I. Numbers and algebra
I. Numbers and formulas
1. Addition and multiplication of rational numbers
The addition of the same number is one-sided, and the addition of different numbers is "big" MINUS "small"; The sign is bigger than the sign, and the absolute value is equal to "zero".
The sign of the same sign is negative and the product of a term is zero. Note that "big" minus "small" refers to the absolute value.
2. Merge similar projects
Don't forget the rules for merging similar projects; Only the algebraic sum of the coefficients is found, and the letters and indexes remain unchanged.
3. Rules for deleting and adding brackets
The key to removing brackets and adding brackets is to look at symbols; The parentheses are preceded by a plus sign, and the parentheses remain unchanged;
There is a minus sign in front of the brackets, which will change when the brackets are deleted and added.
4. Single operation
Addition, subtraction, multiplication, division, multiplication (on), three-level operations can be distinguished; The coefficient is calculated at the same level, and the exponential operation is degraded.
5. Fractional hybrid algorithm
Fractional four operations, sequential multiplication, division, addition and subtraction; For multiplication and division at the same level, the division sign must be changed (multiplied); Multiplication simplification, factorization first; The numerator and denominator meet and then operate; The addition and subtraction of denominator should be consistent, and denominator integration is the key; It is not difficult to find the simplest common denominator.
The sign must be changed in two places, and the result is the simplest.
6. Variance formula
The sum of two numbers multiplied by the difference of two numbers is equal to the square difference of two numbers; Product and difference are two terms, and complete square is not it.
7. Complete square formula
The first square and the last square, the middle is twice that of the first and the last square; The squares of sum are added and then added, and the squares of difference are subtracted and then added.
8. Factorization
By the way, two sets and three groups, cross multiplication is also counted; None of the four methods works, so we have to split the items and add items to reorganize them; There is no hope of reorganization,
Substitution or remainder calculation; Flexible selection of multiple methods, based on the results of continuous multiplication; If the same type of multiplication occurs, this ability means remembering.
Pay attention to mention (common factor formula) two sets (formulas)
9. Factorization of quadratic trinomial
First, think of a completely flat way, followed by cross multiplication; Neither method works, so try to find a root decomposition.
10. Ratio and proportion
The division of two numbers is also called ratio, and the equality of two ratios is called ratio; The first basic attribute, external products and other internal products;
The ratio of the front and back terms and the back terms, and the composition ratio is called the ratio; The composition ratio of the difference between the preceding item and the latter item is the score ratio;
The sum of two items is not as good as two items, and the proportion is equal. The sum of the preceding paragraph is equal to the sum of the following items, and the proportion remains unchanged, which is called equal proportion;
The agreed variables are directly proportional and the product variables are inversely proportional; Judging that four numbers are proportional, the product of two ends is the intermediate product.
1 1. Radical and unreasonable expression
An algebraic expression represen a square root can be called a radical; The radical form is different from the irrational form, and there is no limit to the way it is opened;
Unreasonable types are all radical types, which are distinguished by signs; Only when there are letters in the opened way can it be called unreasonable.
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There are three conditions for the simplest root formula: the denominator contains no sign, the power exponent (number) and the root exponent (number) are coprime, and the power exponent is a little smaller than the root exponent.
II. Equality and inequality
1. Solving linear equations with one variable
Unknown cause separation is known, and separation method is shifting. Addition, subtraction and shift terms should change their signs, and multiplication and division should be reversed.
Remove the denominator first and then the brackets, and move items to merge similar items; The coefficient of 1 is not enough, and the later generation value will be calculated.
2. Solve one-dimensional linear inequality
Remove the denominator and brackets, and change the sign when moving the item; Similar items are merged, and then the coefficients are removed;
Don't forget to change the direction of inequality when both sides are divided by (divided by) negative numbers.
3. Solve one-dimensional absolute value inequality
The big fish takes both sides, and the small fish takes the middle.
4. Solve one-dimensional linear inequalities.
Take the big one, take the small one; Take the big and the small, and there is nowhere to find them.
5. Solve the fractional equation
Multiply the simplest common denominator and write it clearly with algebraic expressions; After the solution is obtained, the root must be tested, the original (root) remains, and the addition (root) is not clear.
6. Solve a quadratic equation with one variable
The equation has no linear term, so it is ideal to find the root directly. If there is no constant term, there is no room for discussion on factorization;
B and c are equal to zero, and the root is also zero. Don't forget; B and c are not both zero, factorization or formula;
You can also set the formula directly and choose a good prescription according to the topic.
7. Solving quadratic inequality in one variable
Firstly, it is transformed into a general formula to construct the second station of the function; If the discriminant value is not negative, the horizontal axis of the curve has an intersection point;
A is to open it, if it is greater than zero, take it from both sides; If the algebraic expression is less than zero, the number of intersections of the solution set;
If the equation has no real root, the big zero solution in the mouth is all; If it is less than zero, there is no solution, and the opening is just the opposite.
ⅲ. Function
Coordinates on the 1. coordinate system
Coordinate plane points (x, y), transverse to the front and longitudinal to the back; Y is 0 on the x axis, and x is 0 on the y axis.
Quadrant bisector has its own characteristics. One, three horizontal and one vertical are equal, and two, four horizontal and one vertical are just the opposite.
A straight line parallel to an axis, the coordinates of points are specific; Parallel to the X axis, with different longitudinal and transverse dimensions; Parallel to the Y axis, horizontal, equal and vertical are different.
Remember the coordinates of the symmetrical point, and don't confuse the opposite position; X axis symmetry is opposite to Y, and Y axis symmetry is opposite to X; It is best to remember that the origin is symmetrical, and the abscissa and ordinate are signed.
2. The value of the function independent variable
Fraction denominator is not zero, and it must be negative under even roots; The base of the zeroth power is not zero, and both algebraic expressions and odd roots will do.
3. Judge the proportional function:
Judge the proportional function and test it in two steps; One quantity means another quantity, yes or no; If there is, it depends on the value. All real numbers must be there.
4. The proportional function () images and attributes
The proportional function is very simple, and the straight line passes through the origin; K is positive one, three, negative two and four, and the changing trend is in the heart;
K is low on the left and high on the right, and the climbing direction is the same; K is negative, the left is high and the right is low, and the mountain is one big and one small.
5. Inverse proportional function () images and attributes
Inverse hyperbola, all except the origin; K is positive one, three, negative two, four and two axes are its asymptotes;
K is high on the left and low on the right, sliding down the mountain in one or three quadrants; K is negative, low left and high right, and the second and fourth quadrants are like climbing mountains.
6. Linear function () images and attributes
The linear function is a straight line, and the image passes through three quadrants; Two coefficients, k and b, whose functions cannot be underestimated;
K is right oblique, X increases or decreases and Y increases or decreases; K is negative to the lower left, and the change law is just the opposite;
K is the included angle of slope, and b intersects the y axis; The greater the absolute value of k, the farther the straight line is from the horizontal axis.
7. Linear function () images and attributes
Quadratic function appears when the quadratic equation changes from zero to y; All real numbers define the domain, and the image is called parabola;
Parabola has an axis of symmetry, and both sides are monotonously opposed; Openings, vertices and intersections that determine the appearance of the image;
The opening and size are broken by a, and the c axis and the y axis intersect; The symbol of B is special, and the symbol is associated with A;
Vertex is either high or low. The height is conspicuous. If you want to draw a parabola, you can also track the point by translation.
Extract the formula and set the vertex, and then choose two ways. It's not difficult to translate. Draw a basic parabola first.
After drawing the list, connect the lines and keep the translation rules in mind. Add left and subtract right in brackets, and add and subtract extra numbers.
8. Trigonometric functions
Increase and decrease of trigonometric function: positive increase and residual decrease.
Special trigonometric function values (30 degrees, 45 degrees, 60 degrees) memory: sine (value), cosine (value) denominator 2, tangent (value), cotangent (value) denominator 3.
Second, space and graphics.
Ⅰ. Lines and angles
1. Lines, rays and line segments
Linear rays are related to line segments and similar shapes; The length of a straight line is uncertain and can extend to both sides indefinitely;
The ray has only one endpoint and extends in a straight line in the opposite direction; The two ends of the line segment are fixed in length and extend in two directions to form a straight line.
The alignment of two points is * * *, and the formation of graphics is the most common.
2. Angle
Starting from one point, two rays form a figure called an angle; The opposite direction of the * * line is a right angle, and half of the right angle is called a right angle;
The right angle is twice that of the fillet, and the one smaller than the right angle is called the acute angle; The straight plane is an obtuse angle, and the flat plane is called the optimal angle;
This sum is called right-angle complementarity, and this sum is called right-angle complementarity.
3. Distance formula between two points
Find the distance between two coaxial points, and the approximate number is it; Two points equidistant from the axis, the same is true for distance calculation;
For any two points on the plane, the horizontal and vertical standard deviations should be found first; The difference square plus square, the distance formula should be kept in mind.
Ⅱ. Plane graphics
Determination of 1. parallelogram
To prove parallelogram, two conditions are needed; The opposite sides of the card are equal, or the opposite sides of the card are parallel;
You can also use a set of opposite sides, which must be equal and parallel;
Diagonal, a treasure, can't run away if you share it equally with each other; It is also useful to have equal diagonals, and you can achieve "two diagonal groups".
2. Determination of rectangle
Any quadrilateral, three right angles form a rectangle; Diagonal lines are divided into equal parts, quadrilateral. It is a rectangle.
Known parallelogram, a right angle is called rectangle; If two diagonal lines are equal, they are naturally rectangles.
3. Determination of diamond shape
Any quadrilateral, four sides are equal to form a diamond; The diagonal of the quadrilateral, the vertical mutual division is a diamond;
It is known that a parallelogram with equal adjacent sides is called a rhombus; If two diagonal lines are perpendicular, it is a diamond in logic.
4. Trapezoidal auxiliary line
Move the trapezoid diagonal, and the two waists will form a line; Move one waist in parallel, with both waists in the "△" position;
Extend the waist intersection a little, there are parallel lines in the "△"; Make two trapezoidal high lines, and the rectangle will be displayed in front of your eyes;
Know the center line of the waist, don't forget to make the center line.
5. Auxiliary lines of triangle
If there is an angular (horizontal) dividing line in the question, it can be vertical on both sides; The perpendicular bisector of the line segment leads to the two ends of the connecting line;
The midpoints of the two sides of the triangle are connected to form a midline; A triangle has a midline, and the midline is doubled.
6. Regular polygons in a circle
Divide a circle equally, the value of n must be greater than three, connect all points in turn, and inscribe a regular n polygon.
7. Proportional line segment in circle
In the case of equal product, change the equal ratio, and find the similarity horizontally and vertically; Not similar, not angry, equal line and equal ratio instead;
Satisfy equal ratio, equivalent product, reference projection and circular power; Parallel lines, scale, and find connecting lines at both ends.
8. Proof of circle
It is not difficult to prove that the radius and diameter are often connected; Chords can be used as the center distance of chords to vertically divide chords;
The diameter is the largest chord of a circle, and the angle of a straight circle stands at the top; If it bisects the chord vertically, the vertical diameter and shooting will affect the ear;
There are also angles related to the circle, don't forget to be related to each other; Perimeter, center of the circle, tangent angle, carefully find the relationship to connect the lines;
The circle angle of the same arc is equal, so it is most commonly used in proving problems; If there is a tangent angle in the circle, it is easy to find the arc;
The circle has an inscribed quadrilateral and the diagonals are complementary. The outer angle is equal to the inner diagonal, and the quadrilateral is inscribed with a circle;
Right angle opposite or * * * chord, try to add auxiliary circle; If you turn the problem around, you can solve it at four o'clock;
Prove that the tangent of the circle and the vertical radius pass through the outer end; There is a * * * point between the straight line and the circle, which proves that the vertical radii are connected;
If the straight line and the circle are not given points, it is necessary to prove that the radius is vertical; A quadrilateral has an inscribed circle, and the sum of the opposite sides is a condition;
If you encounter circle after circle, it is very important to find the right position; The tangents of two circles are common tangents, and the intersections of two circles are common chords.
The tangent is composed of points, and the tangent intersects with n points; When n intersections are vertices, a circumscribed regular N polygon appears.
Regular n polygons are very beautiful, with inscribed circles and circumscribed circles; Both inscribed and circumscribed are unique, and the two circles are concentric circles;
Its graphic axis is symmetrical, and n axes of symmetry all pass through the center of the circle; If the value of n is even, then central symmetry is convenient;
Vertex and radius are the key points in the calculation of regular N-polygon. Change the radius, apome and radius of inscribed circle and circumscribed circle respectively;
Divided into 2n integers of a right triangle, the calculation is simple.
9. Auxiliary lines in geometry
People say that geometry is difficult, and it is difficult in auxiliary lines; Auxiliary line, how to add it? Master theorems and concepts;
We must study hard and find out the rules by experience; There is an angular bisector in the picture, which can be perpendicular to both sides;
You can also look at the picture in half, and there will be a relationship after symmetry; Angle bisector parallel lines, isosceles triangles added;
Angle bisector plus vertical line, try three lines; Perpendicular bisector of a line segment, usually connecting the line to both ends;
It needs to be proved that the line segment is two and a half times, and the extension and shortening can be tested; Two midpoints in a triangle are connected to form a midline;
A triangle has a midline and the midline extends. The parallelogram appears, and the center of symmetry bisects the point;
Make a high line inside the trapezoid and try to translate a waist; It is common to move diagonal lines in parallel to form triangles;
The card is similar, parallel to the line segment, and adding lines is a habit; It is very important to find the line segment by equal product proportional conversion;
Direct proof is difficult, and equivalent substitution is less troublesome; Make a high line above the hypotenuse, which is larger than the middle term.
Calculation of radius and chord length, the distance from the chord center to the intermediate station; If there are all straight lines on the circle, the radius of the center of the tangent point is even;
Pythagorean theorem is the most convenient to calculate tangent length; If you want to prove that it is tangent, carefully distinguish the radius vertical;
Is the diameter, in a semicircle, to connect the chords at right angles; An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely;
There are two strings on the corner of the circle, and the two ends of the strings are connected; Find tangent chord, same arc diagonal, etc. ;
If you want to make a circumscribed circle, make a middle vertical line on each side; Also make an inscribed circle, and the bisector of the inner corner is a dream circle;
If you meet intersecting circles, don't forget to make a chord; Two circles tangent to the inside and outside pass through the common tangent of the tangent point;
If you add a connecting line, the tangent point must be on it; It is less difficult to prove the problem by adding circles at equal angles;
The auxiliary line is a dotted line, so be careful not to change it when drawing; If the graphics are scattered, rotate the experiment symmetrically;
Basic drawing is very important, so you should master it. Pay more attention to solving problems and often sum up methods;
Don't blindly add lines, the method should be flexible; Analysis of the comprehensive selection method will reduce the difficulty no matter how much;
Keep an open mind, study hard and practice hard, and your grades will soar; It is difficult to prove geometric problems, and the key is often the auxiliary line;
Know the midpoint, do the midline, and the director of the midline will look at it twice; The bottom angle divided by half angle is sometimes a long line;
Sum, difference and multiplication of line segments, and the extension of intercepting certificate congruence; Male * * * angle, male * * * edge, implied conditions must be excavated;
Multiple transformation, rotation, translation and folding of congruent graphics; The center line is always connected, and it is easy to be parallel;
Quadrilateral, diagonal and parallel lines with similar proportions; The trapezoidal problem is easy to solve, translate the waist and make a height line;
The two waists are slightly longer, and the diagonal can also be translated; Cosine and cotangent, convenient to have right angles;
Special angles and special edges are solved by using vertical lines; Don't panic when you encounter practical problems, mathematical modeling will help you;
The problems in the circle are not difficult. Take your time. The center distance of chord is the hanging chord, which meets the diameter of fillet company;
The centers of tangent points are closely connected, and the tangent line often adds radius; Two circles are tangent to a common straight line, and two circles intersect at a common chord;
Cut the thread, connect the string, and connect two turns and three turns; Basic graphics should be skilled, and complex graphics should be decomposed;
The above rules are universal, flexible and convenient to apply.