An image problem of quadratic function that bothers me for many years No matter m is any real number, the image of quadratic function y = x * x - (2-m) x + m always passes through (,) The answer is (- 1,3). I don't know how

An image problem of quadratic function that bothers me for many years No matter m is any real number, the image of quadratic function y = x * x - (2-m) x + m always passes through (,) The answer is (- 1,3). I don't know how

We know that this point has nothing to do with m, so we should extract it first. Y = x * x - (2-m) x + M = x ^ 2-2x + m (x + 1) obviously, when x = - 1, M can take any value, and the function value will not change. When x = - 1, y = 3, we should not only pay attention to the common one, but also grasp the key parameter

What's the difference between the three topics of "drawing the image of quadratic function", "drawing the general image of quadratic function" and "drawing the sketch of quadratic function"?

To draw the image of a quadratic function is to draw it with a ruler
To draw a rough picture of a quadratic function is to draw a rough sketch of a quadratic function

Solving a quadratic function problem
m2+m≠0
m2-m=2

m²+m≠0
m(m+1)≠0
M ≠ 0 and m ≠ - 1
m²-m=2
m²-m-2=0
(m-2)(m+1)=0
M = 2 or M = - 1

If f (x) = (- K2 + 3K + 4) x + 2 is an increasing function, then the range of K is?
Oh, thank you, but what I want is the process! This kind of problem will be solved next time!

Because the original equation is an increasing function
So the coefficient of the first term is greater than one
(- K2 + 3K + 4) > 0
-1

A quadratic function problem
As shown in Figure 1, it is known that the line y = - 1 / 2x and the parabola y = - 1 / 4x2 + 6 intersect at two points a and B
1. Calculate the coordinates of a and B points;
2. Find the analytic formula of the vertical bisector of line ab
3. As shown in Figure 2, take a rubber band of the same length as the line AB, and fix the ends at a and B respectively. Pull the rubber band with a pencil to make the pen point P move on the parabola above the line AB, and the moving point P will form countless triangles with a and B. is there a triangle with the largest area among these triangles? If so, find out the largest area, and point out the coordinates of point P at this time; if not, Please give a brief explanation

1.(6,-3)(-4,2)
2. AB: y = - 1 / 2x, AB midpoint coordinates (1, - 1 / 2)
K vertical = 2
y=2x-5/2
three

It is known that the vertex of parabola y = x-4x + m is on the x-axis. The analytic expression of this function and its vertex coordinates are obtained
It is known that the image of the quadratic function y = (M + 6) x + 2 (m-1) x + m + 1 of X always has an intersection with the x-axis, and the value range of M is obtained

The vertex of y = x & sup2; - 4x + M = (X-2) & sup2; + M-4 is on the x-axis, which indicates that the maximum or minimum value of Y is 0y = (X-2) & sup2; + M-4, and the minimum value of Y is y (min) = M-4 when x = 2, so M-4 = 0m = 4Y = x & sup2; - 4x + 4 vertex is (2,0) y = (M + 6) x + 2 (m-1) x + m + 1. If y = 0, it always has focus with x-axis

When x = - 1, the value of the quadratic function is 10, when x = 1, the value of the quadratic function is 4, when x = 2, the value of the quadratic function is 7. Find the analytic formula of the quadratic function. (undetermined coefficient method) 2. The height of a cylinder is equal to the bottom radius, and write out the relationship between its surface area s and radius R
3. When a > 0, the parabola y = AX2______ On the left side of the axis of symmetry, the curve is from left to right______ On the right side of the axis of symmetry, the curve is from left to right______ ,______ Is the lowest point on the parabola

1. Let y = AX2 + BX + C, then A-B + C = 10, a + B + C = 4, 4A + 2B + C = 7 and a, B, C are obtained
2.S=2πR^2+2πR^2=4πR^2
3.m=n(n-1)/2
4. Up down up down down down origin
5. Small

Through (1,0) (- 1,8) and (0,2), the analytic expression of the quadratic function is obtained

Undetermined coefficient method
By substituting each point into the analytic formula of quadratic function y = ax ^ 2 + BX + C, we can get it

The image of a quadratic function passes through points (0,0), (1, - 3), (2, - 8) to find the analytic expression of the quadratic function

Let y = ax & # 178; + BX + C
According to the meaning of the title, we can set up a linear equation of three variables
0=c ①
-3=a+b+c ②
-8=4a+2b+c ③
The solution is: a = - 5, B = 2, C = 0
The analytic expression of the function is
y=-5X²+2X

The image of a quadratic function passes through points (1,0), (1,8) and (0,2) to find the analytic expression of the quadratic function
The second is (- 1,8)

Let y = ax ^ 2 + BX + C
yes
0=a+b+c
8=a-b+c
2=c
There is a solution
a=4,b=-4,c=0
Of course, it can also be set as y = a (x-1) (X-B)