26. 1 quadratic function (1)
Teaching objectives:
(1) can skillfully list quadratic function relations according to practical problems and find out the value range of function independent variables.
(2) Pay attention to students' participation, combine with practice, enrich students' perceptual knowledge and cultivate students' good study habits.
Key points and difficulties:
According to practical problems, we can skillfully list quadratic function relations and find out the range of function independent variables.
Teaching process:
First, give it a try.
1. Let the length of the side AB perpendicular to the rectangular garden wall be xm. First, take some values of x, calculate the length of BC on the other side of the rectangle, and then get the area ym2. Rectangular. Try to fill in the blanks in the table below.
AB length x (m) 1 234 567 89
BC length (m) 12
Area y (m2) 48
2. Can the value of x be arbitrary? Is there a scope limit?
3. We find that when the length (x) of AB is determined, the area (y) of the rectangle is also determined, and y is a function of X. Try to write the relationship of this function.
For 1. Students can fill in the length and area of BC according to the length of AB given in the table, and then guide students to observe the changes of data in the table and ask: (1) What can you find from the table? (2) What guesses can be made about the answers to the above questions? Let the students think, communicate and express their opinions, and draw the conclusion that when the length of AB is 5cm and the length of BC is 10m, the area of the enclosed rectangle is the largest; The maximum area is 50m2.
For 2, students can discuss and communicate in groups, and then each group sends representatives to express their opinions. To form a * * * knowledge, the value of x cannot be taken arbitrarily, and there is a limited range, and its range is 0 < x < 10.
For 3, the teacher can ask, (1) When AB=xm, what is the length of BC? (2) What is the area y? It is pointed out that y = x (20-2x) (0 < x < 10) is the required functional relationship.
Second, ask questions.
A store sells a commodity in 8 yuan at a price of 10 yuan, and it can sell about 100 pieces a day. This store wants to increase profits by lowering the selling price and increasing the sales volume. Through market research, it is found that the sales volume of this commodity can increase by 65,438+00 pieces for every reduction of unit price by 0.65,438+0 yuan. How much can I reduce the price of this commodity in order to maximize the sales profit?
In this question, you can ask the following questions for students to think about and answer:
1. What is the relationship between the profit of commodities and the selling price, purchase price and sales volume?
[Profit = (selling price-purchase price) × sales volume]
2. If the selling price is not reduced, what is the profit of each commodity? What's the total profit for one day?
[10-8 = 2 (yuan), (10-8) × 100 = 200 (yuan)]
3. If the price of each commodity is reduced by X yuan, what is the profit of each commodity? How many items can you sell in a day?
[( 10-8-x)； ( 100+ 100x)]
4. Can the value of x be arbitrary? If you can't take it at will, ask about its scope.
[The value of X cannot be taken arbitrarily, and the value range is 0≤x≤2]
5. If the daily profit of commodities is Y yuan, find the functional relationship between Y and X. ..
[y =( 10-8-x)( 100+ 100 x)(0≤x≤2)]
The functional relationship y = x (20-2x) (0 < x < 10 =) is transformed into:
y =-2 x2+20x(0 < x < 10)………………………………( 1)
The function relation y = (10-8-x) (100+100x) (0 ≤ x ≤ 2) is transformed into:
y =- 100 x2+ 100 x+20D(0≤x≤2)…………………………(2)
Third, observe; summary
1. Teachers guide students to observe the functional relationships (1) and (2), and ask the following questions for students to think and answer;
(1) How many independent variables are there in the function relation (1) and (2)?
(each 1)
(2) How many polynomials -2x2+20 and-100x2+ 100x+200 are there respectively?
(Quadratic polynomials respectively)
(3) What are the * * * characteristics of the functional relations (1) and (2)?
(all expressed by quadratic polynomials of independent variables)
(4) What are the * * * characteristics of the problem in the guide diagram of this chapter and the problem 2 on page P/KLOC-0?
Let the students discuss, exchange and express their opinions, which can be summarized as follows: When the independent variable X is the value, the function Y takes the maximum value.
2. Definition of quadratic function: A function in the form of Y = AX2+BX+C (A, B, C are constants, a≠0) is called the quadratic function of X, A is called the coefficient of quadratic function, B is called the coefficient of linear term, and C is called the constant term.
Fourth, classroom exercises.
1. (Answer) Which of the following functions is a quadratic function?
( 1)y = 5x+ 1(2)y = 4x 2- 1
(3)y=2x3-3x2 (4)y=5x4-3x+ 1
2.P3 exercise 1, 2.
Verb (abbreviation of verb) abstract
Please describe the definition of quadratic function.
2. Many practical problems can be solved by transforming them into quadratic functions. Please make up an application problem of quadratic functions and write out the function relationship according to the real life.
6. Homework: Omission
26. 1 quadratic function (2)
Teaching objectives:
1. Make students draw an image of y=ax2 by tracing points and understand the related concepts of parabola.
2. Make students experience and explore the process of image properties of quadratic function y=ax2, and cultivate students' good thinking habits of observation, thinking and induction.
Key points and difficulties:
Emphasis: It is the focus of teaching to let students understand the related concepts of parabola and draw the image of quadratic function y=ax2 by tracing points. Difficulties: It is difficult in teaching to draw the image of quadratic function y=ax2 by tracing points and explore the properties of quadratic function.
Teaching process:
First, ask questions.
1, students can recall, how are the properties of linear functions studied?
Draw an image of a function first, and then observe, analyze and summarize the properties of a function.
2. Can we analogously study the properties of quadratic function by studying the properties of linear function? If so, what should I learn first?
The properties of quadratic function can be studied by studying the properties of linear function. First, the image of quadratic function should be studied. )
3. What is the image of a linear function? What is the image of quadratic function?
Second, examples
Example 1. Draw the image of quadratic function y=ax2.
Solution: (1) List: List the corresponding value tables of functions within the value range of x:
x…-3-2- 1 0 1 2 3…
y…9 4 1 0 1 4 9…
(2) Drawing points in rectangular coordinate system: using the corresponding values in the table as the coordinates of points, drawing points in plane rectangular coordinate system.
(3) Connection: Connect the points in turn with smooth curves to obtain an image with function y=x2, as shown in the figure.
Question: Observe the image of this function. What are its characteristics?
Let students observe, think, discuss and communicate. To sum up, it has an axis of symmetry, and the axis of symmetry and the image have a little intersection.
Parabolic concept: Curves like this are usually called parabolas.
Concept of vertex: The intersection of parabola and its symmetry axis is called the vertex of parabola.
Third, do it.
1. Draw the images of functions y=x2 and y=-x2 in the same rectangular coordinate system. Observe and compare two images. What do you find in common? What is the difference?
2. Draw the images of functions y=2x2 and y=-2x2 in the same rectangular coordinate system, and observe and compare the images of these two functions. What can you find?
3. What can you find by comparing the images of the four functions?
For 1, while students draw function images, teachers should guide students at lower and middle levels and discuss how to choose several points when commenting. The similarities and differences between the two function images can be discussed in groups. Communicate, let students express different opinions and reach an understanding. The images of the two functions are all parabolas, both are symmetrical about the Y axis, and the vertex coordinates are (0,0). The difference is that the image of function y=x2 opens up and the image of function y=-x2 opens down.
For 2, teachers should continue to patrol and guide students to draw the characteristics of function images and two function images; Teachers can guide students to use 1 for analogy.
For 3, teachers can guide students to draw a conclusion from the discovery of the same point of 1 and 2 * * *: the images of the four functions are all parabolas, which are all symmetrical about the Y axis, and their vertex coordinates are all (0,0).
Fourthly, induction and generalization.
Functions y = x2, y=-x2, y=2x2, y=-2x2 are special cases of function y=ax2. From the * * features of images with functions y = x2, y = 2x2 and y=-2x2, we can guess:
The image of function y=ax2 is a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
If we want to study the characteristics and properties of function y=ax2 images in more detail, how should we classify them? Why?
Ask the students to observe the images with y = x2 and y = 2x2 and fill in the blanks.
When a>0, parabola y=ax2, opening _ _ _ _, the left side of the symmetry axis, the curve goes from left to right _ _.