The image of quadratic function y = x ^ 2-2x 5 is folded along the y-axis to obtain the analytical expression of the parabola

The image of quadratic function y = x ^ 2-2x 5 is folded along the y-axis to obtain the analytical expression of the parabola

By folding the image of quadratic function y = x ^ 2-2x + 5 along the y-axis, the analytic expression of parabola y = x ^ 2 + 2x + 5 is obtained

It is known that the abscissa of the two intersections of parabola y = ax ^ 2 + BX + C and X axis are - 1 and 3, and the ordinate of the intersection of parabola y = ax ^ 2 + BX + C and Y axis is 1

Using the intersection formula y = a (x-x1) (x-x2), we can get y = a (x + 1) (x-3). If the ordinate of the intersection with y axis is 1, we can get a = - 1 / 3, then we can get the analytic formula of the function
Method 2: directly bring in the coordinates of three points, which are (- 1,0) (3,0) (0,1)

It is known that the abscissa of the two intersection points of the parabola y = AX2 + BX + C (a ≠ 0) and X axis are - 1 and 3 respectively, and the ordinate of the intersection point with y axis is - 32; (1) determine the analytical formula of the parabola; (2) state the opening direction, symmetry axis and vertex coordinates of the parabola

(1) Let the analytic formula of parabola be y = a (x + 1) (x-3), substituting the point (0, - 32), we get - 3A = - 32, and the solution is a = 12, so y = 12 (x + 1) (x-3), that is, y = 12x2-x-32; (2) ∵ y = 12x2 − x − 32 = 12 (x − 1) 2 − 2; the opening of parabola is upward, the symmetry axis is a straight line x = 1, and the vertex coordinates are (1, - 2)

As shown in the figure, it is known that the parabola y = AX2 + BX + C (a ≠ 0) passes through three points a (- 2,0), B (0, - 4), C (2, - 4), and the other intersection point with the X axis is e. (1) find the analytical formula of the parabola; (2) find the coordinates and the axis of symmetry of the vertex D of the parabola by the collocation method; (3) find the area of the quadrilateral ABDE

(1) ∵ the parabola y = AX2 + BX + C passes through three points a (- 2, 0), B (0, - 4), C (2, - 4) ∵ 4A − 2B + C = 0C = − 44a + 2B + C = − 4. The solution is a = 12b = − 1C = − 4. The analytical formula of parabola is y = 12x2-x-4. (2) y = 12x2-x-4 = 12 (x-1) 2-92 ∵ vertex coordinates D (1, - 92)