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"The Meaning of Quadratic Function" Quadratic Function PPT Courseware
Review the past and learn the new What is a function?
There are two variables x and y in a certain change process. When the variable x takes a certain value within a certain range, the other variable y always has a unique value corresponding to it.
We call this relationship between two variables a functional relationship.
For the above variables x and y, we call y a function of x. x is called the independent variable and y is called the dependent variable.
Create situations and introduce new lessons
question:
(1) Do you like playing basketball?
(2) Do you know: What is the curve of the basketball route when shooting? How to calculate the height of a basketball when it reaches its highest point?
Discussion and Thoughts:
1. The six faces of the cube are congruent squares. Let the edge length of the cube be x and the surface area be y. Obviously, for every value of x, y has a corresponding value, that is, y is a function of x. Their specific Relationship can express why?
y=6x²
2. What is the relationship between the number of diagonals d and the number of sides n of a polygon?
d=1/2n(n-3)
3. The current annual output of a product in a certain factory is 20 pieces, and it plans to increase production in the next two years. If the output of each year increases by x times compared with the previous year, then the output y of this product two years later will be determined according to the value of x determined by the plan. How should the relationship between y and x be expressed?
y=20(1+x)²
y=20x²+40x+20
Definition of quadratic function:
Generally, a function of the form y=ax2+bx+c (a, b, c are constants, a 0) is called a quadratic function.
1. The highest degree of an independent variable is 2.
2. The coefficient a≠0 of the quadratic term can have no linear term and constant term, but cannot have the quadratic term.
3. The analytical expression of the quadratic function must be an integer.
Among them, x is the independent variable, ax2 is the quadratic term, a is the quadratic coefficient, bx is the linear term, b is the coefficient of the linear term, and c is the constant term.
knowledge application
Which of the following functions are quadratic functions?
(1)y=3x-1 (2)y=3x²
(3)y=3x³+2x² (4)y=2x²-2x+1
(5)y=x-2+x (6)y=x²-x(1+x)
Summary:
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.
Several different representations of y=ax²+bx+c (a, b, c are constants, 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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