1, review the concept of inverse proportional function, find the expression of inverse proportional function and draw a picture.
2. Review the changes and properties of inverse proportional function images and use them to solve practical problems.
Introduction: This section continues to review the chapter on inverse proportional function, and first recalls the overall framework of this chapter:
The concept of knowledge point 1 inverse proportional function
Knowledge point 2 determines the relationship of inverse proportional function
Image and drawing of inverse proportional function of knowledge point 3
Knowledge Point 4 Properties of Inverse Proportional Function
Knowledge point 5 Geometric meaning of proportional coefficient K in inverse proportional function
Application of knowledge point 6 inverse proportional function
Review exercises:
1, to determine whether the following functions are inverse proportional functions:
( 1)y = 3/x(2)y =-0.5x(3)y = 2/x-3
(4)y = 3. 14/x(5)y =-4/x2(6)y = 1/3x
The concept of knowledge point 1 inverse proportional function
Generally, a function in the form of y = k/x (k where k is a constant and k≠0) is called an inverse proportional function, where x is an independent variable, y is a function of x, and k is a proportional coefficient.
Note: The key to judge whether a function is an inverse proportional function is whether the product of two variables is constant.
Knowledge point 2 determines the relationship of inverse proportional function
1. Determine the inverse proportional function relationship in practical problems.
The key: carefully examine the question, find out the meaning of the question and find out the equivalence relationship.
2. Determine the inverse proportional function relationship with the undetermined coefficient method.
Three Expressions of Inverse Proportional Function
Image and drawing of inverse proportional function of knowledge point 3
Let the students recall the images and drawing methods of inverse proportional functions y=6/x and y =-6/x, and the teacher asks the quadrant where the images are located and its symmetry, and then shows them with multimedia.
The image of the inverse proportional function is a hyperbola.
When k > 0, the two branches of hyperbola are in the first and third quadrants respectively; On the axial symmetry of y=-x
When k < 0, the two branches of hyperbola are in the second and fourth quadrants respectively, which are symmetrical about y = X.
The two branches of hyperbola are symmetrical about the origin of coordinates.
Knowledge Point 4 Properties of Inverse Proportional Function
When k > 0, the two branches of hyperbola are in the first and third quadrants respectively, and in each quadrant, y decreases with the increase of x;
When k < 0, the two branches of hyperbola are in the second and fourth quadrants respectively, and in each quadrant, y increases with the increase of X. 。
Basic replication:
1. If the function is inverse proportional, m2+3m+ 1=.
2. If the image of the inverse proportional function y= 1-4m/x is located in the second and fourth quadrants, then the range of m is.
3. It is known that point A (2, y 1) and point B (5, y2) are two points on the inverse proportional function Y = 4/X, please compare the sizes of y 1, y2.
If you add point C (-3, y3), how can you compare the sizes? How many ways are there?
Knowledge point 5 Geometric meaning of proportional coefficient K in inverse proportional function
Exercise:
1. As shown in the figure, point P is the point on the image where the inverse proportional function y=2/x, and the PD⊥x axis is in D, then the area of △POD is
2. As shown in the figure, point A and point B are points on the hyperbola y=3/x, passing through point A and point B respectively, and forming vertical line segments with X axis and Y axis. If the shadow area is 1, then s 1+s2=
Application of knowledge point 6 inverse proportional function
1. As shown in the figure, if the images of the linear function Y 1 = x- 1 and the inverse proportional function Y2 = 2/x intersect at points A (2, 1) and B (- 1, -2), then y/kloc
A.x > 2b.x > 2 or-1 < x < 0.
C.- 1 < x < 2 d.x > 2 or x
2. As shown in the figure, it is known that A (-4,2) and B(n,-4) are the two intersections of the image of a linear function and the image of an 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.
Deformation: As shown in the figure, it is known that A (-4,2) and B(n,-4) are the two intersections of the image of a linear function and the image of an inverse proportional function. Connect AO and BO and find S△AOB.
3. In order to prevent "H 1N 1", a school staff fumigated the classroom. It is known that the drug content y(mg) per cubic meter of indoor air is directly proportional to the time x(min) when the drug burns, and after the drug burns, y is inversely proportional to X. Now it is measured that the medicine burns out in 8 minutes, and the drug content in indoor air is 6 mg per cubic meter. Please answer the following questions according to the information provided in the questions:
(1) the functional relationship between y and x, the value range of independent variable x, and the functional relationship between y and x after drug combustion;
(2) The research shows that only when the drug content per cubic meter is lower than 1.6mg can students enter the classroom, so it takes at least minutes for students to return to the classroom after disinfection;
4. As shown in the figure, point A is a point on the inverse proportional function image, the positive semi-axis of AB perpendicular to the X axis is at point B, and c is the midpoint of OB; The image of a linear function passes through point A and point C, and intersects with the Y axis at point D (0, -2). if
(1) Find the analytical expressions of inverse proportional function and linear function;
(2) Observe the image, please point out the value range of X on the right side of the Y axis.
Course summary:
What did you get from this film?
1, in the calculation of the area product of the image combination graph of linear function and inverse proportional function, we should pay attention to choosing the appropriate decomposition method.
2. In the area calculation of function graph, we should make full use of abscissa and ordinate.
3. Understanding of various mathematical thoughts: classification thoughts, inquiry thoughts, transformation thoughts, and combination of numbers and shapes.
Homework after class:
As shown in the figure, the image of linear function y=kx+b and the image of inverse proportional function y=m/x intersect at a (-2, 1) and b (1, n).
(1) Try to determine the expressions of the above inverse proportional function and linear function;
(2) Find the area of ⊿AOB.