1. Enable students to master concepts, images and properties.

(1) Be able to judge what kind of function is based on the definition, understand the rationality of the restrictions on the base, and have a clear definition domain.

(2) Under the guidance of basic properties, images can be drawn using list tracing methods, and properties can be understood from both the number and shape aspects.

(3) Can use the properties to compare the sizes of certain power numbers, and can use the images to draw images of the shape.

2. Through the study of the nature of conceptual images, students can cultivate their ability to observe, analyze and summarize, and further understand the thinking method of combining numbers and shapes.

3. Through the research on mathematics, students can realize the application value of mathematics and stimulate students' interest in learning mathematics. Make students good at discovering and solving problems in mathematics from real life.

Teaching Suggestions

Textbook Analysis

(1) This study is conducted on the basis that students have systematically learned the concept of functions and basically mastered the properties of functions. It is one of the important basic elementary functions. As a common function, it is not only the first application of the concept and properties of functions, but also the basis for learning logarithmic functions in the future. It is also widely used in life and production practice, so it should be focused on Research.

(2) The teaching focus of this section is to master the images and properties based on understanding the definition. The difficulty is to distinguish the changes in the function value when the base is in and .

(3) is a type of function that is completely unfamiliar to students. How to conduct a more systematic theoretical research on such a function is an important issue faced by students, so it is important to obtain corresponding conclusions from the research process. , but what is more important is to understand the method of systematically studying a type of function, so in teaching, students should especially be allowed to experience the research method so that they can be transferred to the study of other functions.

Teaching Suggestions

(1) According to the textbook, the definition of the definition is a formal definition, that is, what the characteristics of the analytical expression must be, and there cannot be any difference, such as, None of that.

(2) The understanding and recognition of the constraints of the base are also important contents of knowledge. If it is possible, try to let students study the restrictions and requirements on bases and exponents by themselves, and the teacher will provide supplements or explain them with specific examples, because the understanding of this condition is not only related to the understanding and classification discussion of properties, but also to We will learn about the base number in the logarithmic function later, so we must truly understand its origin.

Regarding the drawing of images, although the list drawing method is used, in specific teaching, blind list calculations before drawing points should be avoided, and blind connecting of dots into lines should be avoided. Tables should be listed at key points. Therefore, before drawing the points in the list, you should briefly discuss the properties of the function to obtain a general understanding of the existence range, general characteristics, and changing trends of the image to be drawn, and then This is a guide to list the calculations and draw the points to get the image.

Examples of teaching design

Topics

Teaching objectives

1. Understand the definition, preliminary grasp of images, properties and simple applications.

2. Through the study of images and properties, students can cultivate their ability to observe, analyze and summarize, and further understand the thinking method of combining numbers and shapes.

3. Through the study of the subject, students can master the basic methods of function research and stimulate students' interest in learning.

Key points and difficulties in teaching

The key is to understand the definition and grasp the image and nature.

The difficulty is to understand the impact of the base on the function value.

Teaching tools

Projector

Teaching methods

Inspiring discussion and research style

Teaching process

1. Introducing a new lesson

We have learned exponential operations before. Based on this, today we are going to study a new type of common function——————.

1.6. (Written on the blackboard)

The reason why this type of function is highlighted is that it is a need in real life. For example, let's look at the following question:

Question 1: When a certain cell is *, it changes from 1 * to 2, 2 * becomes 4,... After such a cell * times, we get The number of cells* and between form a functional relationship. Can you write the functional relationship between and?

Answered by students: The relationship between and can be expressed as .

Question 2: There is a 1-meter-long rope. Cut off half of the rope length for the first time, and cut off half of the remaining rope for the second time. ...The remaining length of the rope after cutting it is meters. , try to write the functional relationship between and.

Answered by students: .

In the above two examples, we can see that these two functions are different from the functions we studied earlier. In terms of form, they are in the form of power, and the independent variables are all in the position of the exponent, then Call a function of the form .

1.

The concept of (writing on the blackboard)

1. Definition: A function of the form is called . (Written on the blackboard)

After giving the definition, the teacher will give some explanations on the definition.

2. Some explanations (writing on the blackboard)

(1) Regarding the provisions for:

The teacher first asked the question: Why is it necessary to stipulate that the base is greater than 0 and not equal to 1? (If students find it difficult, they can decompose the problem into what if? For example, at this time, the corresponding function value does not exist in the range of real numbers.

If for is meaningless, if then No matter what value it takes, it is always 1, and there is no need to study it. In order to avoid the occurrence of the above situations,

(2) The definition domain of and (written on the blackboard)

The teacher guides students to review the range of exponents and finds that the exponent can be a rational number. At this time, the teacher can point out that when the exponent is an irrational number, it is also a definite real number. For irrational exponent powers, the properties of rational exponent powers that have been learned are summed up. It is applicable to all algorithms, so the exponential range is expanded to the real number range, so the definition domain is. Another reason for the expansion is to make it more representative and more practical.

About. Judgment of whether it is (written on the blackboard)

We have just understood the requirements of the base number and the exponent respectively. Now let’s understand it from an overall perspective. According to the definition, we know what kind of function is. Please see whether the following function Yes.

(1), (2), (3)

(4), (5).

Students answer and explain the reasons. Comment on the situation and point out that only (1) and (3) are, among which (3) can be written as an exponential image.

Finally, remind students that the definition is a formal definition and must be touched in form. The same will work, and then the problem will be deepened. With the definition domain and the properties of the function initially studied, the key to the study at this time is to draw its image and then summarize the properties in detail.

3. Inductive properties

What method is used to draw the points? The teacher prepares to clarify the properties, and then the students answer the definition domain. :

2. Range:

3. Parity: neither an odd function nor an even function

4. Intercept: None on the axis, It is 1 on the axis.

Regarding properties 1 and 2, we can put them together and ask what role they play (determining the approximate location of the image). We should also prove Article 3. For monotonicity, I suggest looking for some special points, take a look first, and then make a conclusion. The last one is also the basis for drawing the graph of the function (the graph is above the axis and does not intersect with the axis.)

On this basis, teachers can guide students to list and draw points. When picking points, they should also remind students that since there is no symmetry, the values ????should be positive or negative, and because the monotonicity is unclear, the points taken The number of points should not be too small.

Here the teacher can use the computer to list the points and give ten sets of data, and the students can list the points themselves and remind the students when they connect the points to form a line. The changing trend of the image (the smaller the value, the closer the image is to the axis, the larger the value, the faster the image rises), and connects a smooth curve.

2. Image and nature (blackboard writing)

1. How to draw images: list drawing method under the guidance of properties.

2. Sketch:

After drawing the first image, ask students if they need to draw a second one? Is it representative? (The teacher can remind the base that the condition is and , and the value can be divided into two sections.) Let students understand that they need to draw the second one. They might as well take as an example.

At this time, students should be allowed to choose the method of drawing its image. Students should be made aware that list drawing is not the best method, and the image transformation method is simpler. That is, the image between = and is symmetrical about the axis. At this time, the image of is already available, and the conditions for transformation are met. Let the students make their own symmetry, and the teacher draws the picture with the help of a computer to obtain the image in the same coordinate system.

Finally, ask students if they need to draw again. (There may be two possibilities. If the student thinks that there is no need to draw anymore, ask the reason and ask him to tell the nature. If he thinks that he still needs to draw, the teacher can use the computer to draw images like this and compare them together, and then find * **Sexuality)

Since images are characteristics of shapes, we first look at their characteristics from a geometric perspective. The teacher can make a list, as follows:

If students cannot explain the above content, the teacher can appropriately propose an observation angle for students to describe, and then ask students to translate the characteristics of geometry into the properties of functions, that is, Fill in another part of the table with an algebraic description.

After filling in, ask students to follow this example to make another table and fill in the corresponding content. In order to further organize the properties, teachers can propose classification from another angle and organize the properties of functions.

3. nature.

(1) No matter what the value of is, it has a definition domain of , a value range of , and it all passes the point .

　(2) When , it is an increasing function in the domain, and when , it is a decreasing function.

(3) When, , when, .

After the summary, students are reminded to remember the graph of the function. With the graph, the properties can be read from the graph.

Three. Simple application (writing on the blackboard)

1. Exploiting monotonicity ratios. (Written on the blackboard)

After studying its concepts, images and properties of a type of function, the most important thing is to use it to solve some simple problems. First let's look at the following questions.

Example 1. Compare the sizes of the following groups of numbers:

(1) and ; (2) and ;

(3) and 1 . (Writing on the blackboard)

First, let students observe the characteristics of the two numbers. What are the similarities? Have students point out that they have the same base but different exponents. Next, based on this characteristic, what method is used to compare their sizes? Let students associate and propose a method of constructing functions, that is, treating these two numbers as the function values ??of a certain function, and using their monotonicity to compare the sizes. Then take question (1) as an example to give the solution process.

Solution: It is an increasing function on , and

　< . (Written on the blackboard)

The teacher finally emphasized that the process must be clearly written in three sentences:

(1) Construct the function and specify the monotonic interval and corresponding monotonicity of the function.

(2) Comparison of the size of independent variables.

(3) Comparison of function values.

The process of the last two questions is briefly described. Ask students to describe the process following question (1).

Example 2. Compare the sizes of the following groups of numbers

　 (1) and ; (2) and ;

　 (3) and . (Write on the blackboard)

Let students first observe the difference between the numbers in each group in Example 2 and Example 1, and then think about the solution. Guide students to discover that (1) can be written as, so that it can be transformed into a problem with the same base, and then use the method of Example 1 to solve it. For (2), it can be written as, or it can be transformed into a problem with the same base, and (3) The previous method is no longer applicable. Consider new transformation methods and let students think and solve them. (The teacher can remind students that the function value is related to 1, and 1 can be used as a bridge)

Finally, the students say >1.

After solving the problem, the teacher will summarize the method of comparing sizes

(1) Method of constructing functions: The characteristics of numbers are that they have the same base but different references (including those that can be converted into the same base)

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(2) Bridge comparison method: Use special numbers 1 or 0.

Three. Consolidation exercises

Exercise: Compare the size of the following groups of numbers (written on the blackboard)

　(1) and (2) and;

　(3) and; ( 4) with. The solution process is abbreviated

IV. Summary

1. The concept of

2. The image and properties of

3. Simple application

5.