Generally, the function in the form of y=kx(k is constant, k≠0) is called proportional function, where k is called proportional coefficient.
2. Image of proportional function
Generally speaking, the image of the proportional function y=kx(k is constant, k≠0) is a straight line passing through the origin, which we call the straight line y=kx.
When k>o, the straight line y=kx passes through the first and third quadrants;
When k < 0, the straight line y=kx passes through the second and fourth quadrants.
3. Properties of proportional function y=kx
When k>o, y increases with the increase of x;
When k < 0, k decreases with the increase of x 。
Example 1( 1) If the function y=6x-b2+4 is a proportional function, find the value of b. 。
(2) If the function y=(a-2)x+(3+a) is a proportional function, does y increase or decrease with the increase of x?
(3) If the image of the proportional function y=(2-3a) passes through the second and fourth quadrants, find the value of a. 。
The function in (1)(2) is a proportional function, which should satisfy -b2+4=0 and 3+a=0 respectively, while the proportional function in (3) should satisfy a2= 1 and 2-3a respectively.
The answer (1) depends on the meaning of the question.
-b2+4=0
Solve and get b = 2.
(2) From the meaning of the question, 3+a=0.
∴ a=-3。
The proportional function is y=-5x,
∵-5 & lt； 0.
∴ y decreases with the increase of X.
(3) Judging from the meaning of the question, it is
Get a > from ①; 23,
From ②, A = 1,
∴ a= 1。
Example 2 It is known that y- 1 is directly proportional to x+3, and when x=-2, y=-2.
(1) Write the functional relationship between y and x;
(2) Set points (a+ 1, 1-a) on this function image to find the value of a;
(3) If the range of x is-1≤x≤3, find the range of y. 。
Analysis (1) If this proportional coefficient is K, then the value of K can be obtained from generations y- 1=k(x+3), x=-2 and y=-2.
(2) Substitute x=a+ 1 and y= 1-a into the functional relationship between y and x in (1). (3) x is expressed by an algebraic expression containing y, which is transformed from the functional relationship of (1) and substituted into -65433.
The answer (1) can be set to y- 1=k(x+3), and x=-2 and y=-2 are substituted.
-2- 1 = Yes (-2+3),
∴ k=-3。
∴ y- 1=-3(x+3)，
∴ y=-3x-8。
(2)∵ point (a+ 1, 1-a) is on the image of function y=-3x-8,
∴ 1-a=-3(a+ 1)-8。
Solve and get a=-6.
(3)∫y =-3x-8，
∴ x=-8-y3，
∫- 1≤x≤3，
∴ - 1≤-8-y3≤3，
The solution is-17≤y≤-5,
The value range of ∴ y is-17≤y≤-5.
Example 3 shows that y=y 1-y2, y 1 is proportional to x2, y2 is proportional to 1x, and when x= 1, y =-1; When x=3, y= 17. When x=2, find the value of y.
The analysis can set y 1=k 1x2, y2=k2x, then y=k 1x2-k2x, and then substitute the two pairs of values of x and y to get the values of k 1 and k2 respectively, get the functional relationship between y and x, and finally get the functional value y when x=2.
The answer can be set to y 1=k 1x2 and y2=k2x. Then y= k 1x2-k2x,
According to the meaning of the question, you must
Solution:
∴ y=2x2-3x
When x=2, y=2×22-32=6.5.
1. Fill in the blanks and multiple-choice questions (5 points for each small question, * * * 65 points)
1. Clothing self-employed people have bought a batch of clothes, and the purchase price is already 25% off the original price. He is going to set a new price for this batch of clothes, mark it on the price tag and indicate that the price will be reduced by 20%. In this way, he can still get a net profit of 25%. Then the functional relationship between the new price y and the original price x set by this self-employed person is.
2. Among the following functions, () is a proportional function.
a . y = 12x b . y = 4x c . y = 5x-3d . y = 6x 2-2x- 1
3.a It is known that Y is proportional to X. When x=- 1 and y=-6, the functional relationship between Y and X is.
4. The image of a proportional function is shown in the figure, so the analytical formula of this function is ().
a . y = x b . y =-x c . y =-2x d . y =- 12x
5. Given that y=(m-3) is a proportional function, then m=.
6. Regarding the function y= 12x, the following conclusion is correct ().
A. The function image passes through the point (1, 2) B. The function image passes through the second and fourth quadrants.
C.y increases with the increase of X.D. No matter what value X takes, there is always Y >；; 0
7. Please write the resolution function of an image passing point (1, 4).
8. It is known that the image of the proportional function y=kx(k≠0) passes through the second and fourth quadrants, then ().
A.y decreases with the increase of x.
B.y increases with the increase of x.
C. when x
D. no matter how x changes, y remains the same.
9. In a saturated solution at a certain temperature, there are the following relationships among solute, solvent mass and solubility:
Solute mass solution mass = solubility 100. It is known that the solubility of potassium nitrate 100 is 3 1.6g at 20℃. At this temperature, if x g of water can dissolve potassium nitrate yg, then the functional relationship between Y and X is ().
A.y = 0.3 16x b . y = 3 1.6x c . y = 0.3 16x d . x 0.3 16
10. If the image of the proportional function y=(m- 1) passes through the second and fourth quadrants, the value of m is.
1 1.A(-5, y 1) and B (-2, y2) are both on the straight line y=- 12x, so the relationship between Y 1 and y2 is ().
a . yl≤y2 b . yl = y2 c . yl & lt； y2 D.yl & gty2
12. If the images of the scaling functions y=kx and y=2x are symmetric about x, then the value of k is equal to.
The relationship between the mass m(kg) and the volume y(m3) of 13. A, B and C are shown in the figure (ρ represents the density of matter). As can be seen from the image, the following relationship is correct ().
A.ρA & gt； ρB& gt； ρC b .ρA & gt； ρCB & gt； ρB c .ρA & lt； ρB& lt； ρC d .ρA & lt； ρC & lt； ρB
Second, solve the problem (14 questions 8 points, other questions 9 points, ***35 points)
14.( 1998, Heilongjiang) If y-2 is directly proportional to x+2 and x=0, then y=6. Write the functional relationship between y and x, and draw the functional image.
15. (Nanjing, 2004) The cost y (yuan) of holding a table tennis match in a certain place consists of two parts: one part is the fixed cost b (yuan) such as renting a venue, and the other part is in direct proportion to the number of participants. When x=20, y= 1 600, when x=
(1) Find the functional relationship between y and x;
(2) If there are 50 athletes participating in the competition and all the expenses are shared by the athletes, how much does each athlete need to pay?
16. (Hangzhou, 1994) if y+b is directly proportional to x+a (a and b are constants) and x=3, Y = 5；; When x=2 and y=2,
Find the functional relationship between y and x.
17. (Qinghai, 200 1) The image of function y=k 1x passes through point P (2 2,3), which is symmetrical with the image of function y=k2x. Find the functional relationship among y 1, y2 and x, and draw their images in the same coordinate system.