"Using Three Ways to Express Quadratic Functions" Quadratic Function PPT Courseware 4

Description

"Using Three Ways to Express Quadratic Functions" Quadratic Function PPT Courseware 4

learning target

1. Experience the process of expressing the quadratic functional relationship between variables in three ways, and understand the connections and differences between the three ways.

2. Be able to analyze and express the quadratic function relationship between variables and solve problems represented by quadratic functions.

3. Master the different expressions of quadratic functions and study the properties of functions from different aspects.

New course introduction

1. The relationship between the area S and the radius R of the circle in the figure is S=πR²

2. The relationship between daily average temperature and date is as shown in the figure

3. The relationship between the water storage capacity and depth of the reservoir is as follows:

Knowledge explanation

Exploration 1

The perimeter of the rectangle is 12 cm. Let its side length be xcm and its area be ycm². What is the law of how y changes as x changes? Can you express it using functional expressions, tables and images?

(1) Use functional expression to express: y=-x²+6x.

Discuss

(1) In the above question, what is the value range of the independent variable x?

(2) When x takes on what value, the area of ​​the rectangle is the largest? What is its maximum area? How did you get it? Please describe how y changes as x changes.

[Analysis] ⑴∵x is the side length, ∴x is a positive number, x>0.

6-x also takes a positive number, 6-x>0, x<6.

The value range of ∴x is 0＜x＜6.

⑵ First convert the quadratic function y=-x²+6x into the vertex formula y=-x²+6x=-(x²-6x)=-(x²-6x+9-9)=-(x-3)²+9

∴When x=3, y has a maximum value, and the maximum value is 9

When 0

When 3

Exploration 2

The difference between two numbers is 2. Let the larger number be x. How does their product y change with the change of x? Can you express this change using functional expressions, tables and images respectively?

1. Express it as a functional expression: y=x²-2x.

2. Represented in a table:

3． Represented graphically:

4. Answer the following questions based on the above three expressions:

(1) What is the value range of the independent variable x?

(2) What are the symmetry axis and vertex coordinates of the image?

(3) How to describe the change of y as x changes?

(4) Which expression method do you use to answer the above three questions?

Discuss

What are the characteristics of each of the three representations of quadratic functions? What is the connection between them?

The tabular representation of a function can clearly and directly express the numerical correspondence between variables;

The graphical representation of a function can intuitively show the changing process and trend of the function;

The expression of a function can express the relationship between variables more comprehensively, completely and concisely.

Each of these three representation methods has its own advantages and serves different needs. Their connection is that the three ways can be transformed into each other, and expressions can be transformed into tables and images. Each way can be transformed into the other two ways.

Practice in class

1. (Leshan High School Entrance Examination) Suppose a and b are constants, and b>0, and the parabola y=ax²+bx+a2-5a-6 is one of the four images in the figure below, then the value of a is ( )

A.6 or -1 B. -6 or 1

C.6 D.－1

2. (Ordos High School Entrance Examination) It is known that the quadratic function y=-x²+bx+c2. Some of the corresponding values ​​between the function y and the independent variable x are shown in the table, points A(x1, y1), B(x2 , y2) On the graph of the function, when 0

A.y1≥y2 B.y1>y2 C.y1

3. (Zigong·High School Entrance Examination) y=x²+(1-a) x+1 is a quadratic function about x. When the value range of x is 1≤x≤3, y takes the maximum value when x=1, then The value range of the real number a is ( ).

A.a=5 B.a≥5 C.a＝3 D.a≥3

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Update Time: 2024-09-24

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