Addition of rational numbers: addition with the same sign is one-sided; Different symbols add "big" and subtract "small", and the symbols follow big; Absolute values are equal, "zero" is just right. [Note] "Big" minus "Small" refers to the absolute value.
Merge similar items: Do not forget the rules for merging similar items, but only seek the sum of coefficients, and the letters and indexes remain unchanged.
Rules for removing brackets and adding brackets: The key to removing brackets and adding brackets is to look at symbols, with a plus sign before brackets, a minus sign before brackets and a sign before brackets.
Unary linear equation: the known unknowns need to be separated, and the separation method is shift, the addition and subtraction of shift terms need to be changed, and the multiplication and division method needs to be reversed.
Identity transformation: two numbers are subtracted, and the exchange position is the most common. The pros and cons only look at its index, and the odd symbols remain unchanged. (a-b) 2n+1=-(b-a) 2n+1(a-b) 2n = (b-a) 2n square difference formula: There are two square difference formulas, remember that the signs are opposite, and the first and last are multiplied by the first and last, so don't be confused with the complete formula.
Complete Square: There are three complete squares, the first and last symbols are fellow villagers, the first and last squares, and the first and second squares are placed in the middle; The first and last brackets are square, and the symbol of the last item follows the center.
Factorization: One mention (common factor) and two sets (formulas) are divided into three groups. It's not outrageous to look at a few items carefully. Two terms only use square difference, three terms are cross multiplication, and the array method is skillful and not sloppy. Look at the four items carefully. If there are three square numbers (items), use one or three groups, otherwise use two or two groups, five or six items are more, and two or three try to group.
Oral decision of "substitution": dig out letters and replace them with numbers (formulas), and keep both numbers and letters; Replace with fractions or negative numbers, enclose them in parentheses, put (present) brackets in the original brackets, and gradually change brackets (small-medium-large).
Single operation: addition, subtraction, multiplication and division (on), three-level operation is clear, coefficient is calculated at the same level, and exponential operation is degraded.
The general steps to solve the one-dimensional linear inequality problem are to remove the denominator and brackets, change the sign when moving items, merge similar items, and then remove the coefficients. Don't forget to change the direction of inequality when dividing two sides by negative numbers.
The solution set of the unary linear inequality group: take the big one, take the small one, take the middle one, and the small one is nowhere to be found.
The solution set of unary quadratic inequality and unary linear absolute inequality: the big one (fish) takes two sides, and the small one (fish) takes the middle.
Fractional mixed operation rule: Fractional four operations, sequential multiplication and division, addition and subtraction, multiplication and division at the same level, and the division sign must be changed (multiplied); Multiplication simplification, factorization first, numerator and denominator meet, and then operation; 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.
Steps to solve the fractional equation: multiply by the simplest common denominator to become an algebraic expression, and write it clearly. After the solution is obtained, the root must be tested, the original (root) remains, and the addition (root) is unambiguous.
The conditions of the simplest radical: the three conditions of the simplest radical, the denominator contains no sign, the power index (number) root index (number) should be coprime, and the power index is a little smaller than the root index.
Special point coordinate features: coordinate plane point (x, y), transverse to the front and longitudinal to the back; (+,+), (-,+), (-,-) and (+,-), the four quadrants are divided into front and back; Y is 0 on the x axis, and x is 0 on the y axis.
The bisector of the quadrant angle: The bisector of the quadrant angle has its own characteristics. The first and third horizontal and vertical directions are equal, while the second and fourth horizontal and vertical directions are actually opposite.
Straight line parallel to an axis: a straight line parallel to an axis, the coordinates of points are particular, the straight line is parallel to the X axis, and the ordinate is different; The straight line is parallel to the Y axis, and the abscissa of the point remains unchanged.
Symmetrical point coordinates: Remember the symmetrical point coordinates, and don't confuse the positions of opposites. X-axis symmetry is Y-axis symmetry, Y-axis symmetry, and there is a negative sign before X; It is best to remember that the origin is symmetrical, and the abscissa and ordinate are signed.
The range of independent variables: the denominator of the fraction is not zero, and it is not possible to be negative under even roots; The base of the zeroth power is not zero, and both algebraic expressions and odd roots will do.
Motion law of function image;
If an analytic function is written as y=k(x+0)+b, and the analytic formula of a quadratic function is written as y=a(x+h)2+k, the following formula is used: "Move left and right in brackets, and finally move up and down, the right and negative of Zuo Zheng must be remembered, and the up and down negative cannot be wrong".
Linear function image and property formula: linear function is a straight line, and the image passes through three quadrants; The proportional function is simpler, and the straight line passes through the origin; These two coefficients, k and b, play a great role. K is the inclination angle, B intersects with the Y axis, K is positive and 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; The greater the absolute value of k, the farther the straight line is from the horizontal axis.
Quadratic function image and property formula: quadratic function parabola, image symmetry is the key; Openings, vertices and intersections that determine the appearance of the image; The opening and size are broken by A, C intersects the Y axis, and the symbol of B is special, and the symbol is associated with A; First find the vertex position, take the y axis as the baseline, and the difference between the left and right is 0, remember that there is no doubt in your heart; Vertex coordinates are the most important and appear in the general formula. The horizontal scale is the axis of symmetry, and the vertical scale function is the most important. If the position of the symmetry axis is found, the sign will be reversed and different expressions can be interchanged.
The image and property formula of inverse proportional function: inverse proportional function has characteristics and hyperbola deviates far; K is positive, the graph is within the first and third (image) limits, K is negative, and the graph is within the second and fourth (image) limits; The graph is reduced in the first and third functions, and the two branches are reduced respectively. In the figure, two and four are opposite, and the two branches are added respectively; The longer the line is, the closer it is to the axis and it will never touch the axis.
Remember the definition of trigonometric function skillfully: the trigonometric functions learned in junior high school include sine, cosine, tangent and cotangent, which are actually the ratio of triangle sides. You can separate these two words and remember the definition with the following sentence: An unskilled cook taught his apprentice to kill fish and said the following sentence: Cut the fish phosphorus (neighbor) directly. Positive: sine or tangent, right: opposite positive; Remainder: cosine or cosine, adjacent: adjacent edges indicate that the remainder is adjacent; Tangents are right-angled edges.
Addition and subtraction of trigonometric functions: positive increase, subtraction and special trigonometric function value memory: First, remember that the denominators of sine and cosine values of 30 degrees, 45 degrees and 60 degrees are all 2, and the denominators of tangent and cotangent are all 3. Molecules can remember the formula "123,3213927".
Clever numbers: = 1.4 14 (meaning only) = 1.732 1 (three people discuss together) =2.236 (I measured the mountain road) =2.449 (food is wine) =2.645 (second-rate).
Judgment of parallelogram: To prove a parallelogram, two conditions can be met, one is to prove that the opposite sides are equal, the other is to prove that the opposite sides are parallel, and a group of opposite sides can also be proved, and they must be equal and parallel. Slant is a treasure. If you divide it equally, you can't run away. It is also useful to have equal diagonal lines. Only "two diagonal lines" can be achieved.
Auxiliary line of trapezoid problem: move the diagonal of trapezoid to make two waists into 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.
Add auxiliary lines Song: Auxiliary lines, how to add them? Finding the pattern is the key. If there is an angle (horizontal) dividing line in the question, it can be vertical on both sides. The middle perpendicular of the line segment leads to the connecting line at both ends, and the connecting line between the two midpoints of the triangle side forms the middle line; A triangle has a midline, and the midline is doubled.
The proof song of the circle: it is not difficult to prove the circle, and 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 radiation will affect the ear; And the angle related to the circle. Don't forget that they are interrelated, such as circumference, center and tangent angle. Find the relationship carefully and connect the lines. The angles of the circles of the same arc are equal, so it is most commonly used to prove the problem. If there is a tangent angle in the circle, it is easy to find the arc. The circle has an inscribed quadrilateral with complementary diagonals, the outer angle is equal to the inner diagonal, and the quadrilateral is inscribed with the 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; If you want to prove that the tangent of the circle, the vertical radius passes through the outer end, the straight line and the circle have * * * points, the vertical radius is connected, and the straight line and the circle are not given points, you need 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 meet a circle, it is important to know the location. The tangency of two circles is a common tangent, and the intersection of two circles is a common chord.
Proportional line segment in the circle: in the case of equal product, change the equal ratio and find similarity vertically and horizontally; Don't be angry, switch to equal lines and ratios, encounter equal ratios, change equal products, quote projective and circular powers, parallel lines, turn proportions, and find the connection between the two ends.
Song: Divide the circle equally, and the value of n must be greater than three, connecting the points in turn and inscribing a regular N-polygon.
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. The regular N-polygon is very beautiful. It has inscribed circle and circumscribed circle. Both inscribed circle and circumscribed circle are unique. These two circles are concentric circles. Its graph is symmetrical, and n symmetry axes all pass through the center point. 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. Change the inscribed circle, circumscribed circle, apogee and radius respectively, and divide them into 2n right triangle integers, which is simple to calculate.
Function learning decision: the proportional function is a straight line, and the image must pass through a point. The sign of k is the key, which determines the quadrant of the straight line. Negative k passes through two or four limits, x increases and y decreases, and k moves up and down unchanged. Derive to get a straight line, and b decreases upwards. The image passes through three boundaries, and two points determine a line. The key is to choose the coefficient.
The hyperbola of inverse proportional function can be determined by only one point. Positive k is in the range of one or three, x increases and y decreases. At any point on the image, the rectangular area remains unchanged, and the symmetry axis is the angular bisector. The order of x and y can be interchanged.
The selection of parabola of quadratic function needs three points, namely, the positive and negative opening of A, the size of C on Y axis, the simplest sign of △, the counting of intersection points on X axis, the complete calculation of food poisoning knot of B, the parabola with the same sign on the left side of A and B axis translation A unchanged, and the vertices are pulled left and right. The three forms can be transformed, and the matching mode plays the most critical role.