I. Numbers and Algebra A: Numbers and Formulas:
1: rational number
Rational Numbers: ① Integer → Positive Integer /0/ Negative Integer ② Fraction → Positive Fraction/Negative Fraction
Number axis: ① Draw a horizontal straight line, take a point on the straight line to represent 0 (origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction to get the number axis.
② Any rational number can be represented by a point on the number axis.
(3) If two numbers differ only in sign, then we call one of them the inverse of the other number, and we also call these two numbers the inverse of each other.
On the number axis, two points representing the opposite number are located on both sides of the origin, and the distance from the origin is equal.
The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.
Absolute value: ① On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number.
The absolute value of a positive number is himself/the absolute value of a negative number is his opposite number /0 is 0.
Comparing the sizes of two negative numbers, the absolute value is larger but smaller.
Operation of rational number:
Addition: ① Add the same sign, take the same sign, and add the absolute values.
② When the absolute values are equal, the sum of different symbols is 0;
When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value.
(3) A number and 0 add up unchanged.
Subtraction: Subtracting a number equals adding the reciprocal of this number.
Multiplication: ① Multiplication of two numbers, positive sign of the same sign, negative sign of different sign, absolute value.
② Multiply any number by 0 to get 0.
③ Two rational numbers whose product is 1 are reciprocal.
Division: ① Dividing by a number equals multiplying the reciprocal of a number.
②0 is not divisible.
Power: the operation of finding the product of n identical factors A is called power, the result of power is called power, A is called base, and N is called degree.
Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first.
2: Real number
Irrational number: Infinitely circulating decimals are called irrational numbers.
Square root: ① If the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A. If the square of a number X is equal to A, then this number X is called the square root of A..
(3) Positive numbers have two square roots /0 square roots are 0/ negative numbers have no square roots. (4) Find the square root of a number, which is called the square root, where a is called the square root.
Cubic root: ① If the cube of a number X is equal to A, then this number X is called the cube root of A. ..
② The cube root of a positive number is positive /0, and the cube root of a negative number is negative.
The operation of finding the cube root of a number is called square root, where a is called square root.
Real numbers: ① Real numbers are divided into rational numbers and irrational numbers.
② In the real number range, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those of reciprocal, reciprocal and absolute value in the rational number range.
③ Every real number can be represented by a point on the number axis.
3. Algebraic expressions
Algebraic expression: A single number or letter is also an algebraic expression.
Merge similar items: ① Items with the same letters and the same letter index are called similar items. (2) Merging similar items into one item is called merging similar items.
(3) When merging similar items, we add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.
4. Algebraic expressions and fractions
Algebraic expression: ① The algebraic expression of the product of numbers and letters is called monomial, the sum of several monomials is called polynomial, and monomials and polynomials are collectively called algebraic expressions. ② In a single item, the index sum of all letters is called the number of times of the item. ③ In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.
Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items.
Power operation:
Multiplication of algebraic expressions: ① Multiply the monomial with the monomial, respectively multiply their coefficients and the power of the same letter, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. (2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products. (3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.
There are two formulas: square difference formula/complete square formula.
Algebraic division: ① monomial division, which divides the coefficient and the power of the same base as the factor of quotient respectively; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. (2) Polynomial divided by single item, first divide each item of this polynomial by single item, and then add the obtained quotients.
Decomposition factor:
Turning a polynomial into the product of several algebraic expressions is called decomposing this polynomial.
Methods: Common factor method/formula method/grouping decomposition method/cross multiplication.
Fraction: ① Algebraic expression A is divided by algebraic expression B. If the divisor B contains a denominator, then this is a fraction. For any fraction, the denominator is not 0. ② The numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged.
Operation of fraction:
Multiplication: take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product.
Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction.
Addition and subtraction: ① Add and subtract fractions with the same denominator, and add and subtract molecules with the same denominator. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted.
Fractional equation: ① The equation with unknown number in denominator is called fractional equation. ② The solution whose denominator is 0 is called the root increase of the original equation.
B: equations and inequalities
1: equations and equations
Unary linear equation: ① In an equation, there is only one unknown, and the exponent of the unknown is 1. Such an equation is called a one-dimensional linear equation. ② Adding or subtracting or multiplying or dividing (non-0) an algebraic expression on both sides of the equation at the same time, the result is still an equation.
Steps to solve a linear equation with one variable: remove the denominator, shift the term, merge the similar terms, and change the unknown coefficient into 1.
Binary linear equation: An equation that contains two unknowns and all terms are 1 is called binary linear equation.
Binary linear equations: The equations composed of two binary linear equations are called binary linear equations.
A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation.
The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations.
Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method.
2. Inequality and Inequality Group
Inequalities: ① Use symbols > =,; 0, the opening direction is upward, < 0, and the opening direction is downward. You can also decide the size of the opening. The bigger the opening, the smaller the opening, and the smaller the opening. )
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
X is an independent variable and y is a function of X.
Three Expressions of Quadratic Function
General formula: (,b, c are constants, ≠0)
Vertex: [Vertex P(h, k) of parabola] For a quadratic function, its vertex coordinates are
Intersection point: [Only intersection point A(x? , 0) and B(x? 0) parabola
In ...
Note: Among these three forms of mutual transformation, there are the following relations:
h= k=
Quadratic function image
An image of a quadratic function is formed in a plane rectangular coordinate system,
It can be seen that the image of quadratic function is a parabola.
Properties of parabola
1. Parabola is an axisymmetric figure. The symmetry axis is a straight line x = h.
The only intersection of the symmetry axis and the parabola is the vertex p of the parabola.
Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
2. a parabola has a vertex p with coordinate p.
When b=0, p is on the y axis; When δ = the square of b -4ac=0, p is on the x axis.
3. Quadratic coefficient determines the opening direction and size of parabola.
When a > 0, the parabola opens upward; When a < 0, the parabola opens downward.
The greater the absolute value, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic coefficient * * * determine the position of the symmetry axis.
When the signs of A and B are the same (that is, b > 0), the symmetry axis is on the left side of Y axis;
When the signs of A and B are different (that is, b < 0), the symmetry axis is on the right side of the Y axis.
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
When δ = b-4ac > 0, the parabola has two intersections with the X axis.
When δ = b -4ac =0, there are 1 intersections between the parabola and the X axis.
When δ = b squared -4ac < 0, the parabola has no intersection with the X axis. The value of x is an imaginary number (the reciprocal of the value of x =, multiplied by the imaginary number I, and the whole formula is divided by 2).
When a>0, the function obtains the minimum value f(y)=4ac-b squared/4a at x=-b/2a;
When b=0, the axis of symmetry of parabola is the Y axis.
Quadratic function and unary quadratic equation
Especially quadratic function (hereinafter referred to as function),
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
That is, at this point, whether the function image intersects with the X axis means whether the equation has real roots.
The abscissa of the intersection of the function and the x axis is the root of the equation.
The images of 1. quadratic function,,, (various,) have the same shape, but their positions are different. Their vertex coordinates and symmetry axes are shown in the following table:
Vertex coordinates
(0，0) (h，0) (h，k)
axis of symmetry
x=0 x=h x=h x=-b/2a
When h>0, the image can be obtained by moving the parabola parallel to the right by H units.
When h < 0, it is obtained by moving |h| units in parallel to the left.
When h>0, k>0, move the parabola to the right by H units in parallel, and then move it up by K units to get the image;
When h>0, k<0 and parabola move to the right by H units in parallel, and then move down by | k units, an image is obtained;
When h < 0, k >; 0, the parabola moves |h| units to the left in parallel, and then moves up by k units to get the image;
When h < 0, k<0, the parabola moves |h| units to the left in parallel, and then moves |k| units downward to get an image;
Therefore, by studying the image of parabola, the general formula can be transformed into the form of, and its vertex coordinates and symmetry axis can be determined. The approximate position of parabola is very clear, which provides convenience for drawing images.
4. The intersection of parabolic image and coordinate axis:
(1) The image must intersect with the Y axis, and the coordinate of the intersection point is (0, c);
(2) When △ △= > 0, the image intersects the X axis at two points A(x? , 0) and B(x? 0), where is an unary quadratic equation.
(≠0). The distance between these two points AB=|x? -x？ In addition, the distance between any symmetrical points on the parabola can be determined by the following formula.
| 2× ()-A | (A is one of the points)
When △ = 0, the image has only one intersection with the X axis;
When delta < 0. The image does not intersect with the x axis. When >; 0, the image falls above the X axis, and when X is an arbitrary real number, there is y >；; 0; When < 0, the image falls below the X axis, and when X is an arbitrary real number, there is Y.
5. Maximum value of parabola: If >；; 0(& lt； 0), then the minimum (maximum) value of y = when x=.
The abscissa of the vertex is the value of the independent variable when the maximum value is obtained, and the ordinate of the vertex is the value of the maximum value.
6. Find the analytic expression of quadratic function by undetermined coefficient method.
(1) When the given condition is that the known image passes through three known points or three pairs of corresponding values of x and y are known, the analytical formula can be set to the general form: y=ax+BX+C. (A is not equal to 0).
(2) When the given condition is the vertex coordinate or symmetry axis of the known image, the analytical formula can be set as the vertex: y=a(x-h) square+k. 。
(3) When the given condition is that the coordinates of the two intersections of the image and the X axis are known, the analytical expression can be set as two expressions: y = a (x-x 1) (x-x2).
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The above is my serious summary and typesetting.
Maybe the second function part is a bit messy. . I found it online, so I can make do with it.
I hope the students like reading ~ ~ _