Basic formula of high school mathematics

Basic formula of high school mathematics

High school mathematics basic formula parabola: y = ax * + BX + C a > 0 opening up a < 0 opening down C = 0 parabola through the origin B = 0 parabola symmetry axis is Y axis, and the vertex formula y = a (x + H) * + k - H is the vertex coordinate, x k is the vertex coordinate y, generally used to calculate the maximum

Our company served 900 guests in the first month In the second month, 50% of the 900 guests have a return rate. At the same time, each of the 900 guests will introduce two guests, that is to say, the total number of guests served in the second month is 900 * 50% + 900 * 2 = 2250 In the third month, the number of customers served in the second month has a 50% return rate. At the same time, each new guest in the second month can introduce two more guests. Then the number of guests served in the third month is 2250 * 50% + 900 * 2 * 2 = 4725 . What is the number of guest service in the eighth month? How to express this formula The fourth month is equal to the number of guests in the third month * 50% + the number of new customers in the third month * 2, that is (2250 * 50% + 900 * 2 * 2) * 50% + 900 * 2 * 2 * 2, and so on, what will be the eighth month

2250 * 50% + 900 * 2 ^ (n-1) n is the month

A high school mathematics formula application problem I would like to ask each teacher straight line and circle system equation, straight line and straight line system equation, circle and circle system equation how to use? When to use? Best can give examples. Hope

Application of straight line and straight line equation: solving simultaneous equations: x + y = 1 and X-Y = 3. This problem is actually to find the coordinates of the intersection of two straight lines! X = (1 + 3) / 2 = 2 - y = (1-3) / 2 = - 1 (x, y) = (2, - 1)
Equation of straight line and circle: such as intersection point or tangent line of line and circle;
The intersection point of a circle, the distance between the center of a circle, and so on
There are many problems in the exercises of straight line and straight line, circle and circle, straight line and circle

People's education press senior high school mathematics compulsory one last chapter formula

1. Zeros of functions
(1) In general, if the value of a function at the real number a is 0, that is, a is called the zero point of the function
(2) For any function, as long as its image is continuous and uninterrupted, its zero point has the following properties:
① When it passes through zero (not even zero), the sign of function value changes;
② All function values between two adjacent zeros remain sign invariant
(3) The property of function zeros is the basis of studying the distribution of roots of equations. It is derived from the study of zeros of quadratic functions. It is a method from special to general
2. Dichotomy
(1) It is known that the function is continuous on the interval [a, b], and by continuously dividing the interval where the zero point of the function is located, the two endpoints of the interval are gradually approaching the zero point, so as to obtain the approximate value of the zero point, which is called dichotomy
(2) The basis of the definition of dichotomy is the property of the zero point of a function. The definition of dichotomy itself gives the steps to find the approximate value of the zero point of a function. As long as we do it step by step, we can find the zero point of the function with given accuracy
(3) The procedure of finding the approximate value of the zero point of a function by dichotomy is permeated with the idea of algorithm and the consciousness of programming. This step itself is a program for solving problems. This idea of programming has been widely used in computers
3. Several common function models
(1) First order function model:;
(2) Inverse proportional function model:;
(3) Quadratic function model:;
(4) Exponential function model:;
(5) Logarithmic function model:;
(6) Power function model:
（2） Image transformation
1. Drawing method: there are two methods of drawing images with functions expressed by analytic expressions, i.e. tabular point tracing method and image transformation method. Mastering these two methods is the key point of this section. When using the point tracing method to make images, we should avoid blindness before tracing points, and also avoid blindly connecting points into lines. The table should be listed at the key points, To connect lines in proper places, it requires a general study of the existing range, general characteristics and change trend of the image to be drawn. However, this research needs to rely on the theory and means of function property, equation, inequality, etc., which is a difficult point, And determine what kind of transformation. This is also a difficult point
The steps of making function image are as follows:
① Define the domain of function;
② The analytic formula of the simplified function is simplified;
③ The properties of functions are discussed, such as monotonicity, parity, periodicity, maximum value (even change trend);
④ Trace the line and draw the image of the function
2. The so-called geometric transformation method of image is an important way to obtain function image by combining common function image with image geometric transformation knowledge
The transformation of function image includes four kinds: translation transform, stretch transform, symmetry transform and absolute value transform
1. Translation transformation
From y = f (x) → y = f (x + a) + B, it can be divided into lateral translation and longitudinal translation
(1) Lateral translation: from y = f (x) → y = f (x + a)
The image points of y = f (x) are translated by | a | units along the X axis; when a > 0, they are shifted to the left; when a < 0, they are shifted to the right
(2) Longitudinal translation: from y = f (x) → y = f (x) + B
The points in the image of y = f (x) are translated by | B | units along the Y axis; when B ﹥ 0, it moves upward; when B ﹤ 0, it moves downward
2. Telescopic transformation
From y = f (x) → y = AF (Wx) (a ＞ 0, w ＞ 0), it can be divided into transverse and longitudinal stretching
y=f(x)
y=Af(wx)
3. Symmetry transformation
Including symmetry about X axis, Y axis, origin, y = x line
(1) As for X-axis symmetry: y = f (x) and y = - f (x), the characteristic of its analytic formula is: replace y with - y, and the analytic formula can change from one to another
(2) As for Y-axis symmetry: y = f (x) and y = f (- x), the characteristic of the analytic formula is that if x is replaced by - x, the analytic formula can be changed from one to another
(3) As for the origin symmetry: y = f (x) and y = - f (- x), the characteristic of the analytic formula is that x, y are replaced by - x, - y respectively, and the analytic formula can be changed from one to another
(4) As for the straight line y = x, the linear symmetry: y = f (x) and y = F-1 (x), the characteristic of its analytic formula is: replace y with X and X with y, and the analytic formula can change from one to another
4. There are two kinds of absolute value transformation: y = | f (x) | and y = f (| x |)
(1) From y = f (x) → y = | f (x)|
The meaning of absolute value is as follows:
Therefore, the program of geometric transformation can be designed as follows:
① Keep the image above the x-axis
② Flip: the image below the x-axis is symmetrical along the x-axis
③ Remove the image below the x-axis
(2) From y = f (x) → y = f (| x |)
The meaning of absolute value is as follows:
Therefore, the geometric transformation can be designed as follows:
① Keep the image on the right side of the y-axis
② Remove the image to the left of the y-axis
③ Flip: the image on the right side of the Y axis is symmetrical along the Y axis to the left side of the Y axis
2. With the different values of power functions, their definition domain, properties and images are not the same, but they have some common properties
(1) All power functions are defined in (0, + ∞) and the graph is over point (1,1);
(2) The graph of power function passes through the origin and is an increasing function in the interval
In this case, the image of power function is convex;
(3) In the first quadrant, the image of the power function is a decreasing function on the interval
On the right side of the axis, the image approximates the positive half axis infinitely
3. The steps of making power function image are as follows:
(1) First, make the image in the first quadrant;
(2) If the definition domain of power function is (0, + ∞) or [0, + ∞), the drawing has been completed;
If it is also meaningful on (- ∞, 0) or (- ∞, 0], the parity of the function should be judged first
If it is an even function, the image of the second quadrant is made according to Y-axis symmetry;
If it is an odd function, the third quadrant image is made according to the origin symmetry
1. Translation transformation:
2. Symmetry transformation
① Overall symmetry:
② Local symmetry:
3. Telescopic Transformation:
4. The images of two functions which are inverse functions are symmetric with respect to the line y = X
Let's look at how the two transformations work
（1）
（2）
(1) First zoom and then translation: the ordinate of each point on the y = SiNx image remains unchanged, and the abscissa changes to half of the original to get the image of y = sin2x, and then the image of y = sin2x is translated to the left by several units
(2) First translation and then scaling: the image with y = SiNx is shifted to the left by units to get the image, and then the ordinate of each point in the image remains unchanged, and the abscissa changes to half of the original image
（1）
（2）
(1) First symmetry and then translation: the image of y = f (x) is symmetrical about the Y axis, and then the image of y = f (- x) is obtained, and then the image of F (- x) is shifted to the right by 1 unit to get the image of y = f (- x + 1);
(2) First translation and then symmetry: the image of y = f (x) is shifted to the left by 1 unit to get the image of y = f (x + 1), and then the image of y = f (x + 1) is symmetrical about the Y axis to get the image of y = f (- x + 1)
Some abstract function relations are the properties of the functions represented: the properties of one function and the properties of two functions
F (1 + x) = f (1-x) y = f (1 + x) and y = f (1-x)
The image of this function is symmetric about the line x = 1, and the images of these two functions are symmetric about the y-axis
F (x + 1) = f (x-1) y = f (x + 1) and y = f (x-1)
This function is a periodic function with period 2, and the image of these two functions differs by two units (translation)
F (x-1) = f (1-x) y = f (x-1) and y = f (1-x)
This function is even, and the images of these two functions are symmetric with respect to the line x = 1

For senior high school mathematics compulsory one to five and elective (PEP) common formula I'm a senior two liberal arts girl. When I went to elective 1-1, I found that the formulas of compulsory one to compulsory five were widely used. But I liked to be crazy with my girlfriends, so I didn't learn at all Note that it's not 2-1. It's 1-1 Not much, including the best answer 20, 30 points

I am a teacher specializing in mathematics training. I have some information about mathematics in my hand. There are about 10 pages of required formula skills. I can't finish sending them here. Please leave me your email and I will send it to you. There is also a detailed version, including optional ones. If you need it later, I can send it to you
It has been sent to you, and one is too large. I will send it to you when you are online

The quantity product of plane vector and its application It is known that a = (sin θ, 1), B = (1, cos θ), C = (0,3), - π / 2 < θ < π / 2 ① If (4a-c) / / B, find θ ② Find the value range of Ia + B I

（1）4a-c=（4sinθ,1）,b=（1,cosθ）,
Because (4a-c) / / B, 4sin θ cos θ = 1,
That is, sin2 θ = 1 / 2,
Because - π / 2 < θ < π / 2, so - π < 2, θ < π,
SO 2 θ = π / 6 or 2 θ = 5 π / 6,
That is, θ = π / 12 or 5 π / 12
(2) Because a ^ 2 = 1 + (sin θ) ^ 2, B ^ 2 = 1 + (COS θ) ^ 2, a * b = sin θ + cos θ,
Therefore, a + B | 2 = a ^ 2 + B ^ 2 + 2A * b = 2 + 2 (sin θ + cos θ) = 2 + 2 √ 2 * sin (θ + π / 4),
Because - π / 4 < θ + π / 4 < 3 π / 4, so - √ 2 / 2, 0 < (a + b) ^ 2 < = 2 + 2 √ 2,
Therefore, the value range of | a + B | is (0, √ (2 + 2 √ 2)]

What is the scalar product formula of plane vector?

Let the vector be x and Y respectively, and the product (which is a real number) is n
n=xycosα
Where α is the angle between two vectors when the starting point of two vectors is translated to a point

The scalar product of plane vector 1. If vector a and B are not collinear, vector A. vector B ≠ 0, and vector C = vector a - [(vector A. vector a) / (vector A. vector b)]. Vector B, then the angle between vector a and C is? 2. Given that vectors a and B are two mutually perpendicular unit vectors in the plane, if vector C satisfies (vector a-vector C). (vector b-vector C) = 0, then what is the maximum value of the modulus of vector C? The second question doesn't say that C is collinear with a and B

The angle between vector a and C is acosa = A.C / ||||||c | = [A. (a - (a - (A.A / a.a.b) b] / c | = [A.A - (A.A / a.a.b) A.B] / |a|||||c | = 0, so a = π / 22. Let C = Ka + LB from (a a - (A.A / a.b.b) A.B / (A.A / a.a.b) A.B] / | a | C | = 0, so a = π / 22. Let C = Ka + 22. (B-C) = 0, then (a-ka-lb). (b-ka-lb) = 0-k + K ^ 2-L + L ^ 2 = 0 | C ^ 2 = (KA + lb). (KA + lb) = (KA + lb) = (KA + lb) = (k-ka-lb) = (KA + lb) = (KA + lb) = (KA + lb) =

The scalar product of plane vector Given that vectors a and B are not collinear, and | 2A + B | = | a + 2B |, prove: (a + b) ⊥ (a-b)

|2a+b|= |a+2b|
So | 2A + B ^ 2 = | a + 2B ^ 2
That's it=
It's 4 + 4 + = + 4 + 4
That's it=
And = - = 0
So a + B, A-B is vertical

Scalar product of plane vector Given that a and B are not collinear, the vector a + B is perpendicular to 2a-b, and a-2b is perpendicular to 2A + B? These are vectors

If two vectors are perpendicular, then the scalar product is zero
＝＝＝＝＝＝＝＝＝＝＝＝
(a+b)(2a-b)=0
(a-2b)(2a+b)=0
Simplified
2a²+ab-b²=0－－(1)
2a²-3ab-2b²=0－－(2)
From (1) we get AB = B? 2A? And substituting (2) we get
2a²-3(b²-2a²)-2b²=0
Simplified
8a²=5b²
|a|=(√10)|b|/4