1. Definition of fraction: If A and B represent two algebraic expressions, and B contains letters, then this formula is called a fraction.
The condition that a fraction is meaningful is that the denominator is not zero, and the condition that the value of a fraction is zero is that the numerator is zero and the denominator is not zero.
2. The basic nature of the fraction: the numerator of the fraction is multiplied by the denominator or divided by the algebraic expression that is not equal to 0, and the value of the fraction remains unchanged.
3. General and approximate scores of scores: the key is to decompose the factors first.
4. Fractional operation:
Law of fractional multiplication: fractional multiplication, the product of molecules is the numerator of the product, and the product of denominator is the denominator.
Law of fractional division: a fraction is divided by a fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor.
Fractional power law: Fractional power should be numerator and denominator respectively.
Addition and subtraction of fractions: addition and subtraction of fractions with the same denominator and addition and subtraction of molecules with the same denominator. Fractions with different denominators are added and subtracted, first divided by fractions with the same denominator, and then added and subtracted.
Mixed operation: The operation sequence is the same as before. It can be simplified by the operation speed.
5. The zeroth power of any number that is not equal to zero is equal to 1, that is; When n is a positive integer,
6. The operation property of positive integer exponential power can also be extended to integer exponential power (m, n is an integer).
(1) The power of the same base:;
(2) the power of power:
(3) the power of the product:
(4) Power division with the same base: (a ≠ 0);
(5) Power of quotient: (); (b≠0)
7. Fractional equation: an equation with a fraction and an unknown number in the denominator-fractional equation.
The process of solving the fractional equation is essentially to multiply both sides of the equation by an algebraic expression (the simplest common denominator) and transform the fractional equation into an integral equation.
When solving a fractional equation, when both sides of the equation are multiplied by the simplest common denominator, the simplest common denominator may be 0, which increases the root, so the fractional equation must be tested.
Steps to solve the fractional equation:
(1) Simplification before simplification (2) Multiply both sides of the equation by the simplest common denominator and turn it into an integral equation; (3) solving the integral equation; (4) Root inspection.
There are two conditions to add a root: one is that its value should make the simplest common denominator 0, and the other is that its value should be the root of the whole equation after removing the denominator.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
What are the steps of applying the equation? (1) trial; (2) setting; (3) column; (4) solutions; (5) answer.
There are several types of application problems; What is the basic formula? There are basically five kinds: (1) Travel problems: basic formula: distance = speed × time, and travel problems are divided into meeting problems and catching up problems. (2) numerical problems should master the representation of decimals in numerical problems. (3) Basic formula of engineering problem: workload = working hours × working efficiency. (4) The countercurrent problem is smooth = static.
8. Scientific notation: The notation of expressing a number as a form (where n is an integer) is called scientific notation.
When an n-bit integer whose absolute value is greater than 10 is expressed by scientific notation, the exponent of 10 is
In scientific notation, when the absolute value is less than 1, the exponent of 10 is the number of zeros before the first non-zero number (including the zero before the decimal point).
Chapter 17 Inverse proportional function
1. Definition:
2. Image: The image of inverse proportional function belongs to hyperbola. The image of inverse proportional function is both axisymmetric and centrally symmetric. There are two axes of symmetry: straight lines y=x and y =-X, and the center of the axis of symmetry is the origin.
3. Properties: When k > 0, the two branches of hyperbola are located in the first quadrant and the third quadrant respectively, and the y value of each quadrant decreases with the increase of x value;
When k < 0, the two branches of hyperbola are located in the second and fourth quadrants respectively, and the y value of each quadrant increases with the increase of x value. ..
4.| k |: represents the area of a rectangle surrounded by a point on the inverse proportional function image and a vertical line segment formed by two coordinate axes and two coordinate axes.
5. The hyperbola of the inverse proportional function needs only one point to be determined, in which positive K falls within the boundary of one or three, x increases and y decreases, and the rectangular area remains unchanged at any point on the image. The symmetry axis is the angular bisector, and the order of x and y can be interchanged.
1, the concept of inverse proportional function
Generally speaking, the function (k is constant, k 0) is called inverse proportional function. The analytical expression of inverse proportional function can also be written as. The range of the independent variable x is all real numbers of x 0, and the range of the function is all non-zero real numbers.
2. Inverse proportional function image
The image of the inverse proportional function is a hyperbola, which has two branches, which are located in the first and third quadrants, or the second and fourth quadrants respectively, and they are symmetrical about the origin. Because of the independent variable x 0 and the function y 0 in the inverse proportional function, its image does not intersect with the x axis and the y axis, that is, the two branches of the hyperbola are infinitely close to the coordinate axis, but they will never reach the coordinate axis.
3. The properties of inverse proportional function
inverse proportion function
Symbol k > 0k<0
draw
y
O x
y
O x
The value range of property ① x is x 0,
The value range of y is y0; ;
2 when k >; 0, the two branches of the function image are
In the first and third quadrants. In each quadrant, y
It decreases with the increase of x ① the range of x is x 0,
The value range of y is y0; ;
② when k
In the second and fourth quadrants. In each quadrant, y
It increases with the increase of x.
4. Determination of inverse proportion of resolution function.
The method of determining sum is, er, the undetermined coefficient method. Because there is only one undetermined coefficient in the inverse proportional function, only a pair of corresponding values or the coordinates of a point on the image can be used to find the value of k, thus determining its analytical formula.
5. The geometric meaning of the inverse proportional coefficient in the inverse proportional function.
As shown in the following figure, if any point P in the inverse proportional function image is perpendicular to the X axis and the Y axis PM, PN, then the area of the rectangular PMON is S=PM PN=.
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Chapter 17 Inverse proportional function
1. Definition: A function with the shape y = (k is a constant, k≠0) is called an inverse proportional function. Other forms xy=k
2. Image: The image of inverse proportional function belongs to hyperbola. The image of inverse proportional function is both axisymmetric and centrally symmetric. There are two axes of symmetry: straight lines y=x and y =-X, and the center of the axis of symmetry is the origin.
3. Properties: When k > 0, the two branches of hyperbola are located in the first quadrant and the third quadrant respectively, and the y value of each quadrant decreases with the increase of x value;
When k < 0, the two branches of hyperbola are located in the second and fourth quadrants respectively, and the y value of each quadrant increases with the increase of x value. ..
4.| k |: represents the area of a rectangle surrounded by a point on the inverse proportional function image and a vertical line segment formed by two coordinate axes and two coordinate axes.