Analysis of typical examples of quadratic function Quadratic function y = AX2 + BX + C (a is not equal to 0), where a, B, C satisfy a + B + C = 0 and 9a-3b + C = 0, the symmetry axis of quadratic function image is a straight line

a+b+c=0
9a-3b + C = 0. By solving the above equations, we can get the following results:
a=-1/3
b=-2/3
Because the symmetry axis formula of quadratic function is: x = - B / 2A
So the symmetry axis of quadratic function image is:
X=-b/2a
=-（（-2/3）/2（-1/3））
=-1
That is: the symmetry axis of quadratic function image is a straight line x = - 1

Discussion on the classification of the maximum value of quadratic function in Senior High School
If a is a real number, f (x) = the minimum value of 2x & # 178; + (x-a) ‖ x-a ‖, where ‖ represents the absolute value, we hope to have a detailed classification process,

Segment discussion
When x = a
f(x)=2x²=2a²
When x > = a
f(x)=2x²+（x-a）|x-a|=2x²+(x-a)²=3x²-2ax+a²
The axis of symmetry is x = A / 3
If a > = 0, when x = a, the minimum value of F (x) is 2A & # 178;
If a

Exercises of quadratic function
Make some questions,

In general, there is the following relationship between the independent variable x and the dependent variable y: general formula: y = ax ^ 2 + BX + C (a ≠ 0, a, B, C are constants), then y is called the quadratic function of X. vertex formula: y = a (X-H) ^ 2 + K intersection formula (and X axis): y = a (x-x1) (x-x2) important concepts: (...)

A high school quadratic function problem
The image of the function f (x) = ax ^ 2 + BX + C (a is not equal to zero) is symmetric with respect to the straight line x = - B / A. It can be inferred that for any nonzero real number a, B, C, m, N, P, the solution set of the equation m [f (x)] ^ 2 + NF (x) + P = 0 with respect to X can not be?
A {1,2} B {1,4} C {1,2,3,4} D {1,4,16,64}

f(-b/2a+k)=f(-b/2a-k)
M [f (x)] ^ 2 + NF (x) + P = 0, quartic equation, up to 4 solutions
There are only two solutions of F (x), each of which corresponds to two XS, with respect to x = - B / 2A symmetry
So D is absolutely impossible
C symmetry axis X = 2 + 1 / 2

Typical examples of quadratic function
A piece of grassland is a rectangle with a length of 100 meters and a width of 80 meters. If you want to build two mutually perpendicular paths with a width of X meters in the middle, then the area of the lawn becomes y square meters. Find the functional relationship between Y and X, and write out the value range of the independent variable
My function formula is y = 8000-180x + X & sup2;, but I don't know how to take the value range of the independent variable,
But is there an equal sign on the answer? There are equal signs on both sides. Why?

y=8000-180x+x²（0＜x≤80）
When y = 8000-180x + X & sup2; = (x-90) & sup2; - 6300 = 0,
（x-100）（x-80）=0
The solution is x = 100, x = 80
That is, when x ≥ 100, X ≤ 80, y ≥ 0,
The width of the path cannot be wider than that of the rectangle, so there is 0

Y = ax ＾ 2 + BX + C passes through the origin and the second, third and fourth quadrants

Through the origin, so 0 = a * 0 + b * 0 + C, so C = 0
Because it's only the first quadrant, so when x ＞ 0, y

A problem of quadratic function in junior high school
Ask a question:
It is known that a (x12008) and B (x22008) are two points on the image of quadratic function y = ax & sup2; + 5 (a ≠ 0). When x = X1 + X2, the value of quadratic function y is_____________ .
Urgent need answer, who can trouble answer```
Write down the solution
Thank you

y=5
Because the symmetry axis of y = ax & sup2; + 5 is y-axis, so X1 and X2 are symmetric about y-axis, so X1 + x2 = 0, so y = 5

On (the maximum value problem of quadratic function with parameter in closed interval)

Please refer to

Want to ask a high school mathematics about "the maximum value problem of quadratic function on closed interval"
Find the minimum value of F (x) = x square - 2aX + 2 on [2,4]
Please explain the steps and reasons (such as: why take this value, etc.) this is not very good. Thank you

f(x)=x^2-2ax+2
Axis of symmetry:
X = a, opening upward,
When a

High school mathematics -- seeking the maximum value of a quadratic function in an interval
Find the maximum value of quadratic function: y = x ^ 2 + 2T + 5 in closed interval [- 1,1]
The answer should be 8
Thank you very much
But no!
How does y = (x + 1) ^ 2 + 3 come into being?

1、 The formula method was used to calculate the formula
y=x^2+2x+5
y=x^2+2x+1-1+5
y=(x^2+2x+1)-1+5
y=（x＋1）^2+4
When x = 1, ymax = 8
The second method: directly substituting [- 1,1] into y = x ^ 2 + 2x + 5 can also get the answer

Analysis of typical examples of quadratic function Quadratic function y = AX2 + BX + C (a is not equal to 0), where a, B, C satisfy a + B + C = 0 and 9a-3b + C = 0, the symmetry axis of quadratic function image is a straight line