Basic Properties of Teaching Content Function Ⅲ —— Range and Maximum of Function

Research on Quadratic Function with Parameters —— Finding the Maximum Value

The teacher once bowed his head in thought.

Grade and subject-senior one mathematics.

Panyu District Luoxi New City High School

Address: Yuexiu District, Huangpu District, Haizhu District, Baiyun District, Panyu District, Guangzhou

Theoretical basis:

1 This section is based on the quadratic function in junior high school, after learning the monotonicity of the function. Its core idea is to judge the relative position of symmetry axis and interval, from which the idea of combining numbers with shapes and the idea of classified discussion can be realized.

Classified discussion is one of the most obvious differences between high school and junior high school.

The combination of numbers and shapes is the most important idea of functional knowledge learning. It is also the most basic mathematical thought in mathematics.

Analysis of learning situation:

1 Students have learned quadratic function in grade three, and the nature of images is the key learning content.

The students in our school belong to four types of students, and their foundation is relatively weak. Although they have studied in junior and senior high schools for a period of time, their grasp of images is not solid. So the image is somewhat difficult, but it is not difficult.

The idea of classified discussion is the most obvious difference between junior high school and senior high school. Although there is some preparation in junior high school, the classified discussion in senior high school is almost everywhere. The application of letters is a standard questions. Abstract discussion of letters will be one of the difficulties for students.

Research objectives:

1 Analyze the maximum value of nonparametric quadratic function in a fixed interval through images.

2. In "Classification by Images", the maximum value of quadratic function with parameters in a fixed interval (the coefficient of the first term contains parameters) is discussed.

The nonparametric quadratic function and the maximum value of indefinite interval are discussed with images.

Activity 1: warm-up. No parameters, familiar with the image of quadratic function.

Example 1: Find the range of the function in the following interval:

( 1) (2) (3)

Activity 2: Discussion with parameters.

The first category: the symmetry axis of the function is not fixed, and the interval is fixed.

Example 2: Find the minimum value of quadratic function f(x)=x2-2ax- 1 in the interval?

Tips:

Analysis: the symmetry axis x=a is a moving straight line, which may be located on the left side of 0, between 0 and 2, or on the right side of 2.

Variant: Find the maximum value of quadratic function f(x)=-x2+4ax-3 in the interval [-2, 1]?

The second category: Functions have fixed symmetry axes and moving intervals.

Example 3: quadratic function f(x)=x2-2x-3 in [-3, a] (a >; What is the maximum value on -3)?

Variant 1: the function y=x2-2x+3 is the most in the interval [0, a].

Value, find the value of x at this time.

Variant 2: Find the maximum value of the function y=x2-2x+3 in the interval [0, a].

Value, find the value of x at this time.

Extension: It is known that f(x)=x2-2x+3 has a maximum value of 3 and a minimum value of 2 on [0, a], and the range of a is found.

Exercise 1: Known

If the minimum value of f(x) is h(t), find the expression of h(t).

Exercise 2: Known

If the minimum value of f(x) is g(t), find the expression of g(t).

Reflection number one,

1, the teacher should give a hint at an appropriate time.

For example, "activity 1" should prompt students to look at the pictures and make good use of them.

Teachers should emphasize students after learning.

For example, after activity 2, after students' research, no matter what messy answers or standard answers, teachers can use the geometric sketchpad, the axis of symmetry or the interval to slide, and the image is vivid at this time.

Reflection 2,

1 Students often leave images in the first kind of questions, thinking that the smallest independent variable is the smallest function value. The combination of numbers and shapes has become a decoration, which shows that students' image consciousness is too weak.

The discussion of parameters is still a completely abstract behavior and language. Students feel unpredictable and mysterious.

Reflection 3,

The combination of numbers and shapes and classified discussion, as two basic mathematical ideas in high school mathematics, almost run through the whole high school textbook and should be infiltrated from time to time in the classroom. Let the students draw more pictures and draw accurate pictures. Making students think is divergent thinking. This is very important to improve students' ability.

Research problem design table-Zeng Chuiyi

Basic Properties of Teaching Content Function Ⅲ —— Range and Maximum of Function

Research on Quadratic Function with Parameters —— Finding the Maximum Value

The teacher once bowed his head in thought.

Grade and subject-senior one mathematics.

Panyu District Luoxi New City High School

Address: Yuexiu District, Huangpu District, Haizhu District, Baiyun District, Panyu District, Guangzhou