Translate the parabola y = x2 + 2x-8, make it pass through the origin, and write an analytical expression of the parabola after translation______ ．

Let the analytic expression of this function be y = x2 + 2x + C, then (0, 0) is suitable for this analytic expression, and the solution is C = 0. Therefore, there is an analytic expression of the parabola after translation: y = x2 + 2x (the answer is not unique)

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1. The translational parabola y = x ^ 2 + 2x + 8 is an analytical expression of the translational parabola written through the origin
2. Given the parabola y = x ^ 2-2x-8, translate the parabola along the X axis to make it pass through the origin? What does (1) translation along the X axis mean? (2) Does passing through the origin mean that the vertex of the parabola passes through the origin?
3. If the line y = kx-1 is translated upward by 3 units, and then to the right by 2 units, just passing through the origin, then K=
4. The parabola y = - (x + 3) ^ 2 shifts to the right by units just passing through the origin
5. If you translate the parabola y = x ^ 2 + BX + C upward by 2 units, then it passes through the origin. If you translate it upward by 3 units, then it has only one intersection with the X axis, Find parabola
6. 25. The parabola y = - x2 moves upward by 4 units, and then moves to the right by M (M > 0) units, just passing through the origin. The parabola after translation intersects with the straight line y = 2x The parabola y = - x2 moves upward by 4 units, then moves to the right by M (M > 0) units, and then just passes through the origin. The translated parabola and the straight line y = 2x intersect at two points a and B (a is on the left side of B) (1) The analytical formula of parabola after translation (2) Finding the coordinates of two points a and B (3) Point P is a moving point on the parabola above the straight line y = 2x. Whether there is a maximum area of △ ABP, if there is a result, then the coordinates of point P can be obtained. If there is no reason, the area of △ ABP can be calculated, (4) Point q is a point in the coordinate plane. If △ ABQ is an isosceles right triangle and ∠ BAQ = 90 °, write directly the coordinates of the point Q satisfying the condition, and answer directly whether any of these points are on the parabola after translation?
7. It is known that the two roots of the equation AX2 + BX + C = 0 are negative 2 / 3 and 1 / 2 respectively, and the parabola y = AX2 + BX + C has an intersection Q (- 1, - 3) with the straight line passing through the point P (1,2 / 3). The analytic expressions of the straight line and the parabola are obtained
8. There is a parabola y = ax ^ 2 + BX + C, and the point (m, n) is a point on the parabola
9. It is known that f (x) = x ^ 3 + ax ^ 2 + BX + 3, and the tangent equation of curve y = f (x) at point (1, F1)) is 5x + Y-3 = 0 Finding the value of a and B
10. It is known that the two roots of the equation y = AX2 + BX + C are - 1 and 3 respectively. The parabola y = AX2 + BX + C has an intersection n (2, - 3) with the straight line y = KX + m passing through point m (3,2) (1) The analytic formula of straight line and parabola (2) The two intersections of the parabola and X-axis from left to right are a and B respectively, and the vertex is p. if q is the point on the parabola different from a, B and P, and the angle QAP is equal to 90 degrees, the coordinates of q-point can be obtained
11. Translate the parabola y = x2 + 2x-8, make it pass through the origin, and write an analytical expression of the parabola after translation______ ．
12. When a changes, the two tangent points are on the tangent line () respectively A:y=1/2 x²,y=3/2 x² B:y=3/2 x²,y=5/2 x² C:y=x²,y=3x² D:y=3x²,y=5x²
13. It is known that the parabola y = x & # 178; + (2n-1) x + n & # 178; - 1 (n is a constant term) It is known that the parabola y = x & # 178; + (2n-1) x + n & # 178; - 1 (n is a constant term) (1) when the parabola passes through the origin and the vertex is in the fourth quadrant, the corresponding functional relationship is obtained. (2) Suppose that point a is a moving point on the parabola determined by (1), and it is located below the x-axis and on the left side of the symmetric state, passing through point a as a parallel line of the x-axis, intersecting the parabola with another point D, making ab ⊥ x-axis and point B, DC ⊥ X-axis at point C, then (1) when BC = 1, find the perimeter of rectangular ABCD (2) whether there is a maximum perimeter of rectangular ABCD? If it exists, calculate the maximum value and the coordinate of point A. if it does not exist, please explain the reason,
14. As shown in the figure, the parabola y = ax ^ 2 + 3 / 2x + C passes through the origin O and a (4,2), intersects the x-axis with point C, and the point m.n starts from the origin 0 at the same time, and the point m takes two single points The velocity of bit / s moves along the positive direction of y-axis, and point n moves along the positive direction of x-axis at the velocity of 1 unit / s. when one of the points stops moving, the other point stops (1). Find the analytical formula of parabola and the coordinates of point C; (2) in the process of point M. n moving, if the line Mn and OA intersect at point G, it is to judge the position relationship between Mn and OA, and explain the reason; Is there any time t that makes a quadrilateral with O, P, a and C as fixed points an isosceles trapezoid? If so, please explain the reason. {please hurry up,
15. The image of known parabola y = ax ^ 2 + BX + C passes through point a (1,0) B (4,6) (1) Find the analytical formula of the parabola (2) First, translate the parabola in (1) one unit to the left. How many translations up or down can make the parabola and the straight line AB have only one intersection? Write out the analytical formula of the parabola at this time (3) Translate the parabola in (2) 5 / 2 units to the right and t units to the down (T > 0). At this time, the parabola and X-axis intersect at two points m and N, and the line AB and y-axis intersect at point P. when t is the value, the area of the circle passing through M, N and P is the smallest? What is the minimum area? c=2
16. If the parabola y = ax + BX + C (a ≠ 0) passes through the origin, then Write relevant information
17. If Tan α = 2, the abscissa of tangent P is 2, the distance from P to the origin can be obtained
18. Given the parabola y = x2 + 3x-5, find the tangent equation of the parabola at x = 3 Do with derivative
19. Parabola y = 3x & # 178; - X-2 the tangent equation of the intersection of parabola and x-axis is solved by means of Weida theorem
20. The tangent of a point P on the parabola y = x2-3x has an inclination angle of 45 degrees. It intersects with two coordinate axes at two points a and B. the AOB surface of the triangle can be obtained