Finding the inverse function of x = Y-Y ^ 2

x'=1-2y

Y = x / x + 2 (x is not equal to - 2) inverse function

y=x/(x+2)
∴ y(x+2)=x
∴ xy+2y=x
∴ x(1-y)=2y
∴ x=2y/(1-x)
The inverse function is y = 2x / (1-x) (x ≠ 1)

Given that the image of quadratic function y = ax ^ 2 + BX + C passes through (- 1, - 5 / 2) (0, - 4) (4,0), the analytic expression of quadratic function is_ The coordinates of point D are_ The axis of symmetry is , s quadrilateral obdc=___
B(0,-4) C(4,0)

{a-b+c=-2.5,
c=-4
16a+4b+c=0
5, B = - 1, C = - 4
So y = 0.5x ^ 2-x-4
The vertex is (1, - 4.5)
Symmetry axis line x = 1
S=0.5*(4+4.5)*1+0.5*4.5*(4-1)=11

It is known that the vertex coordinates of quadratic function image are (- 2,1) and pass through points (1, - 2), and the relation of quadratic function is obtained

From the vertex formula, the analytic formula of the quadratic function can be set as follows:
y=a﹙x＋2﹚²＋1
The analytic formula of point (1, - 2) is as follows:
－2=a﹙1＋2﹚²＋1
∴a=－1／3
The analytic expression of this quadratic function is:
y=﹙－1／3﹚﹙x＋2﹚²＋1

It is known that a parabola L: y = AX2 + BX + C (where a, B, C are not equal to 0), the coordinates of its vertex P are (& ᦉ 8722; B / 2a, 4ac & ᦉ 8722; B & ᦉ 178 / 4A), and the intersection point with the Y axis is m (0, c). We call m as the vertex, the symmetric axis is Y axis, the parabola passing through P is the adjoint parabola of parabola L, and the straight line PM is the adjoint line of L
(1) Find the analytic expressions of the adjoint line and the adjoint parabola of the parabola y = - 2x & # - 4x + 1
(2) If the adjoint parabola and the adjoint straight line of a parabola are y = - X & # - 3 and y = - x-3 respectively, then the analytical formula of the parabola is?
(3) The analytic expressions of adjoint parabola and adjoint straight line of parabola L: y = AX2 + BX + C (where a, B, C are not equal to 0) are obtained

It is known that a parabola L: y = AX2 + BX + C (where a, B, C are not equal to 0), the coordinates of its vertex P are (& ᦉ 8722; B / 2a, 4ac & ᦉ 8722; B & ᦉ 178 / 4A), and the intersection point with the Y axis is m (0, c). We call m as the vertex, the symmetric axis is Y axis, the parabola passing through P is the adjoint parabola of parabola L, and the straight line PM is the adjoint line of L
(1) Find the analytic expressions of the adjoint line and the adjoint parabola of the parabola y = - 2x & # - 4x + 1
(2) If the adjoint parabola and the adjoint straight line of a parabola are y = - X & # - 3 and y = - x-3 respectively, then the analytical formula of the parabola is?
(3) The analytic expressions of adjoint parabola and adjoint straight line of parabola L: y = AX2 + BX + C (where a, B, C are not equal to 0) are obtained
(3) Analysis: ∵ parabola L: y = AX2 + BX + C (where a, B, C are not equal to 0)
The intersection point of vertex P and y-axis is m (0, c)
Let its adjoint parabola be y = MX ^ 2 + C
∵ over P point
∴y=mb^2/(4a^2)+c=4ac−b²/4a
mb^2/(4a^2)=−b²/4a==>m=-a
The adjoint parabola: y = - ax ^ 2 + C
Adjoint line: k = (- B ^ 2 / (4a) / (- B / (2a)) = B / 2 = = > y = B / 2x + C
(1) Analysis: ∵ parabola y = - 2x & # 178; - 4x + 1
Vertex: P (- 1,3), the point of intersection with y axis is m (0,1)
Adjoint parabola: y = 2x ^ 2 + 1
Adjoint line: k = (3-1) / (- 1) = - 2 = > y = - 2x + 1
(2) Analysis: the adjoint parabola and adjoint straight line of a parabola are y = - X & # 178; - 3 and y = - x-3 respectively
∴M(0,-3)
b/2=-1==>b=-2
-a=-1==>a=1
The analytical formula of this parabola is y = x ^ 2-2x-3

How to solve the geometry problem of quadratic function in junior high school mathematics?

I've just finished my senior high school entrance examination and some junior high school knowledge and experience can tell you. (PS, my score is OK). First of all, if you can determine the analytic formula of quadratic function, you should first determine the analytic formula and eliminate the letters of the analytic formula as much as possible

We need geometry exercises in grade 2 and quadratic function exercises in grade 3. Quadratic function should have answers,

1. Fill in the blanks:
1. As shown in the figure, take the origin of the rectangular coordinate system as the center of the circle and 4 as the diameter to make a circle. The fan-shaped areas of the straight line L passing through the origin and in the positive direction of the x-axis are respectively P and Q. try to write the analytic expression of the function of P with respect to Q________________ .
2. The center of the square ABCD with side length of 2 is at the origin of the plane rectangular coordinate system, the four sides are perpendicular to the coordinate axis, and the point P is a point on the X axis, so that the coordinates of the vertex P of the regular triangle with P as the vertex and the side of the square as the edge are____________ .
3 as shown in the figure, take a rectangular piece of paper, length AB = 10cm, width BC = 5 * radical (3) cm, fold it in half with the dotted line Ce (point E on AD) as the crease, so that point d falls on edge AB, then AE = - - CM, ∠ DCE = - - degree
4. In rectangular coordinate system, circle O and line y = - 4x / 3 + 4 are tangent to point C, then the coordinate of point C is_____
2. Calculation questions:
1. In ⊿ ABC, ab = 8, AC = 6, D is the point on BC, BD: DC = 2:3
Find the value range of AD
2. As shown in the figure, square ABCD, points m and N are on BC and CD respectively, so that Mn = BM + DN can find the size of ∠ man
3. Comprehensive questions:
1. It is known that the x-axis of the image of quadratic function y = - x2 + 8x-12 intersects at two points a and B, and the image of primary function passes through two points a and C (3,3)
(1) find the analytic formula of the first-order function
(2) when x is the value, the value of the first function is less than that of the second function
(3) can we find a point P on the symmetry axis of quadratic function image to minimize the value of PA + PC
2. As shown in the figure, in the plane rectangular coordinate system, the straight line AB intersects the x-axis and y-axis at points a and B respectively, and OA = ob = B. take point o as the center of the circle, a (a < b) as the radius, draw the circle, intersect the straight line AB at points c and D respectively, and make CF ⊥ OA, CE ⊥ ob, and the perpendicular feet are points F and e respectively
(1) find the function analytic formula of line ab;
(2) find the perimeter of the rectangular Office (expressed by the algebraic formula containing a and b);
(3) set point P as any moving point on the straight line AB, then make PF ⊥ X axis and PE ⊥ Y axis through point P respectively, and the perpendicular feet are points F and e respectively. Try to explore whether the perimeter of rectangular ofPE is a fixed value? And explain the reason
3. As shown in Figure 3, in the plane rectangular coordinate system, the line AB intersects the x-axis and y-axis at points a and B respectively, with OA = 6 and ob = 8. Then, take point o as the center of the circle and 5cm long as the radius, draw the circle intersection line AB at points c and D, and intersect the negative half axis of x-axis at point M
(1) find the analytic formula of line AB; (2) find the coordinate of point C;
(3) find the analytical formula of parabola passing through points a, C and m;
(4) is there a point P on the parabola of (3) such that the area of △ PAM is 11? If so, ask for the coordinates of point P. if not, explain the reason

A quadratic function problem in the third grade of junior high school~
As shown in the figure, the image of the quadratic function y = ax ^ 2 + BX + C (a = / = 0) passes through the point (- 1,2), and the abscissa of the intersection point with the X axis are X1 and X2 respectively, where - 2 < x1

Select (d)
1. It is known from the problem that the opening of the image of the quadratic function y = ax & sup2; + BX + C (a ≠ 0) is downward, that is, a < 0,
When x = - 2, y = 4a-2b + C, the point (- 2,4a-2b + C) is in the third quadrant, so it is correct
2. Also - 2 < x1

A problem of quadratic function in the third grade of junior high school
It is known that the image of quadratic function y = - 1 / 2x + m passes through point a (- 2,3) and intersects with X axis at point B. the image of quadratic function y = AX2 + bx-2 passes through points a and B
1. The analytic expressions of these two functions are obtained respectively
2. If the image of quadratic function is translated along the positive direction of y-axis, the translated image intersects with the image of primary function at point P, and intersects with Y-axis at point Q, but when PQ ‖ x-axis, how many units is the image of quadratic function translated
Mainly the second question!
It's better to have a problem-solving process, and if not, explain how and why
There is only one answer to refusal

Mathematics forecast volume of 2009 Pudong New Area senior high school entrance examination (question 24)

If you translate the parabola y = x square 4 units to the left, then 2 units to the down, and then rotate 180 degrees clockwise with the vertex as the center, then the analytical formula of the parabola is?
My answer is y = - (x + 4) square - 2, but the answer is y = - 4 (x + 4) square - 2
So am I wrong

It's a wrong answer. Remember that the left-right translation of the image is the change of X, left plus right minus. Up and down translation is y translation, up plus down minus. Rotating 180 degrees is to add a negative sign to a in the general formula of quadratic function. You're right

Finding the inverse function of x = Y-Y ^ 2