Function learning formula
The proportional function is a straight line, and the image must pass through the origin.
The symbol of k is the key, which determines the quadrant of the straight line.
Negative k goes through two or four limits, x increases and y decreases,
The up-and-down translation k is a constant, and the linear is obtained by derivation.
Add b up and subtract b down, and the image passes three limits.
Two points determine a line, and the selection coefficient is the key.
The hyperbola of inverse proportional function can be determined by only one point.
Positive k falls within one or three limits, x increases and y decreases,
At any point on the image, the rectangular area remains unchanged.
The symmetry axis is the angular bisector, and the order of x and y can be interchanged.
Quadratic parabola, choose three points,
Judging the positive and negative opening of a, the size of c, y axis,
The sign of △ is the simplest, and the number of intersections on the X axis,
On the left side of the same sign axis of A and B, the parabola translation A remains unchanged.
Vertex guides the image to turn, which can be changed in three forms.
Matching method plays the most important role.
Regular polygon stunt song
To bisect a circle, the value of n must be greater than three.
Connect all the points in turn and connect a regular N polygon in front of your eyes.
Tangent line consists of dividing points, and the tangent line intersects with n points.
When n intersections are vertices, a circumscribed regular N polygon appears.
Regular n polygons are beautiful, with inscribed circles and circumscribed circles.
Both inscribed circle and circumscribed circle 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, central symmetry is convenient.
Vertex and radius are the key points in the calculation of regular N-polygon.
The radius, apome and radius of inscribed circle and circumscribed circle change respectively,
Divided into 2n integers of a right triangle, the calculation is simple.
Proportional line segment in a circle
In the case of equal product, change the equal ratio, and find the similarity horizontally and vertically;
Not similar, don't be angry, equal line and equal ratio instead,
Keywords encounter equal ratio, variational equal product, reference projection and circular power,
Parallel lines, scale, and find connecting lines at both ends.
Function and sequence
Sequence function is divided into mother and son, and arithmetic difference ratio is self-contained.
What is the sum of series? The idea of general recursion is open;
There is no difference between the separation of variables and the synthesis of functions.
The same increase and different decrease monotony, and the interval digs the maximum.
binomial theorem
How much binomial power knows, the source of Wan Li is the general term;
Expand the trinomial index system, and the combination coefficient is Yang Hui angle.
Divisibility proves that the bottom is wonderful, binomial and unique;
Who is the most symmetrical at both ends? The main peak is all the other peaks that look short in the sky. ..
solid geometry
Multi-point * * * lines cross on both sides, and the multi-line * * * surface is ingenious;
Three vertical upper chords are large in space, and two lower arcs on the sphere are small.
Line-to-line relationship line opposite search, face-to-face angle line table;
Equal product transformation is continuous projection and can cut bridges.
Equality and inequality
Unequal roots of functional equations often lead to parameter ranges;
One positive, two definite and three equal phases, the mean value theorem is the best.
The parameter ratio is uncertain, and the two formulas are different.
There is no absolute equality and inequality, only constantly separated variables.
According to years of practice, sum up the rules and simplify the complex;
Generalization knowledge is hard to change, and high school mathematics is clever to remember.
Concise and catchy, combined with teaching materials is better.
The first meeting will be ugly.
Shorthand formula
I. Settings and functions
Content intersection and complement set, and power exponential pair function.
Parity and increase and decrease are the most obvious observation images.
The appearance of compound function, the discrimination of property multiplication law,
If you want to prove it in detail, you must master the definition.
Exponential function and logarithmic function are reciprocal functions.
Cardinality is not a positive number of 1, and 1 increases or decreases on both sides.
The domain of the function is easy to find. The denominator cannot be equal to 0.
Even roots must be non-negative, and zero and negative numbers have no logarithm;
The tangent function angle is not straight, and the cotangent function angle is uneven;
The real number sets of other functions have intersection in many cases.
Two mutually inverse function have that same monotone property;
The images are symmetrical with y = x as the symmetry axis;
Solve the very regular inverse solution of substitution domain;
The domain of inverse function, the domain of original function.
The nature of power function is easy to remember, and the index reduces the score;
Keywords exponential function, odd mother and odd son odd function,
Even function with odd mother and even son, even mother non-parity function;
In the first quadrant of the image, the function is increased or decreased to see the positive and negative.
Second, trigonometric functions
Trigonometric functions are functions, and quadrant symbols are labeled.
Function image unit circle, periodic parity increase and decrease.
The same angle relation is very important, and both simplification and proof are needed.
At the vertex of the regular hexagon, cut the chord from top to bottom;
The numb 1 records that triangle connecte the vertices in the center;
The sum of the squares of the downward triangle, the reciprocal relationship is diagonal,
Any function of a vertex is equal to the division of the last two.
The inductive formula is good, negative is positive and then big and small,
It is easy to look up the table when it becomes a tax corner, and it is essential to simplify the proof.
Half of the integer multiple of two, odd complementary pairs remain unchanged,
The latter is regarded as an acute angle, and the sign is judged as the original function.
The cosine of the sum of two angles is converted into a single angle, which is convenient for evaluation.
Cosine product minus sine product, angular deformation formula.
Sum and difference products must have the same name, and the complementary angle must be renamed.
The calculation proves that the angle is the first, pay attention to the name of the structure function,
Keep the basic quantity unchanged, from difficult to simple.
Guided by the principle of reverse order, the product of rising power and falling power and difference.
The proof of conditional equality, the idea of equation points out the direction.
Universal formula is unusual, rational formula is ahead.
The formula is used in the right and wrong direction, and the deformation is used skillfully;
1 add cosine to consider cosine, 1 subtract cosine to consider sine,
The power-on angle is halved, and the power-on angle is reduced by one norm;
The essence of the inverse function of trigonometric function is to find the angle.
First find the trigonometric function value, and then determine the angle range;
Using right triangle, the image is intuitive and easy to rename.
The equation of a simple triangle is reduced to the simplest solution set;
Three. inequality
The solution to inequality is to use the properties of functions.
The unreasonable inequality of the opposite side is transformed into a rational inequality.
From high order to low order, step-by-step transformation should be equivalent.
The mutual transformation between numbers and shapes helps to solve problems.
The method of proving inequality is powerful in real number property.
Difference is compared with 0, and quotient is compared with 1.
Comprehensive method with good direct difficulty analysis and clear thinking.
Non-negative common basic expressions, positive difficulties are reduced to absurdity.
There are also important inequalities and mathematical induction.
Graphic function help, drawing modeling construction method.
Fourth, "series"
Arithmetic ratio two series, the sum of n terms in the general formula.
Two finites seek the limit, and four operations are the other way around.
The problem of sequence is changeable, and the equation is simplified as a whole calculation.
It is difficult to sum series, but it is skillful to eliminate dislocation and transform.
Learn from each other's strong points and calculate the sum formula of split terms.
Inductive thinking is very good, just do a program to think about it:
Counting two and seeing three associations, guessing proves indispensable.
There is also mathematical induction to prove that the steps are programmed:
First verify and then assume that 1 is added from k to k,
The reasoning process must be detailed and affirmed by the inductive principle.
Verb (abbreviation of verb) plural
As soon as the imaginary unit I came out, the number set was expanded into a complex number.
A complex number and a logarithm, the real and imaginary parts of horizontal and vertical coordinates.
Corresponding to a point on the complex plane, the origin is connected with it in the form of an arrow.
The axis of the arrow is opposite to the X axis, and the resulting angle is a radial angle.
The length of the arrow shaft is a model, and the numbers are often combined.
Algebraic geometric triangles, try to transform each other.
The essence of algebraic operation is I polynomial operation.
The positive integer of I is the second time, and there are four numerical periods.
Some important conclusions, cleverly remember the results.
The ability of mutual transformation between reality and reality is great, and complex number equals transformation.
Solve with equations and pay attention to the whole substitution.
On the geometric operation diagram, the addition parallelogram,
Subtractive triangle rule judgment; Multiplication and division operations,
Rotate back and forth to expand and contract the length of the annual module.
In triangular operation, it is necessary to distinguish between radiation angle and mode.
It is very convenient to take a square and make a square by using Demofo formula.
The radial angle operation is very strange, and the product quotient is used to sum the difference.
These four properties are inseparable, such as equality module and yoke,
Two will not be real numbers, and the comparison size is not allowed.
Complex numbers are very close to real numbers, so we should pay attention to the essential differences.
Six, permutation, combination, binomial theorem
The two principles of addition and multiplication are the laws that run through.
What has nothing to do with order is combination, but what needs order is arrangement.
Two formulas, two properties, two ideas and methods.
Arrange and combine the summary, and the application questions must be transformed.
It is common sense to arrange and combine together and choose the back row first.
Special elements and positions should be considered first.
Don't worry too much, and don't miss too much. Punching is a skill.
Arrange combinatorial identities and define proof modeling tests.
On binomial theorem, China Yang Hui Triangle.
Two properties, two formulas, function assignment transformation.
Seven. solid geometry
The trinity of point, line and surface is represented by cone billiards.
All distances start from points, and all angles are made of lines.
Vertical parallelism is the key point, and the concept must be clear in the proof.
Line, line, plane, plane, triple cycle.
When the whole idea of the equation is solved, it becomes consciousness.
Before calculation, it is necessary to prove and draw the removed figure.
Auxiliary lines of solid geometry, usually vertical lines and planes.
The concept of projection is very important and the key to solving problems.
The dihedral angle and volume projection formulas of lines with different planes are vivid.
Axioms are naturally three vertical lines, which solve many problems.
Eight, "plane analytic geometry"
Keywords directed segment straight circle, elliptic hyperbolic parabola,
The combination of polar coordinates and numbers and shapes of parametric equations is called normal form.
Descartes' viewpoint pair, point and ordered real number pair,
The two correspond to each other, creating a new geometric way.
The two ideas reflect each other and turn into ideas to fight the front line;
The undetermined coefficient method is actually the idea of equations.
Combine the three types and draw a curve to solve the equation.
The equation is given by curves, and the relationship between curves is judged.
Four tools are magic weapons with good coordinate parameters;
Plane geometry can not be lost, find the complex number of rotation transformation.
Analytic geometry is geometry, and you can't get carried away.
Graphics are intuitive and detailed, and mathematics is mathematics.