In junior high school textbooks, the research on quadratic function is more detailed. Because junior high school students have a weak foundation and limited acceptance, this part of learning is mostly mechanical and difficult to understand in essence. After entering senior high school, especially in the review stage of senior three, we need to further study quadratic functions in order to flexibly use their basic concepts and properties (visualization, monotonicity, parity and boundedness).
First, further understand the concept of function
The definition of function has been described in junior high school. After entering high school, I learned mapping on the basis of learning set, and then I learned the concept of function, mainly to clarify the function from the perspective of mapping. At this time, I can use the functions that students already know, especially quadratic functions, as an example to understand the concept of functions more deeply. Quadratic function is the mapping from set A (domain) to set B (range). : A→B, so that the element y=ax2+bx+c(a≠0) in the set b corresponds to the element x in the set a, which is denoted as? (x)=ax2+ bx+c(a≠0) Here ax2+ bx+c represents the corresponding law, and also represents the image of the element X in the definition domain in the value domain, so that students can have a clear understanding of the concept of function. Students can further deal with the following problems after mastering the marks of function values:
Type I: known? (x)= 2x2+x+2, what? (x+ 1)
Can't you put it here? (x+ 1) is understood as the function value when x=x+ 1, and can only be understood as the function value when the independent variable is x+ 1.
Type ⅱ: Set? (x+ 1) = x2-4x+ 1, what? (10)
This question is understood as, known correspondence law? Next, the image of the element x+ 1 in the domain is x2-4x+ 1. The essence of finding the image of element X in the definition domain is to find the corresponding law.
Generally speaking, there are two methods:
(1) Represents the given expression as a polynomial of x+ 1
(x+ 1) = x2-4x+1= (x+1) 2-6 (x+1)+6, and then x+1 (x)=x2-6x+6
(2) Variable replacement: it has strong adaptability and can be applied to general functions.
Let t=x+ 1, then x = t-1∴ (t) = (t-1) 2-4 (t-1)+1= T2-6t+6. (x)= x2-6x+6
Monotonicity, maximum and image of quadratic function.
When studying monotonicity in senior high school, students must be strictly demonstrated with the definition of monotonicity of quadratic function y=ax2+bx+c in the interval (-∞,-] and [-,+∞), so as to have a strict theoretical basis. At the same time, we should make full use of the intuition of function images and give students appropriate exercises, so that they can gradually and consciously use images to learn quadratic functions.
Type ⅲ: Draw the images of the following functions, and study their monotonicity through the images.
( 1)y = x2+2 | x- 1 |- 1
(2)y = | x2- 1 | 0
(3)= x2+2|x|- 1
Here, students should pay attention to the differences and connections between these functions and quadratic functions. Master the function marked with absolute value as piecewise function, and then draw its image.
Type Ⅳ setting? (x) = x2-2x- 1 The minimum value in the interval [t, t+ 1] is g(t).
Find: g(t) and draw an image with y=g(t)
Solution:? (x)= x2-2x- 1 =(x- 1)2-2。 When x= 1, the minimum value is -2.
When 1∈[t, t+ 1] represents 0≤t≤ 1, g (t) =-2.
When t > 1, g(t)=? (t)=t2-2t- 1
When t < 0, g(t)=? (t+ 1)=t2-2
t2-2，(t & lt0)
g(t)= -2，(0≤t≤ 1)
t2-2t- 1，(t & gt 1)
First, let the students understand the meaning of the question. Generally, a quadratic function has only a minimum value or a maximum value on the real number set r, but when the definition domain changes, the situation of taking the maximum value or the minimum value will also change. In order to consolidate and be familiar with this knowledge, students can be supplemented with some exercises.
For example: y = 3x2-5x+6 (-3 ≤ x ≤- 1), find the range of this function.
The knowledge of cubic and quadratic functions can accurately reflect students' mathematical thinking;
Type ⅴ: Set quadratic function? (x)= ax2+bx+c(a & gt； 0) equation? Two roots of (x)-x = 0 x 1, and x2 satisfies 0.
(i) When X∈(0, x 1), prove X 0, that is? (x)-x > 0。 At this point, it is proved that X.