A very complex function problem, a function problem

The axis of symmetry is. b/2a=1 => b=-2a

After (-1,0), 0,-3).

0=a-b+c

3=ca-b=3

a=1, b=-2, c= -3

y=x^2-2x-3

With respect to x=1 do a point symmetry point a' (3,0), connect a'c to cross x=1 to m(1,-3 2).

This m is the point sought.

point b is (3,0) and p is (1,y).

bp 2 is y 2+4

CP 2 is (Y+3) 2+1

BC 2 is 18

Triangular PCB:

pc^2+cb^2=pb^2

y+3)^2+1+18=y^2+4

6y+21=0

y=-7/2

Therefore p is (1,-7 2).

Let this line be y=kx+b

mk+b=n①

nk+b=m②

m-n)k+b=n-m③

Get (m-n)k=n-m

b=0k=-1

The function passes. 2. Four quadrants.

As we know, as shown in the figure (not allowed in the above figure, you will draw a picture by yourself), the image of the primary function y=kx+b intersects with the x-axis and y-axis respectively at the points a(4,0),b(0,2), and the line segment ab is used as one side to make an equilateral triangle abc, and the point c is on the image of the inverse proportional image y=m of negative x. (The position of c is not stated in the original question, there should be two points).

1) Find the value of m.

2) O is the origin, whether there is a point P on the vertical bisector of the ** segment ob, so that the area of the triangle ABP is equal to one-half m, if so, find the coordinates of the P point; If not, please explain why.

1) Analysis: The image of the function y=kx+b intersects with the x-axis and y-axis at the points a(4,0) and b(0,2) respectively

Function y=-1 2x+2

Let c(x,y).

x-4) 2+y 2=x 2+(y-2) 2==>y=2x-3, i.e. the ab perpendicular equation.

ab|^2=4+16=20

x-4)^2+y^2=20==>5x^2-20x+16=0==>x1=-1/2, x2=4

c(-1, 2,-4) or c(4,5).

Point c is on the image of the inverse proportional image y=-m x.

m1=-2, or m2=-20

When m=-20.

The intersection of the perpendicular bisector of ob and ab is (2,1).

s(⊿abp)=1/2*(x-2)*2=x-2=-10==>x=-8

When m=-2.

s(⊿abp)=1/2*(x-2)*2=x-2=-1==>x=1

There is a problem: the m values obtained are all negative, and the triangle area cannot be negative.

It should be said that the p-point does not exist.

Find the value of k b according to ab. ab:y=-x 2+2 set c (a,b) line segment|ab|= 20 under the root number, abc is an equilateral triangle, so the distance between the two points of c and ab is equal to the 20 under the root number, so there is.

A 2+(b-2) 2=(a-4) 2+b 2 is simplified to show that b=2a-3 is 15 below the root number

So there is |x/2+y-2|(1 4+1) = 15 under the root number is simplified to obtain: |a/2+b-2|= 5/2 times the root number 3 gives the value of ab. c (2 + root number 3, 2 times root number 3 plus 1) and (2-root number 3, 1-2 times root number 3).

Point c is on the inverse proportional function, and the value of m can be obtained by substitution.

They are (8-5 times root number 3) < 0, 8 + 5 times root number 3.

The perpendicular bisector of ob is y=1, and the distance from p(x,1)p to ab can be set to |x/2+1-2|The area of the triangle apb under the root number (1 4 + 1) = 1 2|ab|Multiply the distance from p to ab Question requirement = m 2 to find x

m has two and one is negative).

It's too hasty, I don't know if it's right.

It is known that by f(x+1)=f(2-x)=f[3-(x+1)]f(x)=f(3-x).

So getting f(x) is about x=3 2 symmetry.

So the distribution of these 101 roots is also symmetrical with respect to the symmetry.

That is, one root is 3 2, and the remaining 100 roots can be divided into 50 pairs, and the two roots of each pair are symmetrical with respect to x=3 2.

Using the midpoint coordinate formula, the sum of these 100 roots is equal to 3 2 100 150 and the sum of all 101 roots is 3 2 101 303 2

Easy to know f(

So the function is about symmetry.

So if a certain x makes f(, then there must be f(

i.e. and both are the roots of the equation.

And the sum of these two roots is.

Since a total of 101 roots are odd, x= is also the root of the equation.

Then the original equation has 50 pairs of symmetry roots and a root of x=.

The sum of all roots is 50*3+

Let the point on f(x) be (x,y).

The point of symmetry with respect to (1,0) is (2-x,-y).

2-x, -y) on y=x+1 x.

Generations get -y=2-x+1 (2-x).

y=x-2+1/(x-2)