Analytic expression of quadratic function I'm a bit at a loss when I learn this quadratic function. I don't know how to start. Please give me some examples of different methods, that is, different conditions, and then write the answers to show me. Thank you very much,

Analytic expression of quadratic function I'm a bit at a loss when I learn this quadratic function. I don't know how to start. Please give me some examples of different methods, that is, different conditions, and then write the answers to show me. Thank you very much,

Let me tell you a universal formula:
The properties of quadratic function
The vertex is (B / - 2A, 4ac-b ^ 2)
The axis of symmetry is a straight line x = B / - 2A
The intersection point with y axis is (0, c)
When the intersection point with X axis is (x1, x2) discriminant > = 0, the two roots of the equation AX ^ 2 + BX + C = 0 are x1, x2

Solving analytic expression of quadratic function
Known axis of symmetry = 1, over (3,0), (0,3)
Known axis of symmetry = - 1, over (1,5), (2,10)
Each vertex is solved in three ways

(1) Because axis of symmetry = 1
Let the function be y = a (x-1) ^ 2 + C
Because (3,0), (0,3)
So substituting into the solution, we get a = - 1, C = 4
So vertex: y = - (x-1) ^ 2 + 4
Intersection: y = - (x-3) (x + 1)
General: y = - x ^ 2 + 2x + 3
(2) Because axis of symmetry = - 1
Let the function be y = a (x + 1) ^ 2 + C
Because (1,5), (2,10)
So substituting into the solution, we get a = 1, C = 1
So vertex: y = (x + 1) ^ 2 + 1
Intersection: None
General: y = x ^ 2 + 2x + 2

Know three points and find the analytic expression of quadratic function
Given that the quadratic function y = AX2 + BX + C passes through a (- 2,7) B (6,7) C (3, - 8), find the analytic expression of the quadratic function?

Set X = - 2, y = - 3
x=6 y=7
X = 3, y = - 8, you can get the solution
Although it's a bit troublesome to solve the linear equation with three variables, it's OK to practice writing more

As shown in the figure, which of the following functions is linearly independent? Why?
 

A: The latter is twice as large as the former
C: The latter is 2 times of the former
D: The latter is four times of the former
These are the basic concepts of logarithm

Is complete linear correlation a functional relation?
When the correlation coefficient r = 1, each data point is on the regression line, at this time, the two variables are completely linear correlation, and the correlation is an uncertain relationship. The problem is that each data point is on the regression line, and the relationship between the two variables is a certain function, isn't that right?

The correlation coefficient r = 1 can only show that the existing data is completely linear correlation, but it does not rule out the possibility that some data points are not on the regression line, but you did not find it
In other words, this is incomplete induction, and the conclusion that the relationship between two variables is certain is unreliable
In addition, even the existing data may not be completely linear correlation, but the accuracy of the data you get is relatively low. If you improve the accuracy of the data, R may be equal to 0.99999

How to judge the linearity, time invariance and causality of signal and system?
（9）y(n)=f(n)*g(n)
The following is my personal judgment. I hope you can criticize and correct me
(1) Linear, time invariant, causal
(2) Nonlinear, time invariant, causal
(3) Linear, time invariant, causal (N0 > 0)
(4) Linear, time-varying, causal
(5) Nonlinear, time invariant, causal
(6) Linear, time invariant, non causal
(7) Linear, time-varying, causal
(8) Linear, time-varying, causal
(3) Linear, time-varying, causal

How to judge whether a system is linear and time invariant?
As shown in the figure, the formula

Generally, the linear system represented by linear constant coefficient ordinary differential equation is called time invariant linear system

3. Determine whether the following systems are linear shift invariant
Determine whether the following systems are linear shift invariant systems

No. It's a shift invariant system, but it's not a linear system
Definition of linear system: if for two excitations X1 (n) and X2 (n), t [ax1 (n) + bx2 (n)] = at [X1 (n)] + Bt [X2 (n)], where a and B are arbitrary constants. The nonlinear system does not satisfy the above relation
It is proved that any Y 1 (n) = x 1 (n) ^ 2, y 2 (n) = x 2 (n) ^ 2, a * y 1 + b * y 2 = a * x 1 ^ 2 + b * x 2 ^ 2 is not equal to (a * x 1 + b * x 2) ^ 2, so it is not a linear system

Signal and system, judge whether the system is linear system, causal system and stable system

Ordinary differential equation problem ~ linear differential equation~
Let F 1 (x) f 2 (x) f 3 (x) be three linearly independent solutions of the linear nonhomogeneous equation y '' + P (x) y '+ Q (x) y = 0, and find its general solution
The answer is y = c1y1 + c2y2 + (1-c1-c2) Y3

Handsome guy, the first two are general
Solution,
The last term is the special solution
Just look. Homogeneous power
The general solution of the equation and non-homogeneous equation
The solution is the same, so it's the same
It's the first two
The last term is the homogeneous equation
The special solution of is given by the non-homogeneous power
The method of solving homogeneous equation
The form of special solution is the most important
The latter item has been changed