The first three quadratic functions are a problem, and now wait for 20

Solution: The image of the quadratic function y=ax 2+bx+c passes through the points a(1,5),b(-1,9),c(0,8).

List a system of ternary equations.

Disarm abc a=-1 b=-2 c=8

The quadratic function is analytically y=-x 2-2x+8

To find the coordinates of point d of point D of the symmetry axis of the quadratic function image at point a, then you first need to find the symmetry axis, there are two ways to find the symmetry axis, one is to directly write that when y=0, find the intersection point of the quadratic function and the x-axis, and you can find the symmetry axis. The second method is to match y=-x 2-2x+8 as a vertex, then I will write it as a vertex).

Received the above.

y=-x^2-2x+8

x^2+2x-8）

x+1)^2+9

The axis of symmetry of this quadratic function is the straight line x=-1

a(1,5) with respect to the coordinates of point d on the axis of symmetry of this quadratic function image is (-3,5).

A=-1, b=-2, c=8, y=-x 2-2x+8, respectively

So the axis of symmetry is x=-1

d(-3,5)

Substituting the three points a(1,5)b(—1,9)c(0,8) into the formula, the three formulas are: 5=a+b=c; 9=a-b+c;c=8 combines the above equations to obtain a=—2;b=—1；c=8.According to the axis of symmetry formula x=—b 2a; The coordinates with the axis of symmetry x=—1 are obtained as (1,5).

Because the two points a and d are symmetrical with respect to the axis of symmetry, the distance from x to the axis of symmetry is equal. The abscissa of point D is the distance from the abscissa of point A to the axis of symmetry plus 1 4 to obtain the abscissa of point D. The ordinate number should be the same as point a.

The coordinates of point d can be obtained as (—6 4,5).

Solution: 1, k + k-4 + 2x+3 of y=(k+2)x is a quadratic function, so k+2 0, and k +k-4=2. So k=-3

k = 2 is not suitable for the topic). 2, so y=-x +2x+3=-(x -2x+1)+4=-(x-1) +4, so the vertex coordinates of the parabola are (1,4) and the axis of symmetry is the straight line x=1. 3. Draw slightly.

4, because -x +2x+3=0 is x=-1, x=3, that is, the parabola y=-x +2x+3 intersects a(3,0) and b(-1,0) with the x-axis. So when -1 x 3 y 0. When x=-1, x=3, y=0...

When x -1, or x 3 y 0.

5. When x is 1, y increases as x increases.

6. Because the parabola intersects with the y-axis at C(0,3), in ABC, the bottom edge Ab=4, and the high oc is 3, so S abc=1 2ab oc=6.

7. From the title, e(0, root 3), so the analytic formula of the function after d (1, 0) and e (0, root 3) is y = - root 3x + root 3, and the tangent nature knows that the straight line tangent to the circle d is perpendicular to de, so the analytical formula is y = root 3 3x + root 3.

First, the analytic formula (intersection formula) of the function is obtained: y=a(x-0) (x-40), and the vertex is (20,16) from the image, and the analytical formula is substituted to obtain 16=a20 (-20). Get a=

i.e. y = x = 15 according to the meaning of the question(Is it 15 or 5?) If it's 15, substitute it. x=15.Get y=15If it is x=5, y=7

So the iron pillar is 15 meters (x = 5 is 7 meters).

Answer: The coordinates of the parabolic vertices are: 20,16, and the analytical formula can be set as:

y=a x 20 16, the parabola passes through the origin, and the origin coordinates are substituted into: a= 1 25, y= 1 25 x 20 16, and the coordinates of the m point are: m 15,0, so that x=15 is substituted into the analytic formula:

y=15, the length of this iron pillar = 15 meters.

(0,0) (40,0)(20,16) three-point determination parabolic analytic formula y=-1 25 x 2+8 5 x

f(40/2-5)=15

The original area is 2 2=4, and the changed area is (x+2) 2

So the increased area s= x+2) 2-4 = x 2+4x).

At x=2, the increased area is 12