, and when, it decreases with the increase of.

(Wenzhou, 2007) It is known that point P(- 1, a) is on the image of inverse proportional function, so the value of a is ().

A.- 1 B. 1 C. -2 D. 2

(Jinhua City, 2007) Among the following functions, the inverse resolution function of the image passing point (1,-1) is () b.

A, B, C, D,

Huzhou (2007) in the following four points, the point on the hyperbola y= is ().

a 、( 1， 1) B 、( 1，2) C 、( 1，-2) D 、( 1，2)

(Nanjing, 2007) The image of the inverse proportional function (constant,) is located at ().

A. First and second quadrants B. First and third quadrants

C. Second and fourth angular limits D. Third and fourth quadrants

(Lushunkou District, 2007) After the point moves down 1 unit and falls on the image of the function, the value is ().

A.B. C. D。

(Shi Yan, 2007) According to the research results of physicist Boyle 1662, the product of the pressure p(pa) in a balloon and its volume v(m3) is a constant k, that is, PV = k (k is a constant, k > 0), and the following image can correctly reflect the functional relationship between p and v is ().

(Binzhou, 2007) As shown in Figure 5, the point is a moving point on the inverse proportional function, the axis is at this point, and the area is, then the image of the function is ().

(Jingzhou City, 2007) As shown in the figure, the symmetrical center of a square ABCD with a side length of 4 is the coordinate origin O, AB‖ axis and BC‖ axis, and the inverse proportion function and the image intersect with the sides of the square ABCD, then the sum of the areas of the shaded parts in the figure is ().

A.2 B.4 C.6 D.8

(Taian, 2007) It is known that three points are all on the image of inverse proportional function. If, then the following formula is correct ().

A.B. C. D。

(Lin Yi, 2007) It is known that the image of the inverse proportional function is in the second quadrant and the fourth quadrant, and there are two points on the image of the function, so the relationship with the size is ().

A.b.c.d is not sure.

(Lushunkou District, 2007) Draw approximate images of functions and functions in the same coordinate system, and the correct function image is ().

(Zunyi City, 2007) In the following figure, the shadow area is 1 ().

(Shenzhen, 2007) In the same rectangular coordinate system, the image of function sum is roughly ().

There are six points in the plane rectangular coordinate system (Guiyang 2007),,,, five of which are on the same inverse proportional function image, and the point that is not on this inverse proportional function image is ().

A.b., c.d.

(Zhuzhou City, 2007) As shown in the figure, the images of linear function and inverse proportional function intersect at two points A and B. If one intersection point is known as A(2 1), the coordinate of the other intersection point B is ().

A.B.

C.D.

(Mianyang City, 2007) If A(a? 1, b 1), B(a2, b2) are two points on the inverse proportional function image, and A 1 < A2, then the relationship between b 1 and b2 is

A.b 1b2？ D. Scale uncertainty

(Yiyang City, 2007) It is known that the image corresponding to the direct proportion function and the inverse proportion passes through the point (2, 1), so the values of sum are () respectively.

A.=，=2 B. =2，= C. =2，=2 D. =，=

(Foshan, 2007) If it is the radius of the bottom of the cylinder, it is the height of the cylinder. When the lateral area of a cylinder is constant, the image of its functional relationship with the cylinder is roughly ().

(Huanggang City, 2007) It is known that the life of the monitor of a certain brand computer is about hours, and the working days of the monitor are d (days), and the average daily working hours are t (hours), so the image that can correctly represent the functional relationship between d and t is ().

Meishan City (2007) as shown in the figure, two points on the image are inverse proportional functions, both perpendicular to the axis, and the extension lines with vertical feet intersect with this point. If the coordinates of are, the ratio of the areas of is ().

A.B. C. D。

(Ningbo City, Zhejiang Province, 2007) As shown in the figure, it is a linear function y=kx+b and an inverse proportional function y=, so the solution of the equation kx+b= about x is ().

(A)xl= 1，x2=2 (B)xl=-2，x2=- 1

(C)xl= 1，x2=-2 (D)xl=2，x2=- 1

(Weifang City, 2007) Let the function be at any point on the first quadrant image, and the symmetrical point of this point about the origin is, if it is parallel to the axis, if it is parallel to the axis, if it intersects with this point, then the area of ().

A. equal to 2 b equals 4

C equals 8 d, which varies with the number of points.

(Zhuzhou City, 2007) As shown in the figure, the images of linear function and inverse proportional function intersect at point A and point B. If one intersection point is known as A(2 1), the coordinate of the other intersection point B is ().

A.(2，- 1) B. (-2，- 1)

C.(- 1，-2) D. ( 1，2)

Zhuji (2007) shows that the side length of square ABCD is 1, e, f, g and h are points on each side, and AE=BF=CG=DH. Let the area of the small square EFGH be y and AE be x, then the function image of y about x is roughly ().

(Weihai, 2007) As shown in the figure, a straight line and a hyperbola intersect at one point. The passing point is used as the axis and the vertical foot is used as the point connection. If so, the value is ().

A.B. C. D。

Zhuji (2007) If the voltage across the constant resistor R is 5 volts and the current passing through it is 1 amp, then the image of the current I passing through this resistor changing with the voltage across it is ().

Qingdao (2007) balloon is filled with a certain amount of gas. When the temperature is constant, the pressure P (kPa) of the gas in the balloon is an inverse proportional function of the gas volume V (m3), as shown in the figure. When the pressure in the balloon is greater than 120 kPa, the balloon will explode. For safety reasons, the volume of the balloon should be ().

A. not less than m3B. Less than M3C. Not less than M3D. Less than M3.

Answer: c

Analysis: This topic investigates the inverse proportional function image and its properties. The image of inverse proportional function is a special curve, which consists of two branches, called hyperbola, and its proportional coefficient k is equal to the result of abscissa and ordinate of any point on hyperbola. Because the hyperbola of this question passes through (1.6,60), we can know that the inverse proportional decomposition function is that when the air pressure in the balloon is 120 kPa, that is, Y = 120, x=, so this question chooses C. Here, the proportional coefficient k in the inverse proportional function is designed as:

As shown in the figure, point P is any point on the hyperbola. If the intersection point P is defined as the PA⊥x axis of point A, the PB⊥y axis of point B, and the coordinates of point P are (x, y), then PA =, Pb =.

=PM PN= =

*,∴,∴s=

That is, the vertical line segment with any point on the hyperbola as the coordinate axis, and the area of the rectangle surrounded by two vertical line segments and two coordinate axes is.

The image of inverse proportional function (Zaozhuang, 2007) is shown in the figure. Point M is a point on the function image, MN is perpendicular to the X axis, and the vertical foot is point N. If s △ mon = 2, the value of k is ().

(a) Articles 2 (b) to 2 (c), 4 (d) to 4

Chongqing (2007) as shown in the figure, in the rectangular ABCD, AB = 3, BC = 4, the point P moves on the side of BC, connecting DP, the intersection point A is AE⊥DP, the vertical foot is E, assuming DP =, AE =, the approximate image that can reflect the functional relationship between sum is ().

(A) (B) (C) (D)

Second, fill in the blanks:

(Shuangbai County, 2007) It is known that point A (m, 2) is on a hyperbola, then m =.

(Harbin, 2007) Given the image passing point of the inverse proportional function, the analytical formula of this inverse proportional function is.

(Taizhou, 2007) The coordinate of a point on the inverse proportional function image is.

(Nanchong, 2007) It is known that the image of the inverse proportional function passes through points (3, 2) and (m, -2), so the value of m is _ _.

(Chongqing, 2007) If the image of inverse proportional function (≠0) passes through point A (1, -3), the value is.

Shaoyang (2007) As shown in Figure (4), if the point is a function on the image, then.

Guangzhou (2007) has known that the total land area of Guangzhou is 7434, and the per capita land area S (unit: person) changes with the change of the population n (unit: person) of the city, so the functional relationship between S and N is.

(Shaoguan City, 2007) Please write the inverse proportional function relation of an image in the second quadrant and the fourth quadrant _ _ _ _ _ _ _.

(Wuhu City, 2007) In the case of doing some work on an object, the force F (N) is inversely proportional to the distance S (m) that the object moves in the direction of the force. As shown in the figure, P (5, 1) is on the image, so when the force reaches 10 N, the distance the object moves in the direction of the force is _ _.

(Wuxi, 2007) The image passing point of inverse proportional function, with the value of.

(Xiantao City, Qianjiang City, 2007) As shown in the figure, the image of the inverse proportional function intersects with the straight line at point A and point B, and intersects with the AC‖ axis and BC‖ axis, then the area of △ABC is equal to one area unit.

Xiangtan City (2007) If the image of inverse proportional function is over-point, then.

(Shaoxing, Zhejiang, 2007) Write an analytical formula of inverse proportional function like in the first quadrant and the third quadrant.

(Lianyungang, 2007) Xiaoming's home is far from school, and Xiaoming needs to walk to school, so Xiaoming's walking speed can be expressed as; If the contact area between the weight on the horizontal ground and the ground is 0, then the pressure of the object on the ground can be expressed as; Functional relationships can also represent the relationships between variables in many different situations. Please give another example of 1.

(Shaanxi Curriculum Reform 2007) Among the three vertices, the point that may be on the inverse proportional function image is.

(Zhou Su, 2007) It is known that the point P is on the image of the function (x > 0), and the vertical feet of the PA⊥x axis and the PB⊥y axis are A and B respectively, so the area of the rectangular OAPB is _ _ _ _ _ _ _ _.

(Qingliu County, 2007) It is known that the images of inverse proportional function y= 0 are distributed in the second and fourth quadrants, so y in linear function y = kx-2 is _ _ _ _ _ _ _ (fill in "increase" or "decrease" or "unchanged").

(Meizhou, 2007) The degree of myopia glasses is inversely proportional to the focal length (m) of the lens. It is known that the focal length of 400-degree myopia glasses is 0.25m, so the functional relationship between the degree of glasses and the focal length of lenses is.

(Zhuji, 2007) Xiaoming designed an electronic game: an electronic flea starts from a point P 1 whose abscissa is t (t(t>0), increases 1 in turn according to the law of the abscissa of the point, and jumps to the right on a parabola > 0 to get points P2 and P3. At this time, the area of △P 1P2P3 is.

(Henan Province in 2007) Write an expression of function crossing point (1,-1).

(Wuhan, 2007) As shown in the figure, it is known that the hyperbola (x > 0) passes through the midpoint f of the right-angled OABC side AB, intersects with BC at point E, and the area of the quadrilateral OEBF is 2, then k = _ _ _ _ _ _ _ _ _ _ _ _

(Deyang, 2007) If there are two points on the image of the inverse proportional function, then _ _ _ (fill in "or" or ").

(Yiwu, Zhejiang, 2007) It is known that the image of the inverse proportional function passes through point P (A+ 1, 4), then a = _ _ ▲ _ _

(Bazhong, 2007) As shown in Figure 5, if the points are on a hyperbola and the points are symmetrical, the analytical formula of this hyperbola is.

Third, answer questions:

(Yongzhou, 2007) It is known that the images of linear function and inverse proportional function pass through (-2,-1) and (n, 2).

(1) Find the analytical expressions of these two functions.

(2) Draw the image sketches of these two functions.

(Beijing, 2007) In the plane rectangular coordinate system, the image of the inverse proportional function and the image of the inverse proportional function are symmetrical about the axis and intersect with the straight line at this point, so try to find a certain value.

(Leshan City, 2007) As shown in Figure (12), the image of inverse proportional function and that of linear function intersect at two points.

(1) Find the analytical expressions of inverse proportional function and linear function;

(2) Answer according to the diagram: When taking any value, the value of the inverse proportional function is greater than that of the linear function.

(Jingzhou City, 2007) As shown in the figure, D is a point on the image of inverse proportional function, the axis of DE⊥ in E, the axis of DC⊥ in C, and the image of the sum of linear functions passes through point C, which intersect the axes at point A and point B respectively, and the area of quadrangle DCAE is 4, so the value is obtained.

Changzhou (2007) known sum is an inverse proportional function of two points on the image.

( 1);

(2) If there is a point, is there a point on the inverse proportional function image, which makes the quadrilateral with four vertices a trapezoid? If it exists, find out the coordinates of the point; If it does not exist, please explain why.

(Yancheng, 2007) As shown in the figure, Xiaohua designed an experiment to explore the lever balance condition: hanging a heavy object at a fixed position on the left side of the midpoint of a homogeneous wooden pole, pulling it down with a spring scale on the right side of the midpoint, changing the distance (cm) between the spring scale and the point, and observing the change of the pointer (n) of the spring scale. The experimental data are recorded as follows:

(cm) 10

15 20 25 30

30 people

20 15 12 10

(1) Take the corresponding values in the above table as the coordinates of points, draw the corresponding points in the coordinate system, connect these points with smooth curves and observe the obtained images, guess the functional relationship between sum and find the functional relationship;

(2) When the pointer of the spring rod is 24N, what is the distance between the spring scale and this point? As the distance between the spring scale and the point decreases, what will happen to the indication on the spring scale?

(Zhongshan, Guangdong, 2007) As shown in the figure, in the rectangular coordinate system, the image of the linear function and the image of the inverse proportional function intersect at two points.

(1) Find the analytical formula of linear function;

(2) the area to be searched.

(Taizhou City, 2007) Through market survey, the demand (kg) of an agricultural and sideline product in a certain area has the following relationship with the market price (yuan/kg) ():

(Yuan/kg)

5 10 15 20

(kg)

4500 4000 3500 3000

It is also assumed that the production quantity (kg) of such agricultural and sideline products in this area is directly proportional to the market price (yuan/kg): (1). Regardless of other factors, if the demand quantity is equal to the production quantity, then the market is in equilibrium at this time.

(1) Please explore the functional relationship between and by tracing, and find out the functional relationship;

(2) According to the above market survey, please analyze: when the market is in a balanced state, what is the market price of the agricultural and sideline products in this area, and what is the total sales income of farmers during this period?

(3) If the functional relationship between the production quantity and the market price changes after the local farmers have finished processing this kind of agricultural and sideline products, but the functional relationship between the demand quantity and the market price remains unchanged, then when the market is in a balanced state, the total sales income of local farmers will increase by 17600 yuan compared with that when the market is not finished processing. What is the market price of this agricultural and sideline products at this time?

(Jining in 2007)

(1) Given that the length and width of rectangle A are 2 and 1 respectively, is there another rectangle B whose perimeter and area are twice that of rectangle A? For the above problems, Xiao Ming solved them from the perspective of "graphics" by using function images. The process of Xiao Ming's argument begins like this: if X and Y are used to represent the length and width of a rectangle, then rectangle B satisfies X+Y = 6 and XY = 4. Please complete the following demonstration process according to Xiao Ming's argument.

(2) Given that the length and width of rectangle A are 2 and 1 respectively, is there a rectangle C whose perimeter and area are half of rectangle A respectively? Xiao Ming thinks this question is affirmative. Do you agree with Xiao Ming? Why?

(Chengdu, 2007) As shown in the figure, the image of the linear function and the image of the inverse proportional function intersect at point A and point B,

(1) Try to determine the expressions of the above inverse proportional function and linear function;

(2) Find the area of △AOB.

(Ziyang, 2007) As shown in Figure 6, it is known that A (-4,2) and B(n,-4) are the two intersections of the image of the linear function y=kx+b and the image of the inverse proportional function.

(1) Find the analytical expressions of inverse proportional function and linear function;

(2) According to the image, write the value range of x that makes the value of linear function smaller than the value of inverse proportional function.

(Shanghai, 2007) As shown in Figure 9, in the rectangular coordinate plane, the image of function (,is a constant) passes through, where the intersection point is the vertical axis, the vertical foot is, the intersection point is the vertical axis, and the vertical foot is, connecting,,.

(1) If the area of is 4, find the coordinates of this point;

(2) Verification:

(3) If, find the resolution function of a straight line.

(Fuzhou, 2007) As shown in the figure, it is known that a straight line and a hyperbola (k > 0) intersect at point A and point B, and the abscissa of point A is 4.

(1) Find the value of k;

(2) If the ordinate of point C on the hyperbola (k > 0) is 8, find the area of △AOC;

(3) another straight line l passing through the origin o intersects the hyperbola (k > 0) at two points p and q (point p is in the first quadrant). If the area of the quadrilateral composed of points A, B, P and Q is 24, find the coordinates of point P. ..

Solution: (1) ∵ The abscissa of point A is 4, ∴ when = 4, = 2.

The coordinate of point A is (4,2).

Point a is a straight line and a hyperbola (k >；; 0),

∴ k = 4 ×2 = 8。

(2) Scheme 1: As shown in figure 12- 1,

∵ Point C is on the hyperbola when = 8, = 1.

The coordinate of point ∴c is (1, 8).

Intersections a and c are perpendicular to the axis, and the vertical feet are m and n, respectively, and a rectangular DMON is obtained.

S rectangle ONDM= 32, S△ONC = 4, S△CDA = 9, S△OAM = 4.

S△AOC= S rectangle ondm-s △ onc-s △ CDA-s △ OAM = 32-4-9-4 =15.

Scheme 2: as shown in figure 12-2,

The intersection points c and a are perpendicular to the axis, and the vertical feet e and f,

∵ Point C is on the hyperbola when = 8, = 1.

The coordinate of point ∴c is (1, 8).

Points c and a are on hyperbola,

∴ S△COE = S△AOF = 4.

∴ S△COE+S trapezoid CEFA = S△COA+S△AOF

∴ S△COA = S trapezoidal CEFA.

∫S trapezoid CEFA = ×(2+8)×3 = 15

∴ S△COA = 15。

(3)∫ The inverse proportional function image is a centrally symmetric graph about the origin o,

∴ OP=OQ，OA=OB。

∴ Quadrilateral APBQ is a parallelogram.

∴ S△POA = S parallelogram APBQ = ×24 = 6.

The abscissa of point p is (> 0 and),

Get a p (,).

The intersection points p and a are perpendicular to the axis respectively, and the vertical feet are e and f,

∵ points p and a are on hyperbola, ∴S△POE = S△AOF = 4.

If 0 < < 4, as shown in figure 12-3,

∫S△POE+S trapezoidal PEFA = S△POA+S△AOF

∴ trapezoid PEFA = S△POA = 6.

∴

.

The solution is = 2, =-8 (truncation).

∴ P(2，4)。

If > 4, as shown in figure 12-4,

∫S△AOF+S trapezoidal AFEP = S△AOP+S△POE

∴ trapezoid PEFA = S△POA = 6.

∴ ,

The solution is = 8, =-2 (truncation).

∴ P(8， 1)。

The coordinates of point P are p (2,4) or p (8, 1).