I. Function definition and definition:

Independent variable x and dependent variable y have the following relationship:

y=kx+b

It is said that y is a linear function of x at this time.

In particular, when b=0, y is a proportional function of x.

Namely: y=kx(k is a constant, k≠0)

Second, the properties of linear function:

The change value of 1.y is directly proportional to the corresponding change value of x, and the ratio is k.

That is: y=kx+b(k is any non-zero real number b, take any real number)

2. When x=0, b is the intercept of the function on the y axis.

Iii. Images and properties of linear functions:

1. Practice and graphics: Through the following three steps.

(1) list;

(2) tracking points;

(3) The connection can be the image of a function-a straight line. So the image of a function only needs to know two points and connect them into a straight line. (Usually find the intersection of the function image with the X and Y axes)

2. Property: (1) Any point P(x, y) on the linear function satisfies the equation: y = kx+b (2) The coordinate of the intersection of the linear function and the y axis is always (0, b), and the image of the proportional function always intersects the origin of the x axis at (-b/k, 0).

3. Quadrant where K, B and function images are located:

When k>0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X;

When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.

When b>0, the straight line must pass through the first and second quadrants;

When b=0, the straight line passes through the origin.

When b<0, the straight line must pass through three or four quadrants.

Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.

At this time, when k>0, the straight line only passes through the first and third quadrants; When k < 0, the straight line only passes through the second and fourth quadrants.

Fourth, determine the expression of a linear function:

Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..

(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.

(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, we can list two equations: y 1 = kx 1+b … ① and y2 = kx2+b … ②.

(3) Solve this binary linear equation and get the values of K and B. ..

(4) Finally, the expression of the linear function is obtained.

Senior High School Mathematics Compulsory 3 Knowledge Points Summary Part II

Mathematics (literature) in senior high school includes 5 compulsory books and 2 elective books, and (science) includes 5 compulsory books and 3 elective books, and 2 books are studied each semester.

Compulsory 1: 1, the concepts of set and function (this part of knowledge is abstract and difficult to understand) 2, basic elementary function (exponential function and logarithmic function) 3, the nature and application of function (abstract and difficult to understand).

Compulsory 2: 1, solid geometry (1), proof: vertical (multi-plane vertical), parallel (2), solution: mainly included angle, including line plane angle and plane angle.

This part of knowledge is the difficulty of senior one students, such as: an angle is actually an acute angle, but what is shown in the picture is an obtuse angle, etc., which requires students to have a strong three-dimensional sense. This part of the knowledge college entrance examination accounts for 22-27 points.

2. Linear equation: it is not a separate proposition in the college entrance examination, but it is easy to combine with conic curve.

3, the circle equation:

Compulsory course 3: 1, preliminary algorithm: required content of college entrance examination, 5 points (choose or fill in the blanks) 2, statistics: 3, probability: required content of college entrance examination, science accounted for 15 in 2009, liberal arts mathematics accounted for 5 points.

Compulsory 4: 1, trigonometric function: (image, nature, high school, emphasis and difficulty) must be tested: 15-20, often mixed with other functions.

2. Plane vector: NMET is not a separate proposition, but it can be easily combined with trigonometric functions and conic curves. In 2009, science accounted for 5 points, and liberal arts accounted for 13 points.

Compulsory course 5: 1, triangle solution: (sine, cosine theorem, trigonometric identity transformation) In the college entrance examination, science accounts for about 22 points, and liberal arts mathematics accounts for about 13 points. 2. Series: required for college entrance examination, 17-22. 3. Inequality: (Linear programming is easy to understand in class, but difficult to do. Inequality is not a separate proposition, but is generally combined with function to find the maximum value and solution set.

Senior high school mathematics compulsory 3 knowledge points summary 3

I. The concept of set

Characteristics of elements in (1) set: certainty, mutual difference and disorder.

(2) The relationship between sets and elements is represented by the symbol =.

(3) Symbolic representation of common number sets: natural number set; Positive integer set; Integer set; Rational number set, real number set.

(4) Representation of sets: enumeration method, description method and Wayne diagram.

(5) An empty set refers to a set without any elements.

An empty set is a subset of any set and a proper subset of any non-empty set.

function

I. Mapping and function:

The concept of (1) mapping: (2) one-to-one mapping: (3) the concept of function:

Second, the three elements of function:

The judgment method of the same function: ① correspondence rule; (2) Domain (two points must exist at the same time)

Solution of resolution function (1):

① definition method (patchwork method): ② substitution method: ③ undetermined coefficient method: ④ assignment method:

(2) The solution of functional domain:

(1) The universe with parameters should be discussed by classification;

(2) For practical problems, after finding the resolution function; We must find its domain, and the domain at this time should be determined according to the actual meaning.

(3) The solution of function value domain:

① Matching method: transform it into a quadratic function and evaluate it by using the characteristics of the quadratic function; Often converted into:;

(2) Reverse solution: the value range used to represent, and then the value range obtained by solving the inequality; Commonly used to solve, such as:

(4) Substitution method: transforming variables into functions of assignable fields and returning to ideas;

⑤ Triangular Bounded Method: Transform it into a function containing only sine and cosine, and use the boundedness of trigonometric function to find the domain;

⑥ Basic inequality methods: transformation and modeling, such as: using the average inequality formula to find the domain;

⑦ Monotonicity method: The function is monotonous, and the domain can be evaluated according to the monotonicity of the function.

⑧ Number-shape combination: According to the geometric figure of the function, the domain is found by the method of number-shape combination.

Third, the nature of the function:

Monotonicity, parity and periodicity of functions

Monotonicity: Definition: Note that the definition is relative to a specific interval.

The judgment methods are: definition method (difference comparison method and quotient comparison method)

Derivative method (for polynomial function)

Composite function method and mirror image method.

Application: compare sizes, prove inequalities and solve inequalities.

Parity: Definition: Pay attention to whether the interval is symmetrical about the origin, and compare the relationship between f(x) and f(-x). F(x)-f(-x)=0f(x)=f(-x)f(x) is an even function;

F(x)+f(-x)=0f(x)=-f(-x)f(x) is odd function.

Discrimination methods: definition method, image method and compound function method.

Application: function value transformation solution.

Periodicity: Definition: If the function f(x) satisfies: f(x+T)=f(x) for any x in the definition domain, then t is the period of the function f(x).

Others: If the function f(x) satisfies any x in the domain: f(x+a) = f (x-a), then 2a is the period of the function f (x).

Application: Find the function value and resolution function in a certain interval.

Fourth, graphic transformation: function image transformation: (key) It is required to master the images of common basic functions and master the general rules of function image transformation.

Regularity of common image changes: (Note that translation changes can be explained by vector language, which is related to vector translation)

Translation transformation y = f (x) → y = f (x+a), y = f (x)+b.

Note: (1) If there is a coefficient, first extract the coefficient. For example, the image of the function y=f(2x+4) is obtained by translating the function y=f(2x+4).

(2) Combining with the translation of vector, understand the meaning of translation according to vector (m, n).

Symmetric transformation y=f(x)→y=f(-x), which is symmetric about y.

Y=f(x)→y=-f(x), which is symmetrical about x.

Y=f(x)→y=f|x|, keep the image above the X axis, and the image below the X axis is symmetrical about X.

Y=f(x)→y=|f(x)| Keep the image on the right side of the Y axis, and then make the right part of the Y axis symmetrical about the Y axis. (Note: it is an even function)

Telescopic transformation: y=f(x)→y=f(ωx),

Image transformation of Y=f(x)→y=Af(ωx+φ) reference trigonometric function.

An important conclusion: if f(a-x)=f(a+x), the image of function y=f(x) is symmetrical about the straight line x=a;