There is an image of a quadratic function. Three students have described some of its characteristics: A: the axis of symmetry is a straight line x = 4; B: the abscissa of the two intersections with the X axis is an integer, and the ordinate of the intersection with the Y axis is also an integer; C: the area product of the triangle with the three intersections as the vertex is 12______ ．

There is an image of a quadratic function. Three students have described some of its characteristics: A: the axis of symmetry is a straight line x = 4; B: the abscissa of the two intersections with the X axis is an integer, and the ordinate of the intersection with the Y axis is also an integer; C: the area product of the triangle with the three intersections as the vertex is 12______ ．

According to the meaning of the question, let y = a (X-2) (X-6), ∵ the area of the triangle with the three intersection points of the coordinate axis as the vertex is 12, the intersection coordinates of the parabola and the coordinate axis can be (0, 6), ∵ a (0-2) (0-6) = 6, the solution is a = 12, so y = 12 (X-2) (X-6). So the answer is: y = 12 (X-2) (X-6)

There is an image of a quadratic function. Three students have given some characteristics of it
A: the axis of symmetry is a straight line, x = 4
B: the abscissa of the two intersections with the X axis is an integer
C: the ordinate of the intersection with the Y axis is also an integer, and the area of the triangle with the three intersections as its vertices is 3
Please write an analytic expression of quadratic function satisfying all the above characteristics

Since the axis of symmetry is a straight line x = 4, we can take the intersection coordinates with the X axis as (1,0) and (7,0), then the distance between the two is 6
If the area of the triangle is 3, the height should be 1, so the intersection coordinate of the function and the Y axis can be set as (0,1)
According to the coordinates of these three points, the analytic formula of quadratic function can be set as: y = ax ^ 2 + BX + C
0=a+b+c
0=49a+7b+c
1=c
The solution is a = 1 / 7
b=-8/7
c=1
So the analytic expression of this quadratic function is: y = 1 / 7 * x ^ 2-8 / 7 * x + 1

If the coefficient of quadratic function y = ax ^ 2 + BX + C satisfies A-B + C = 6, then the function image must pass through the point?
Ax ^ 2 is the square of ax, the answer is (- 1,6), but I don't know why (- 1,6), please help me to analyze
These answers are not very good,

When x = - 1, y = A-B + C
Naturally, it's y = 6. In fact, my answer is the same as I didn't answer. This question tests your observation thinking

How to judge quadratic function image according to A.B.. C

Three expressions of quadratic function: ① general formula: y = ax ^ 2; + BX + C (a, B, C are constants, a ≠ 0); ② vertex formula [vertex P (h, K)]: y = a (X-H) ^ 2 + K; ③ intersection formula [only limited to..]

What is the relationship between the root of quadratic function and coefficient

Thank you

How to judge the relationship between the coefficients of quadratic function
For example, in a quadratic function y = ax & # 178; + BX + C, a + B + C is greater than 0 or less than 0. What I don't understand now is how to judge the size relationship of 4A + B

Substitution numerical method
for example
Take x = 2 and substitute Y3 = 4A + 2B + C
Take x = 1 and substitute it into Y1 = a + B + C
Take x = 0 to substitute y2 = C
a. B, C, three unknowns, three equations, three solutions
A, B.C
Naturally, 4A + B will come out

On the problems of "the relationship between the root and coefficient of quadratic equation of one variable" and "quadratic function",
1. Given that real numbers x, y, Z satisfy x + y = 4 and xy = Z + 4, find the value of X + 2Y + 3Z
2. When x is greater than or equal to 0, find the value range of function y = - x (2-x)
3. When t is less than or equal to X and less than or equal to t + 1, find the minimum value of the function y = 1 / 2 X-5 / 2 (where t is a constant)

1. Z2 is always greater than 0, so XY ≥ 4, and X + y = 4 (x + y) 2 = 16 = x2 + Y2 + 2XY, so XY ≤ 4, so XY can only be 4. When x = y = 2, z = 0x + 2Y + 3Z = 62, y = - x (2-x) = x2-2x = (x-1) 2-1x ≥ 0, y can get the minimum value - 1, so the value range of Y is [- 1, positive infinity) 3, when t ≤ x ≤ T + 1, y =

What is the method of undetermined coefficient to find the analytic expression of quadratic function?
Be specific! I didn't go to that class! Please be specific!

Let y = ax & sup2; + BX + C
According to the given conditions, we can calculate a, B, C
For example, if the given coordinates are known, the
The vertex of y = ax & sup2; + BX + C is at (- B / 2a, (4ac-b & sup2;) / 4A)
Two equations can be obtained. And so on

Using undetermined coefficient method to find the analytic expression of quadratic function,
It is known that the axis of symmetry of the parabola is x = - 2 and it passes through two points (- 1, - 1) (- 4,0),

Let the equation of this quadratic function be y = ax ^ 2 + BX + C, (C is not equal to 0), from the known: - B / 2A = - 2, A-B + C = - 1, 16a-4b + C = 0, the solution is a = 1 / 3, B = 4 / 3, C0 =, y = 1 / 3x ^ 2 + 4 / 3x

How to use the specific coefficient method to find the analytic expression of quadratic function
Generally speaking, let the quadratic function be [y = ax & # 178; + BX + C], and let the function image pass (- 1,10), (1,4), (2,7)
We can get the equations: A-B + C = 10
a+b+c=4
4a+2b+c=7
To solve this system of equations, we have to
a=2 ,b=-3,c=5
How to calculate a = several, B = several, C = several

a-b+c=10
a+b+c=4
4a+2b+c=7
Isn't this a system of linear equations with three variables?
Subtract the above two expressions
2b=-6
b=-3
Form 3 - form 2
3a+b=3
3a-3=3
3a=6
a=2
Replace the first style again
c=10-3-2 =5