If you want anything else, please leave me a message.
Example 1 The image of the inverse proportional function passes through point (2,5). If the point (1, n) is on the image of the inverse proportional function, the value of n is.
This question examines how to determine the analytical formula through the points on the inverse proportional function image and how to determine the coordinates of the points through the analytical formula.
Because the image of the inverse proportional function passes through the point (2,5), we can substitute the coordinates of the point (2,5), determine the analytical formula by finding k, and then substitute the point (1, n) into the analytical formula to find the value of n, or the property of the inverse proportional function, that is, the product of the horizontal and vertical coordinates of the points on the image is a constant k, which can be directly used to find n, and 2× can be obtained from the meaning of the question.
Fill in 10.
Through the deformation of inverse proportional resolution function, it can be concluded that the product of abscissa and ordinate of points on the inverse proportional function image is constant because k is constant. According to this conclusion, it is easy to get the result of this kind of problem.
Example 2 is shown in Figure 3- 1. It is known that the coordinate of point A is (1, 0), and point B moves in a straight line. When the line segment AB is the shortest, the coordinates of point B are
A. (0,0) BC
This topic examines the knowledge of functions, line segments and right triangles. The combination of numbers and shapes is one of the important mathematical methods.
When the line segment AB is the shortest, AB⊥BO, starting from the point B on the straight line, we can know that ∠ AOB = 45, OA= 1, and the intersection point B is the vertical line of the X axis. According to the isosceles "three lines in one" and the right triangle "the center line of the hypotenuse is equal to half of the hypotenuse", the coordinates of the point B can be easily obtained.
Choose B.
Some students can find out where point B moves, but they can't find out the specific coordinates. Breakthrough method: Knowing the analytical formula of straight line BO, we can get the coordinates of points according to the intersection of two straight lines, then we can get the analytical formula of straight line AB, and we can get the coordinates of the intersection by using the equation.
The key to solve the problem: in the analytical formula of two mutually perpendicular straight lines, the coefficient of the first term is reciprocal, and according to this, the analytical formula of straight line AB can be obtained by combining the coordinates of point A.
A publishing house published a popular science book suitable for middle school students. If the first print run of the book is not less than 5000, the corresponding data between input cost and print run are as follows:
Print run x (quantity) 5000 80001000015000 …
Grade y (yuan) 28500 36000 4 1000 53500 …
(1) after exploring the data in the above table, it is found that the input cost y (yuan) of such reading materials is a linear function of the number of copies x (copies). Find the analytical expression of this linear function (it is not required to write the range of x);
(2) If the publishing house invests 48,000 yuan, how many books can it print?
This topic studies the determination and application of resolution function.
(1) Let the first resolution function be, then the solution is, so the relationship between functions is.
(2) x= 12800 because.
Can print 12800 copies of this book.
The key is to select a suitable pair of values from the data in the table given by the topic and substitute them into the set analytical formula to find the analytical formula.
Example 4 If M, N and P are all on the image of the function (k < 0), then the size relation of is ().
A, B, C, D, D.
This topic investigates the properties of inverse proportional function and compares the function value with the function image.
When k < 0, the image of the inverse proportional function is located in two or four quadrants, and in each quadrant, y increases with the increase of x, which shows that > >,
Choose B.
Some students can't correctly understand the nature of the inverse proportional function image, which is easily misunderstood as "when k < 0, the image is located in the second and fourth quadrants, and y increases with the increase of x". Breakthrough method: not simply judge by nature, but draw an image and judge by a sketch.
The key to solving the problem: when describing the image and properties of the inverse proportional function, because it is a hyperbola, the premise of "in each quadrant" must be explained.
Example 6 Figure 3-2 shows a partial image of a known parabola. If y < 0, the value range of x is
A.- 1 0, > 0, < 0, so < 0, so < 0; Because the point (1, 2) is on a parabola, it can be obtained by substituting (1, 2) into the analytical formula; It can be seen from the image that when X =- 1, the corresponding y is below the x axis, so it is < 0; Parabola and x axis have two intersections, so > 0).
Third, answer questions.
2 1. solution: (1) it is observed that all points are distributed on a straight line, ∴ let (k≠0).
Obtained by undetermined coefficient method
(2) If the daily sales profit is Z, then =
When x=25, z is as high as 225,
Therefore, when the sales price of each product is set as 25 yuan, the maximum daily sales profit is 225 yuan.
22. solution: (1) ∫ s △ FAE: s quadrilateral aoce =1:3, ∴ s △ FAE: s △ foc =1:4.
∫ Quadrilateral AOCB is a square, ∴AB‖OC, ∴△FAE∽△FOC, ∴ AE: OC = 1: 2.
∵ OA = OC = 6, ∴ AE = 3, and the coordinate of point E is (3,6).
(2) Let the analytical formula of the straight line EC be y = kx+b,
The straight line y = kx+b passes through e (3,6) and c (6 6,0).
∴, solution:
The analytical formula of ∴ linear EC is y =-2x+ 12.
23. Solution: (1) Let it be a linear function, and the analytical formula is
At that time,; When = 3,6.
An analytic function of the solution ∴
When represents this analytic function, left ≠ right. ∴: It is not a linear function.
Similarly, it is not a quadratic function.
Suppose it is an inverse proportional function. The analytical formula is. At that time,
The inverse proportional function of the available solution is.
Verification: When =3, it conforms to the inverse proportional function.
Similarly, it can be verified that 4 o'clock, 0 o'clock and 0 o'clock are true.
It can be expressed by an inverse proportional function.
(2) Solution: ① When it is 50,000 yuan, (ten thousand yuan),
∴: The production cost is 4000 yuan lower than that in 2004.
(2) at that time. Ⅷ
∴ (ten thousand yuan)
∴ About 6300 yuan will be invested.
24. Solution: (1)∵,
When x=2.
(2) As shown in the figure, the image is a parabola with an upward opening.
The symmetry axis is x=2 and the vertex is (2, -3).
(3) According to the meaning of the question, x 1 and x2 are two of the equations x2-4x+ 1=0.
∴x 1+x2=4,x 1x2= 1.
∴
25. the solution: (1) is known as OC=0.6, AC=0.6, and the coordinates of point a are (0.6, 0.6).
Substituting y=ax2, we get the analytical formula of a=, ∴ parabola as y=x2.
(2) The abscissas of points D 1 and D2 are 0.2 and 0.4, respectively.
Substituting y=x2, we get the point D 1, and the ordinate of D2 is y 1=×0.22≈0.07, and y2=×0.42≈0.27.
∴ column c1d1= 0.6-0.07 = 0.53, c2d2 = 0.6-0.27 = 0.33,
Because the parabola is symmetrical about the Y axis, the total length of columns required for the fence is:
2(c 1d 1+C2 D2)+oc = 2(0.53+0.33)+0.6≈2.3m .
26. Solution: (1) It is known that the other side of the rectangle is
Then = =, the range of the independent variable is 0 < < 18.
(2)∵ ==
∴ When =9 (0 < 9 < 18), the nursery has the largest area, with the largest area of 8 1.
Another solution: ∫=- 1 < 0, take the maximum value,
∴ when = (0 < 9 < 18), ()