Rational number 1. Addition operation of rational numbers
Two numbers with the same sign are added, and the absolute value is added with the same sign.
Different symbols increase or decrease, large numbers determine and symbols.
Add up the opposites of each other, and the result is that zero must be remembered well.
"Big" minus "small" refers to the absolute value.
2. Subtraction operation of rational numbers
Negative is equal to plus negative, and reducing the burden is equal to plus positive.
Symbolic law of rational number multiplication.
The sign of the same sign is negative and the product of a term is zero.
3. Four operation skills of rational number mixed operation
Conversion methods: one is to convert division into multiplication, the other is to convert multiplication into multiplication, and the third is to convert decimals into decimals for reduction calculation in the mixed operation of multiplication and division.
Rounding method: In the mixed operation of addition and subtraction, two numbers with zero sum, two numbers with the same denominator, two numbers with integer sum and two numbers with integer product are usually combined to solve.
Split method: first split the band score into the sum of an integer and a true score, and then calculate it.
Clever use of arithmetic: Clever use of addition arithmetic or multiplication arithmetic in calculation will often make the calculation easier.
Cycle 1. Symmetry of circle
The (1) circle is an axisymmetric figure, and its symmetry axis is the straight line where the diameter lies.
(2) A circle is a figure with a symmetrical center, and its symmetrical center is the center of the circle.
(3) A circle is a rotationally symmetric figure.
2. Vertical diameter theorem
(1) bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.
(2) Inference:
Bisect the diameter (non-diameter) of a chord, perpendicular to the chord and bisecting the two arcs opposite the chord.
Bisect the diameter of the arc and bisect the chord of the arc vertically.
3. The degree of the central angle is equal to the degree of the arc it faces. The degree of the circle angle is equal to half the radian it subtends.
(1) The circumferential angles of the same arc are equal.
(2) The circumferential angle of the diameter is a right angle; The angle of a circle is a right angle, and the chord it subtends is a diameter.
4. In the same circle or equal circle, as long as one of the five pairs of quantities, namely two chords, two arcs, two circumferential angles, two central angles and the distance between the centers of two chords, is equal, the other four pairs are also equal.
5. The two arcs sandwiched between parallel lines are equal.
(1) The center of the circle passing through two points must be on the vertical line connecting the two points.
(2) Three points that are not on the same straight line determine a circle, the center of which is the intersection of the perpendicular lines of three sides, and the distances from this point to these three points are equal.
The outer center of a right triangle is the midpoint of the hypotenuse. )
6. The positional relationship between a straight line and a circle. D represents the distance from the center of the circle to a straight line, and r represents the radius of the circle.
A straight line and a circle have two intersections, and the straight line and the circle intersect; There is only one intersection point between a straight line and a circle, and the straight line is tangent to the circle; There is no intersection between a straight line and a circle, but a straight line and a circle are separated.
Mathematical theorem 1. There is only one straight line when two points intersect.
2. The line segment between two points is the shortest.
3. The complementary angles of the same angle or equal angle are equal.
4. The complementary angles of the same angle or equal angle are equal.
5. There is one and only one straight line perpendicular to the known straight line.
6. Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.
7. The axiom of parallelism passes through a point outside a straight line, and one and only one straight line is parallel to this straight line.
8. If two straight lines are parallel to the third straight line, the two straight lines are also parallel to each other.
9. The same angle is equal and two straight lines are parallel.
10. The internal dislocation angles are equal and the two straight lines are parallel.
1 1. The inner angles on the same side are complementary and the two straight lines are parallel.
12. Two straight lines are parallel with the same included angle.
13. Two straight lines are parallel and the internal dislocation angles are equal.
14. Two straight lines are parallel and complementary.
15. Theorem The sum of two sides of a triangle is greater than the third side.
16. It is inferred that the difference between two sides of a triangle is smaller than the third side.
17. The sum of the interior angles of the triangle and the sum of the three interior angles of the theorem triangle are equal to 180.
18. It is inferred that the two acute angles of 1 right triangle are complementary.
19. Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
20. Inference 3 The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
When the linear function is proportional, the quotient of x and y is certain. In the inverse proportional function, the product of x and y is definite. In y=kx+b(k, b is constant, k≠0), when x increases by m times, the function value y increases by m times; On the contrary, when x is reduced by m times, the function value y is reduced by m times.
1. Find the k value of the function image: (y 1-y2)/(x 1-x2).
2. Find the midpoint of the line segment parallel to the X axis: |x 1-x2|/2.
3. Find the midpoint of the line segment parallel to the Y axis: |y 1-y2|/2.
4. Find the length of any line segment: √ (x 1-x2) 2+(y 1-y2) 2 (note: the sum of squares of (x1-x2) and (y1-y2) under the root sign).
Quadratic function 1. Properties of quadratic function
Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c(a≠0).
When y=0, the quadratic function is a unary quadratic equation about x (hereinafter referred to as the equation), that is, ax? +bx+c=0(a≠0)
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.
2. The range of quadratic function
Vertex coordinates (-b/2a, (4αc-b? )/4α)
The basic form of quadratic function is y=ax? +bx+c(a≠0)
When a > 0, the parabolic opening is upward and the image is above the vertex, so the range y≥(4ac-b? ) /4a, that is [(4ac-b? )/4a，+∞).
When a < 0, the parabola opens downward, and the range of the function is (-∞, (4ac-b? )/4a]
When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is an even function, and the analytical expression is deformed into y=ax? +c(a≠0).
Solving application problems with column equations (groups) is an important aspect of integrating mathematics with practice in middle schools. The specific steps are as follows:
(1) review the questions. Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems.
⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both). Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them.
⑶ Use algebraic expressions containing unknowns to express related quantities.
(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, the number of unknowns is the same as the number of equations.
5] Solving equations and testing.
[6] answer.