"Quadratic Function" PPT Courseware 2

Description

"Quadratic Function" PPT Courseware 2

learning target:

1. Based on the specific situation and existing knowledge and experience, discuss the relationship between two variables and realize the meaning of the quadratic function.

2. Be able to write analytical expressions of some simple functions and determine whether they are quadratic functions.

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. Among them: a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.

Notice:

(1) The left side of the equal sign is the variable y, and the right side is the integer about the independent variable x.

(2) The highest degree of the right side of the equation is 2.

(3) a, b, c are constants, and a≠0.

(It can be without linear terms and constant terms, but it cannot be without quadratic terms.)

(4) The value range of x is any real number.

The general form of a quadratic function:

y＝ax2＋bx＋c (where a, b, c are constants, a≠0)

Special forms of quadratic functions:

When b=0, y=ax2+c

When c=0, y=ax2+bx

When b=0, c=0, y=ax2

Example explanation

Example 1. Which of the following functions are quadratic functions? If yes, indicate the quadratic term coefficient, linear term coefficient, and constant term respectively.

(1)y=3(x-1)²+1 (2)y=x+1/x

(3)s=3-2t² (4)y=(x+3)²-x²

(5)y=1/x²-x (6)v=10πr²

illustrate:

To determine whether a function is a quadratic function, see whether it is simplified into the form y=ax2+bx+c (a, b, c are constants and a≠0).

Example 2. Known function y= (m+3)x m2-7

(1) When m takes on what value, this function is a quadratic function?

(2) When m takes what value, is this function a proportional function?

(3) When m takes on what value, this function is an inverse proportional function?

Do it:

Known function y=(k²- k)x²+kx+√2

(1) When k has a value, y is a linear function of x?

(2) For what value of k, y is a quadratic function of x?

Solution (1) According to the question, k²- k=0

When k=1, y is a linear function of x. k≠0

2) When k²- k≠0, y is a quadratic function of x.

k≠0 and k≠1

Give it a try:

1. Among the following functions, (x, t are independent variables), which ones are quadratic functions? ( )

A y=ax2+bx+c B y2=x2-4x+1

C y=x2 D y=2+ √x2+1

2. The condition that function y=(m-n)x2+ mx+n is a quadratic function is ( )

A, m, n are constants, and m≠0 B, m, n are constants, and n≠0

C, m, n are constants, and m≠n D, m, n are any real numbers

Class summary:

What did you gain from studying this class?

1. 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.

2. Methods and steps to determine whether a function is a quadratic function:

(1) First organize the function so that the right side is the algebraic expression containing the independent variable, and the left side is the dependent variable;

(2) Determine whether the algebraic expression containing independent variables is an integer;

(3) Determine whether the highest degree of an item containing an independent variable is 2;

(4) Determine whether the coefficient of the quadratic term is 0.

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Update Time: 2024-09-02

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