## It is known that the parabola y = ax ^ 2 + BX + C intersects with the X axis at a and B, intersects with the Y axis at point C (0,3), and the axis of symmetry is a straight line x = 2 (1) (2) Let p be a fixed point on the axis of symmetry, and find the minimum value of the perimeter of the triangular APC (3) Let d be a point on the parabola and E be a point on the axis of symmetry. If the quadrilateral with points a, B, D and E as vertices is a diamond, the coordinates of point D are_____ (both points a and B are on the positive half axis of X axis) That's what it says on the paper. I'm also an excellent student. I really don't have any extra conditions to ask

[reference answer] children's shoes, do you think the title information is complete? You should know at least one coordinate of a and B. It is obtained from the intersection of the function and Y axis at C (0,3): C = 0, so y = ax ^ 2 + BX. Since the axis of symmetry is x = 2 = - B / (2a), that is, B = - 4A, the parabolic analytical formula is y = ax ^ 2 - 4ax. It is required that the analytical formula of the function must have 3

### It is known that the image of function f (x) is symmetric about the y-axis, the image of function g (x) is symmetric about the origin, and f (x) + G (x) = x times of 10 Find f (x) and G (x)

Analysis: such problems should be solved by skillfully constructing a system of binary first-order equations about f (x) and G (x),

Ontology should grasp the image characteristics of odd function and even function to skillfully construct equations

Meaning of the question:

The graph of function f (x) is symmetric about the Y axis,

Therefore, the function f (x) is an even function, that is, f (x) = f (- x)

The image of function g (x) is symmetric about the origin,

Therefore, the function g (x) is an odd function, that is, G (x) = - G (- x)

F (x) + G (x) = 10 ^ x (where 10 ^ X represents the x power of 10). (1)

The above formula can be obtained by substituting - X:

f(-x)+g(-x)=f(x)-g(x)=10^(-x) .(2)

The solution of simultaneous equations (1) and (2) is as follows:

f(x)=[10^x+10^(-x)]/2

g(x)=[10^x-10^(-x)]/2

### The symmetry axis of the image of function f (x) = 3sin (2x - π / 3)

Let 2x - π / 3 = π / 2 + K π (K ∈ z)

The solution is x = 5 / 12 π + K π / 2 (K ∈ z)

Therefore, the axis of symmetry of the image with function f (x) = 3sin (2x - π / 3) is x = 5 / 12 π + K π / 2 (K ∈ z)

### Let the function f (x) = √ 3 / 2 - √ 3sin ^ 2wx sinwxcoswx, and the distance from one symmetry center of the image to the nearest symmetry axis is π / 4 1. Find w 2. Find the maximum and minimum values of F (x) on [π, π / 2]

f(x)=√3/2-√3sin ² wx-sinwxcoswx

=(√3/2)*(1- 2sin ² wx）- (1/2)*2sinwxcoswx

=(√3/2)*cos(2wx）- (1/2)*sin(2wx)

=cos(2wx)*cos(π/6) - sin(2wx)*sin(π/6)

=cos(2wx + π/6)

Given that the distance between a symmetry center of the function image and the nearest symmetry axis is π / 4, there are:

Minimum positive period T = 2 π / (2W) = 4 * π / 4

The solution is: w = 1

Then the analytical formula of the function can be written as: F (x) = cos (2x + π / 6)

If π / 2 ≤ x ≤ π, then π ≤ 2x ≤ 2 π

There are: 7 π / 6 ≤ 2x + π / 6 ≤ 13 π / 6

Therefore, when 2x + π / 6 = 2 π, that is, x = 11 π / 12, the maximum value of the function is 1;

When 2x + π / 6 = 7 π / 6, i.e. x = π / 2, the minimum value of the function is - (√ 3) / 2

### Y = x + 1 / (x-1) proves that the image of the function is a centrosymmetric graph, and its symmetry center is obtained one 2. And write clearly to prove that a function is a centrosymmetric graph, and the general method and general method of finding its center

The function is y = x + 1 / (x-1);

Move the function graph down one unit to get:

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

Move the figure one more unit to the left:

y=x+1/x;

The function is an odd function, which is a centrosymmetric figure, and the symmetry center is the origin,

Therefore, the given function is also a centrosymmetric graph, and the center of symmetry is (1,1);

Note: for the function y = f (x), move m downward, and the unit is y = f (x) - M;

Move m units to the left, y = f (x + m);

Moving up and right is similar

### It is proved that the function image of x-3 / X is a centrosymmetric graph, and the centrosymmetric graph is obtained~~~ The process should be more detailed. Thank you

y = x - 3/x

Y = x is symmetrical about the origin (a line bisecting one or three quadrants)

Y = - 3 / X is symmetric about the origin (hyperbola in the second and fourth quadrants)

The addition is still symmetrical about the origin