It is known that the vertex of the parabola y x 1 2 4 is A,

This question a(1,4) b(-1,0) c(3,0) bc=4 This should be the case.

According to the image, if the point p is below the x-axis, then the point q must also be below the x-axis, and only when pq is parallel and equal to bc, a parallelogram can be formed, and it can be seen from q(0,m) that the ordinate of the point p must be m, so the abscissa of p is x=1-under the root number (4-m), on the left side of the y-axis or under the 1+ root number (4-m), on the right side of the y-axis. So pq = under the root number (4-m)-1 (left) or pq = under the root number (4-m) + 1 (on the right side).

That is, when x=1-under the root number (4-m), under the root number (4-m)-1=4, the solution gives m=-21 and x=-4

When x=1+(4-m) and (4-m)+1=4, m=-5 and x=4 are solved

So, in the above case p(-4,-21) or p(4,-5).

If the point p is above the x-axis, then the point q must also be below the x-axis, and a parallelogram can only be formed if bq is parallel and equal to pc.

So there must be bqo cph (o is the origin, h is p as the perpendicular foot of the x-axis) to have bq parallel and equal to pc (because bq=cp, and obq= hcp, so that bq cp).

Because the length of oq is 0-m=-m, the length of hp is also -m, that is, the ordinate of p is -m, because bo=1, hp = 1, so h(2,0) i.e., the abscissa of p is 2, and p(2,-m) is brought into the parabola y=-(x-1) 2+4 to get p(2,3).

So the coordinates of the point p are p(-4,-21),p(4,-5),p(2,3).

x=-(-4)/2=2

x=2 is substituted for y=-4x-1

y=-9x=2, y=-9 is substituted into y=x 2-4x+m

This gives -9=4-8+m

m=-5 The analytic formula of this parabola y=x -4x-5

From the inscription, the vertex coordinates of the parabola y=x 2-4x+m can be found as (2,m-4), and the coordinates of this point are brought into the line y=-4x-1.

m-4=-4*2-1

m=-5, so the analytic formula for the parabola is y=x, 2-4x-5

The calculations aren't necessarily correct, but that's the way of thinking.

x=-b/2a =2

So the x=2 of the vertex is then substituted y=-4x-1 to get y=-9, so the vertex (2,-9).

Then substitute the parabola to find m

Hit slowly--

i.e. 2=a*(-1).

a=2, so y=2x

So y=4 then 2x =4

x = 2, so x = 2

The coordinates of points A and B are Ling Trap A (1, -13 4), B (0, -9 4), then Zao Fan.

The equation for the straight line ab is y=(-9 4+13 4) (0-1)*x-9 4, which is reduced to y=-x-9 4

Let the line ab intersect the translational parabola at two points c(x1,y1),d(x2,y2).

Let the downward translation distance be k, then the parabolic equation for C2 is y=x 2-2x-9 4-k

Substituting the AB equation into the C2 equation yields it.

x-9 4=x 2-2x-9 4-k, which is reduced to x 2-x-k=0

Since c and d hail on both the straight ab and the parabola c2, the curve equations of both are satisfied.

From Vedic theorem there is x1+x2=1, x1x2=-k; y1+y2=-(x1+x2)-9/2=-11/2

y1y2=(-x1-9/4)*(x2-9/4)=x1x2+9/4*(x1+x2)+(9/4)^2=-k+117/16

then cd= [x1-x2) 2+(y1-y2) 2].

(x1+x2)^2-4x1x2+(y1+y2)^2-4y1y2]

1^2-4*(-k)+(11/2)^2-4*(-k+117/16)]

It is easy to find ab= [1-0) 2+(-13 4+9 4) 2] = 2

BC+AD=AB, CD=BC+AD+AB=2AB, i.e., CD2=4AB2

1^2-4*(-k)+(11/2)^2-4*(-k+117/16)]=4*2

Solve the equation and get k=3 4

The analytic formula for parabola c2 is: y=x 2-2x-3

2) The coordinates of the points e, f, and g are e(1,-4),f(1,0),g(0,-3).

The coordinates of the point m,n are m(m,0),n(1,n), where -4 n 0

mng=90°, then k(mn)*k(gn)=-1

i.e. (n-0) (1-m)*(n+3) (1-0)=n(n+3) (1-m)=-1

That is, m=n(n+3)+1, substituting the value range of n can get -5 4 m 5

That is, the value range of the real number m is [-5, 4,5].

y2=-3(x- 2) 5 The analysis and detailed explanation are as follows:

y1 = 3x, that is, x = y1 3 = 2py1, so, p = 1 6, y1 is the same shape (p must be the same), and the direction is opposite (the parabola in front of p with "-" is: x = -y 3

The vertex of this parabola is (0,0) and the parabola is (2, 5) which is the parabola sought, so according to the translation of the coordinates, { x=x' - h{ y=y'- k,x, y) is the point on the original image, i.e., the point on the image before translation; i.e. x = -y 3(x.)', y') is a point on the translated image, that is, a point on the translated image; i.e. the equation for the required requirements.

h, k)=(2, 5)

So, obtainable,-(y' - 5)/3 = (x'- 2) Since the parabola after translation is y2=f(x), put x'with Generation X; y'If you change it to y2, then you get, -(y2 - 5) 3 = (x-2) i.e. y2 = -3(x- 2) 5

This is the parabolic equation that is being sought.

Answer: y2=-3(x-2) 2+5;

Analysis: The parabolic equation can be set to y2=a(x-2) 2+5; ......This method of setting equations is called: vertex equations.

Law ......: If the vertices of the parabola are (m,n), then the equation for the parabola is y=a(x-m) 2+n).

Because the parabola shape and size are the same as y1, the opening direction is opposite, so a=-3

That is, the relation of the parabola y2 is y2=-3(x-2) 2+5.

Thank you.

y1=3x^2

Y2 is the same size as Y1 and the opening direction is opposite, so Y2=-3(X-2) 2+5

This is due to the fact that the shape and size of the quadratic parabola are only related to the 2nd order coefficient, and the opening direction is only related to the sign of the 2nd order coefficient.

y=(1 4)x +1, x =4(y-1), focal point (0,2), alignment y=0 (i.e., x-axis);

According to the definition of parabola, the distance from the point to the focal point and the alignment line is equal, so ac=af;

Find the l equation first, y-2=[(2 -5 4) (0+1)]x=3x 4;

Find the intersection point b of the line and the parabola: 3x 4=(1 4)x +1, x=4 (because x1=1, and x1+x2=(3 4) (1 4));

Let the slope of the line l after rotation be k, and its equation is: y=kx+2, then the abscissa of the intersection point of the x-axis with the loss is x=-2 k;

According to the title: |(2/k)+1|*|4-(-2/k)|=8；

4 = 3x x = 3 4 (miscalculated here).

x= 2/3 of the root number