Senior one mathematics 1 induction of useful and compulsory knowledge points
I. Definition and definition of expressions
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
y=ax^2+bx+c
(a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Two. Three Expressions of Quadratic Function
General formula: y = ax 2+bx+c (a, b and c are constants, and a≠0).
Vertex: y = a(x-h)2+k[ vertex P(h, k) of parabola]
Intersection point: y=a(x-x? )(x-x？ ) [only when it is related to the x axis A(x? , 0) and B(x? 0) parabola]
Note: Among these three forms of mutual transformation, there are the following relations:
h=-b/2ak=(4ac-b^2)/4ax？ ，x？ =(-b √b^2-4ac)/2a
Three. Quadratic function image
The image of quadratic function y = x 2 in plane rectangular coordinate system,
It can be seen that the image of quadratic function is a parabola.
Four. Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line
x=-b/2a .
The intersection of symmetry axis and parabola is the vertex p of parabola.
Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
2. The parabola has a vertex p, and the coordinates are
P(-b/2a,(4ac-b^2)/4a)
-b/2a=0, p is on the y axis; When δ = b 2-4ac = 0, p is on the x axis.
3. Quadratic coefficient A determines the opening direction and size of parabola.
When a>0, the parabola opens upwards; When a<0, the parabola opens downward.
The larger the |a|, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
When A and B have the same number (ab>0), the symmetry axis is on the left side of Y axis;
When a and b have different numbers (i.e. AB
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
δ=b^2-4ac>； 0, parabola and x axis have two intersections.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
δ=b^2-4ac<； 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 =-b √ b 2-4ac, multiplied by the imaginary number I, and the whole formula is divided by 2a).
Induction of useful and compulsory knowledge points in mathematics in senior one.
I. Collection of related concepts
1, meaning of set: some specified objects are set together into a set, where each object is called an element.
2. Three characteristics of elements in a set:
1. element determinism; 2. Mutual anisotropy of elements; 3. The disorder of elements
Description: (1) For a given set, the elements in the set are certain, and any object is either an element of the given set or not.
(2) In any given set, any two elements are different objects. When the same object is contained in a set, it has only one element.
(3) The elements in the set are equal and have no order. So to judge whether two sets are the same, we only need to compare whether their elements are the same, and we don't need to examine whether the arrangement order is the same.
(4) The three characteristics of set elements make the set itself deterministic and holistic.
3. Representation of assembly: {} For example, {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean and Arctic Ocean}
1.Set is expressed in Latin letters: A={ basketball player of our school}, B={ 1, 2, 3, 4, 5}
2. Representation methods of sets: enumeration and description.
Note: Commonly used number sets and their symbols:
The set of nonnegative integers (i.e. natural number set) is recorded as n.
Positive integer set N_ or N+ integer set z rational number set q real number set r
On the concept of attribution
Elements in a collection are usually represented by lowercase Latin letters. For example, if A is an element of set A, it means that A belongs to set A, and it is denoted as AA; On the other hand, if a does not belong to the set a, it is recorded as a? A
Enumeration: enumerate the elements in the collection one by one, and then enclose them in braces.
Description: A method of describing the common attributes of elements in a collection and writing them in braces to represent the collection. A method to indicate whether some objects belong to this set under certain conditions.
① Language Description: Example: {A triangle that is not a right triangle}
② Description of mathematical expression: Example: The solution set of inequality x-32 is {x? R|x-32} or {x|x-32}
4, the classification of the set:
1. The finite set contains a set of finite elements.
2. An infinite set contains an infinite set of elements.
3. An example of an empty set without any elements: {x | x2 =-5}
Second, the basic relationship between sets
1. contains a subset of relationships.
Note: There are two possibilities that A is a part of B (1); (2)A and B are the same set.
On the other hand, set A is not included in set B, or set B does not include set A, which is marked as AB or BA.
2. Equation relation (55 and 55, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} elements be the same.
Conclusion: For two sets A and B, if any element of set A is an element of set B and any element of set B is an element of set A, we say that set A is equal to set B, that is, A = B.
(1) Any set is a subset of itself. alcoholic anonymous
② proper subset: If AB, and A 1B, then set A is the proper subset of set B, and it is recorded as AB (or BA).
③ If AB, BC, then AC
(4) If AB is also BA, then A = B.
3. A set without any elements is called an empty set, and it is recorded as
It is stipulated that an empty set is a subset of any set and an empty set is a proper subset of any non-empty set.
Third, the operation of the set.
Definition of 1. intersection: Generally speaking, the set consisting of all elements belonging to A and B is called the intersection of A and B. 。
Write AB (pronounced a to b), that is, AB={x|xA, and xB}.
2. Definition of union set: Generally speaking, a set consisting of all elements belonging to set A or set B is called the union set of A and B, which is marked as AB (pronounced as A and B), that is, AB={x|xA, or xB}.
3. the nature of intersection and union: AA=A, A=, AB=BA, AA=A,
A=A，AB=BA。
4. Complete works and supplements
(1) Complement set: Let S be a set and A be a subset of S (that is, a set composed of all elements in S that do not belong to A), which is called the complement set (or complement set) of subset A in S..
(2) Complete set: If the set S contains all the elements of each set we want to study, it can be regarded as a complete set, usually expressed by U. 。
(3) Properties: ⑴ cu (cua) = a ⑴ (cua) ⑴ (cua) a = u.
Induction of useful and compulsory knowledge points in mathematics 3 of senior one.
Definition:
From the point of view of plane analytic geometry, a straight line on a plane is a graph represented by a binary linear equation in a plane rectangular coordinate system. To require the intersection of two straight lines, we only need to solve these two binary linear equations simultaneously. When simultaneous equations have no solution, two straight lines are parallel. When there are infinite solutions, two straight lines coincide; When there is only one solution, two straight lines intersect at one point. The angle between the upward direction of a straight line and the positive direction of the X axis (called the inclination angle of the straight line) or the tangent of the angle (called the slope of the straight line) is often used to indicate the inclination degree of the straight line (for the X axis) on the plane. Slope can be used to judge whether two straight lines are parallel or perpendicular, and also to calculate their intersection angle. The coordinates of the intersection of the straight line on the coordinate axis and the coordinate axis are called the intercept of the straight line on the coordinate axis. The position of a straight line on a plane is completely determined by its slope and intercept. In space, when two planes intersect, the intersection line is a straight line. Therefore, in the spatial rectangular coordinate system, two three-dimensional first-order equations representing the plane are simultaneous as the equations of the straight line obtained by their intersection.
Expression:
Inclined type: y=kx+b
Two-point formula: (y-y1)/(y1-y2) = (x-x1)/(x1-x2)
Point oblique type: y-y 1=k(x-x 1)
Interception formula: (x/a)+(y/b)=0.
Supplement: don't forget the most basic standard equation, AX+BY+C=0,
Because the above four linear equations do not include the case that the slope k does not exist, such as x=3, this straight line cannot be expressed in the above four forms, and special attention should be paid to the case that k does not exist in the process of solving problems.
Exercise questions:
1. Given that the equation of a straight line is y+2=-x- 1, then ().
A. The straight line passes through the point (2,-1) with a slope of-1.
B The straight line passes through the point (-2,-1) with a slope of 1.
C. The straight line passes through the point (-1, -2) with a slope of-1.
D. The straight line passes through the point (1, -2) with a slope of-1.
Analytically, C. Because the linear equation y+2=-x- 1 can be transformed into y-(-2)=-[x-(- 1)], the straight line passes through the point (-1, -2) with a slope of-/kloc-0.
2. The slope of the straight line 3x+2y+6=0 is k, and the intercept on the y axis is b, so there is ().
A.k=-，b=3B.k=-，b=-2
C.k=-，b=-3D.k=-，b=-3
Analytically, C. Transform the linear equation 3x+2y+6=0 into oblique y=-x-3, then k=- and b=-3.
3. Given that the equation of straight line L is y+ 1=2(x+), the slope of L is a, and the intercept on Y axis is b, then the value of logab is ().
A.B.2C.log26D.0
Choose B. From the meaning of the question, a=2, x=0 and b=4, so logab=log24=2.
4. If the inclination angle of the straight line L: y- 1 = k (x+2) is 135, the intercept of the straight line L on the y axis is ().
A. 1B。 - 1C.2D.-2
Analytically choose B. Because the dip angle is 135, k=- 1,
So the straight line l: y- 1 =-(x+2),
Let x=0 and y=- 1.
5. When the crossing point is (-1, 1), the straight line whose slope is twice that of the straight line y=x-2 is ().
A.x=- 1B.y= 1
c . y- 1 =(x+ 1)d . y- 1 = 2(x+ 1)
Analytically select c from the known slope k=2×=.
Then the linear equation is y- 1=(x+ 1).
Senior one mathematics useful knowledge points induction related articles;
★ Summary of Mathematics Knowledge Points in Senior One (must see before the exam)
★ Summary of knowledge points in the general examination of senior one mathematics.
★ Summarize and sort out the required knowledge points of high school mathematics.
★ The summary of mathematics knowledge points in senior one must be at the end of the term.
★ Summary of mathematics knowledge points in senior one.
★ Summary of knowledge points in the final exam of senior one mathematics.
★ Small induction of mathematics knowledge points in senior one.
★ Comprehensive summary of mathematics knowledge points in senior one.
★ Summary and induction of mathematics knowledge points in senior one.
★ Formula Summary of Key Knowledge Points in Senior One Mathematics