Related concepts of 1. score

Let a and b represent two algebraic expressions. If b contains letters, the formula is called a fraction. Note that the value of denominator b cannot be zero, otherwise the score is meaningless.

A fraction with no common factor between numerator and denominator is called simplest fraction. If the numerator and denominator have common factors, they should be simplified and differentiated.

2, the basic nature of the score

(m is an algebraic expression that is not equal to zero)

3. Fraction operation (the algorithm of fraction is similar to that of fraction).

(Sum of different denominators, divided first);

4. Zero index

5. Negative integer index

Pay attention to the operational properties of positive integer power

It can be extended to the exponential power of integers, that is, m and n in the above formula can be O or negative integers.

6. General steps to solve the fractional equation: multiply the simplest common denominator on both sides of the equation, remove the denominator, and become an integral equation. Solving this integral equation .. checking the root is to substitute the root of the integral equation into the simplest common denominator and see if the result is zero. If the result is not 0, it means that this root is the root of the original equation; If the result is 0, it means that this root is an additional root of the original equation and must be discarded.

7. General steps to solve application problems with column fractional equation:

(1) examine the meaning of the question; (2) Set an unknown number (with unit); (3) List the formulas according to the quantitative relationship in the topic, find out the equal relationship and list the equations; (4) Solve the equation and check the root to see if the solution of the equation meets the meaning of the question; (5) Write the answer (there must be a unit).

Direct ratio, inverse ratio, linear function

The first quadrant (+,+), the second quadrant (-,+), the third quadrant (-,-) and the fourth quadrant (+,-);

The ordinate of a point on the X axis is equal to 0; conversely, all points with ordinate equal to 0 are on the X axis, and the abscissa of a point on the Y axis is equal to 0; conversely, all points with abscissa equal to 0 are on the Y axis.

If a point is on the bisector of the first and third quadrants, its abscissa is equal to its ordinate; If a point is on the bisector of the second and fourth quadrants, its abscissa and ordinate are opposite;

If two points are symmetrical about x, the abscissa is equal and the ordinate is opposite; If two points are symmetrical about y axis, the ordinate is equal and the abscissa is opposite; If two points are symmetrical about the origin, the abscissa and ordinate are mutually opposite numbers.

Definition of 1, linear function and proportional function

(1) if y=kx+b(k, b is constant, k≠0), then y is called a linear function of x.

(2) When b = 0, the linear function y=kx+b is y=kx(k≠0). At this point, y is called a direct proportional function of X. ..

Note: Proportional function is a special linear function, and linear function contains proportional function.

2. The image and properties of the proportional function.

(1) The image of the proportional function y=kx(k≠0) is a straight line passing through (0,0) (1,k).

(2) when k >; At 0, y increases with the increase of x, and the straight line y=kx rises from left to right through the first and third quadrants.

When k < 0, y decreases with the increase of x, and the straight line y = kx decreases from left to right through the second and fourth quadrants.

3. Images and properties of linear functions

The image of (1) linear function y=kx+b(k≠0) is a straight line passing through (0, b) (-, 0).

Note: (0, b) is the coordinate of the intersection of the straight line and the Y axis, and (-,0) is the coordinate of the intersection of the straight line and the X axis.

(2) when k >; 0, y increases with the increase of x, and the straight line y=kx+b(k≠0) rises.

When k < 0, y decreases with the increase of x, and the straight line y = kx+b (k ≠ 0) decreases.

4. The influence of the symbols of k and b in the linear function y=kx+b(k≠0, k b is constant) on the image.

( 1)k & gt； 0, b>0 straight line passes through the first, second and third quadrants.

(2)k & gt； 0, b<0 straight line passes through the first, third and fourth quadrants.

(3)k & lt； 0, b>0 straight line passes through the first, second and fourth quadrants.

(4)k & lt； 0, b<0 straight line passes through two, three and four quadrants.

5. Understanding of the coefficients k and b of the linear function y = kx+b.

(1)k(k≠0) is the same, and all straight lines are parallel when b is different, that is, straight lines; Straight line (non-zero, constant)

(2) When k (k ≠ 0) is different and b is the same, all straight lines intersect at a fixed point (0, b) on the Y axis, for example, straight lines y = 2x+3 and y =-2x+3 intersect at a point (0, 3) on the Y axis.

6. Straight line translation: The so-called translation means to move a straight line left and right (or up and down) in parallel. The straight line k obtained by translation remains unchanged, and how many units the straight line translates along the Y axis can be obtained by the formula, where b 1, b2 is the ordinate of the intersection of two straight lines and the Y axis, and how many units the straight line translates along the X axis can be obtained by the formula, where X 1.

7. The relationship between straight line y=kx+b(k≠0) and equations and inequalities.

(1) the straight line y=kx+b(k≠0) is a binary linear equation about y.

(2) Finding the intersection of two straight lines is to solve the equation about X and Y..

(3) If y & gt0kx+b & gt；; 0。 If y < 0, kx+b

(4) One-dimensional linear inequality, y 1≤kx+b≤y2( y 1, y2 is a known number, y 1

(5) the solution set of the unary linear inequality kx+b≤y0 (or kx+b≥y0)( y0 is a known number) is the range of the independent variable corresponding to the ray satisfying y≤y0 (or y≥y0) on the straight line y = kx+b.

8. Determine the conditions of analytic expressions of proportional function and linear function.

(1) Because there is only one undetermined coefficient k in the proportional function y=kx(k≠0), the value of k can be obtained by only one condition (such as a pair of values of x and y or a point).

(2) There are two undetermined coefficients k and b in the linear function y = kx+b, and two independent conditions are needed to determine the two equations about k and b, so as to find the values of k and b.. These two conditions are usually two points or two pairs of values of X and Y. ..

9. Inverse proportional function

(1) inverse proportional function and its image

So, if y is an inverse proportional function of x.

The image of the inverse proportional function is a hyperbola with two branches. The image of inverse proportional function can be drawn by tracing points.

(2) Properties of inverse proportional function

When K>0, the two branches of the image are in the first and third quadrants respectively, and in each quadrant, Y decreases with the increase of X;

When k < 0, the two branches of the image are in two and four quadrants respectively, and in each quadrant, y increases with the increase of x.

(3) Because there is only one undetermined coefficient k in the proportional function, the value of k can be obtained by only one condition (such as a pair of x, y values or a point).

Similar triangles's judgment method:

(1) if de ‖ BC (type A and X), △ADE∽△ABC.

(2) Projective Theorem If CD is the height on the hypotenuse of Rt△ABC (Double Right Angle Graph)

Solving right triangle