Given that the parabola y = ax2-2x + 1 has no intersection with the X axis, then the quadrant of the vertex of the parabola is () A. Fourth quadrant B. third quadrant C. second quadrant D. first quadrant

∵ the parabola y = ax2-2x + 1 has no intersection with the x-axis, ∵ △ = 4-4a < 0, the solution is: a > 1, ∵ the opening of the parabola is upward, and ∵ B = - 2, ∵ B2A > 0, ∵ the symmetry axis of the parabola is on the right side of the y-axis, and the vertex of the parabola is in the first quadrant

The parabola y = x ^ 2-2 (M + 1) x + n passes through the point (2,4), and its vertex is on the straight line y = 2x + 1
(1) Find the analytical expression of the parabola
(2) Find the area of the triangle formed by the line y = 2x + 1 and the symmetry axis and X axis of the parabola

(1) The parabola y = x ^ 2-2 (M + 1) x + n passes through the point (2,4), so 2 ^ 2 - 2 (M + 1) × 2 + n = 4, so n = 4m + 4, so y = x ^ 2-2 (M + 1) x + n can be changed into y = x ^ 2 - 2 (M + 1) x + 4m + 4 to form the vertex formula y = [x - (M + 1)] ^ 2-m ^ 2 + 2m + 3, and the vertex coordinates are (M + 1, - m ^ 2 + 2m + 3) because the vertex is on the straight line y = 2x + 1, so 2

Given that the vertex of the parabola y = - (x + m) ^ 2 is on the straight line y = - 2x + 6, then M=

Solution
The parabolic equation is as follows
(x+m)²=-y.
Vertex (- m, 0)
According to the question set, 2m + 6 = 0
∴m=-3

It is known that the vertex of the graph of a first-order function passing through the parabola y = - 2x & # - 4x is a point (1,0), then the analytic expression of the first-order function is

Parabola y = - 2 (x ^ 2 + 2x + 1) + 2 = - 2 (x + 1) ^ 2 + 2,
Vertex coordinates: (- 1,2),
Let the analytic formula of the straight line be y = KX + B, and the equations are obtained
2=-K+b
0=K+b、
The solution is k = - 1, B = 1,
The analytic formula of a function: y = - x + 1

Given that the parabola y = ax2-2x + 1 has no intersection with the X axis, then the quadrant of the vertex of the parabola is () A. Fourth quadrant B. third quadrant C. second quadrant D. first quadrant