Categorical Discussion of Absolute Values 15, Categorical Discussion of Absolute Values

1. When x<3a.

y=(a-x)(3a-x)=3a^2-4ax+x^2=(x-2a)^2-a^2

When x=2a, y is minimal.

0<=x<=1 3a=1 a=1/3

y=-1/9

2. When x=3a.

y=03, when x>3a.

y=(a-x)(x-3a)=-x^2+4ax-3a^2=-(x-2a)^2+a^2<0

Because a>0

In other words, if x>0 is in this case, x must be less than 1

Back to the first scenario at this point.

Classification discussion, the absolute value is directly open if it is positive, and it is negative, and it is preceded by a negative sign.

At x<3a, y=(x-2a) 2-a2, the axis of symmetry is x=2a, at 0 3a, y=-(x-2a) 2+a 2, when 2a=1 2 i.e., a=1 4, the minimum value is f(0)=f(1)=-3a 2=-3 16When 01 4, the minimum value is f(0)=-3a2

1. If x>=3a

y=-(x-2a)^2+a^2

Discuss a, if a>=1 2,x is not in the range of [0,1], so this situation does not exist.

If 1 3 = a

The axis of symmetry of y is x=2a, y increases first and then decreases at [0,1], comparing the values of x=0 and x=1, y=-3a 2 when x=0, and y=4a-3a 2 when x=1, so the minimum value is y=-3a 2

2. If x<3a

y=(x-2a)^2-a^2

If a>=1 2

The axis of symmetry of y is x=2a, y is monotonically decreasing at [0,1], and the minimum value is x=0, y=3a2

If 1 3 = a

The axis of symmetry of y is x=2a, y decreases first and then increases at [0,3a], and the minimum value is the minimum value of y, i.e., x=2a, y=-a2

To sum up the results discussed.

When x>=3a, the minimum value of the function y=-3a2

When x<3a, the minimum value of the function is y=-a2

The absolute value is the distance from the point to the origin of a number on the number line, denoted by "|".to dedicate.|b-a|or |a-b|Represents the distance between a point representing a and a point representing b on the number line.

In mathematics, absolute or modulus | x |, regardless of its sign, i.e., |x |x represents positive x, | x |x means negative x (-x is positive in this case), |0 | 0。For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. The absolute value of a number can be considered as the distance from zero.

The generalization of the absolute value of real numbers occurs in a wide variety of mathematical settings, such as complex numbers, quaternions, ordered rings, fields, and vector spaces to define absolute values. Absolute values are closely related to the concepts of size, distance, and norm in a variety of mathematical and physical environments.

Concept: Absolute value refers to the distance from the point to the origin of a number on the number line, with "|".to dedicate.|b-a|or |a-b|Represents the distance between a point representing a and a point representing b on the number line.

1) The absolute value of any rational number is a number greater than or equal to 0, which is the non-negativity of the absolute value.

2) There is only one number whose absolute value is equal to 0, which is 0

3) There are two numbers whose absolute value is equal to the same positive number, and these two numbers are opposite to each other.

4) The absolute values of two numbers that are opposite to each other are equal.

The absolute value is only positive, which refers to the distance from the point corresponding to the point of a number on the number line to the origin, with "|".to dedicate.|b-a|or |a-b|Represents the distance between a point representing a and a point representing b on the number line.

The absolute value of a number should be the numerical representation of the distance from the point to the origin. For example, the absolute value of 3 is 3, because the distance from the origin of this point representing 3 on the number line is 3. The absolute value of -3 is also 3, because the position of the point -3 on the number line is 3 from the origin.

And so on.

Absolute value, many children who have just entered junior high school, they don't know what the absolute value is? A moment is positive and a time is negative, to put it bluntly, as long as you add an absolute value in front of the symbol, then the number is positive, whether it is a positive number, when you start to explain to the children, as long as you say so, they initially understand this, and it will be easier to say much later.

On the number line, the distance from the point to the origin of a number is called the absolute value of the number. Absolute values are marked with "||".To the table slip group modeling. On the number line, the value that represents the distance between a point of number a and the point of number b is called the absolute value of a-b and is denoted as |a-b|。

On the number line, the distance from a number to the origin is called the absolute value of the number For example, 5 refers to the distance between the point and the origin of the number 5 on the number line, and this distance is 5, so the absolute value of 5 is 5.

The absolute value of a non-negative number (positive number and 0,) is itself, and the absolute value of a non-positive number is its opposite. The absolute values of two numbers that are opposite to each other are equal. The absolute value of a is given by "|".a |" means read as "the absolute value of a".

The absolute value of the real number a is always a non-negative number, i.e. |a |≥0。The absolute values of two numbers that are opposite to each other are equal, i.e., |-a|=|a|。If a is positive, then the |x|x of =a has two values a, such as |x|=3, then x= 3

The absolute value refers to the distance from the corresponding point to the origin of the number of fronts on the number axis, which is called the absolute value of this number, and the absolute value is ".

to dedicate.|On the number line, assuming that a>b, and a>0, and tanzai b>0, then the value representing the distance between the point of number a and the point of number b, read as the absolute value of a-b, is denoted as.

a-b|。

Solution: When a=-4, a+4=0, a-2 0

Original = 0+[-a-2)].

0-a+22-a

When -4 a 2, a + 4 0, a -2 0

Original = a+4+[-a-2)].

a+4-a+2

6 When a=2, a+4 0, a-2=0

Original = A+4+0

A+4 In summary: |a+4|+|a-2|The value of 2-a or 6 or a+4 is discussed in classification.

It is discussed in three cases:

The maximum value greater than or equal to 0.

The minimum value is less than or equal to 0.

Take the middle interval of the two values.

For example: |a+4|+|a-2|value.

a>2a<-4

4<=a<=2

The distance between the point corresponding to a number on the number line and the origin (point o) is called the absolute value of the number. Absolute values can only be non-negative. Algebra Definition: |a|=a（a>0） |a|=-a（a<0） |a|=0（a=0）

Geometric significance. On the number line, the distance from a number to the origin is called the absolute value of the number For example, it refers to the distance between the point and the origin represented on the number line, this distance is 5, so the absolute value is 5, and it refers to the absolute value of the point and the origin represented on the number line.

, this distance is, so the absolute value is.

Algebraic meaning. The absolute value of a positive number and 0 is itself, the absolute value of a negative number is its opposite, and the absolute value of 0 is 0 The absolute value of two numbers that are opposite to each other is equal The absolute value of a is used as "|a |" means read as "the absolute value of a".

The absolute value of a positive number is itself. The absolute value of a negative number is its opposite. , the absolute value is non-negative, 0. The absolute value of 0 is still zero. The absolute value of the particular zero is both his own and his opposite, written |0|=0

The distance between the point corresponding to a number (set to a) and the origin (0) on the number line is called the absolute value of the number and is denoted as |a|。

The absolute value of a positive number is itself.

The absolute value of a negative number is its opposite.

Compared with two negative numbers, the absolute value is larger than the smaller;

The absolute value of 0 is 0.

Algebraic definition: a|=a（a>0）

a|=-a(a<0) (Note: -a is not a negative number)|a|=0（a=0）

To put it simply, today's temperature is 30 degrees, yesterday is 25, tomorrow is 35, today and yesterday, tomorrow are 5 degrees, high 5, low 5 are 5, we say that the absolute difference is 5, from this concept, extend a geometric concept, define an origin, the distance between the point and the origin represented on the number line is the absolute value.

7. Isn't there a math book?

The distance between the point corresponding to a number on the number line and the origin (point zero) is called the absolute value of the number.

Do you understand the definition and nature of absolute values? Explain absolute values in the most scientific way.

1 Algebraic Definition of Absolute Value:

The absolute value of a positive number is itself; The absolute value of a negative number is its opposite; The absolute value of zero is zero.

2 Geometric Definition of Absolute Values:

The distance from the origin of a number is represented on the number line, which is called the absolute value of the number.