"Relationships Described by Quadratic Functions" Quadratic Function PPT Courseware

Description

"Relationships Described by Quadratic Functions" Quadratic Function PPT Courseware

learning target:

1. Go through the process of exploring and representing quadratic functions, and gain experience in using quadratic functions to express the relationship between variables.

2. Be able to determine whether a function is a quadratic function.

3. Be able to transform a quadratic function into a general form and determine the values ​​of a, b, and c.

Creation & Situation

The radius of the circle is xcm and the area of ​​the circle is ycm². Write the functional relationship between y and x;

A rectangular site is surrounded by a fence with a total length of 60m. Write down the relationship between the site area y (m²) and the length of one side of the rectangle x (m).

There are 100 orange trees in an orchard, and each tree bears an average of 600 oranges. Now we are going to plant more orange trees to increase the yield, but if we plant more trees, the distance between the trees and the amount of sunlight each tree receives will be reduced. According to empirical estimates, every time a tree is planted, each tree will bear an average of 5 fewer oranges.

①What variables are there in the problem? Which of these are independent variables? What are the dependent variables?

② Assuming that x orange trees are planted in the orchard, how many orange trees are there in the orchard? How many oranges does each tree bear on average at this time?

③If the total output of oranges in the orchard is y, then please write the relationship between y and x.

think about it

The bank's savings interest rate changes with time, that is to say, the interest rate is a variable. In our country, the adjustment of interest rates is determined by the People's Bank of China based on the development of the national economy.

Assume that the annual interest rate of RMB one-year regular savings is The expression of y (yuan) (without considering interest tax).

y=100(x+1)²=100x²+200x+100.

Thinking and summarizing

y=-5x²+100x+60000,

y=100x²+200x+100.

Is y a function of x?

y is a linear function of x? Is it an inverse proportional function?

Definition: Generally, a function of the form y=ax²+bx+c (a, b, c are constants, a≠ 0) is called the quadratic function of x.

Small scale chopper

The radius of the circle is 4cm. Assume that when the radius increases by xcm,

The area of ​​the circle increases by ycm².

(1) Write the expression of the functional relationship between y and x;

(2) When the radius of the circle increases by 1cm, √2cm, and 2cm respectively, how much does the area of ​​the circle increase?

Summary and expansion

Several issues that should be noted in the definition:

1. Definition: Generally, a function of the form y=ax²+bx+c(a, b, c are constants, a≠0) is called the quadratic function of x. y=ax²+bx+c(a, b, c is a constant, several different representations of a≠0):

(1)y=ax² --------- (a≠0,b=0,c=0,).

(2)y=ax²+c ------ (a≠0,b=0,c≠0).

(3)y=ax²+bx ---- (a≠0,b≠0,c=0).

2. The essence of the definition is: ax²+bx+c is an integer, the highest degree of the independent variable x is quadratic, and the value range of the independent variable x is all real numbers.

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Update Time: 2024-07-29

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