Given that the square of the parabolic equation y = MX (M belongs to R, and M is not equal to 0) 1, if the focal coordinate of the parabola is (1,0), the equation of the parabola is solved

Given that the square of the parabolic equation y = MX (M belongs to R, and M is not equal to 0) 1, if the focal coordinate of the parabola is (1,0), the equation of the parabola is solved

y^2=2px,
The parabolic focal coordinates are (1,0) P / 2 = 1, P = 2
Parabolic equation:
y^2=4x

It is known that the parabola y = - x2 + 2x + M-1 has two intersections A and B with the x-axis. (1) find the value range of M; (2) if the coordinate of point a is (- 1,0), find the analytical formula of the parabola and the coordinate of the top point C; (3) is there a point P (not coincident with point C) on the parabola in question (2) so that s △ PAB = s △ cab? If it exists, find out the coordinates of point p; if not, explain the reason

(1) There are two intersections between the parabola and the x-axis, that is, b2-4ac = 22-4 × (- 1) × (m-1) = 4 + 4m-4 = 4m & gt; 0; 0; (2) ∵ A's coordinates are (- 1, 0), ∵ - (- 1) 2 + 2 × (- 1) + M-1 = 0, the solution is m = 4, ∵ parabolic analytic formula is y = - x2 + 2x + 4-1 = - x2 + 2x + 3, ∵ y = - x2 + 2x + 3 = - (x2-2x + 1) + 3 + 1 = - (x-1) 2 + 4, the coordinates of vertex C are (1, 4); (3) there are points P (1-22, - 4) or (1 + 22, - 4) According to (2), the coordinate of point C is (1,4), and the distance from point C to AB is 4. We can find the point P below the x-axis, so that s △ PAB = s △ cab. At this time, the ordinate of point P is - 4, - x2 + 2x + 3 = - 4. We can sort out that x2-2x-7 = 0. We can get the solution that x = - B ± b2-4ac2a = - (- 2) ± (- 2) 2-4 × 1 × (- 7) 2 × 1 = 1 ± 22, and there is point P (1-22, - 4) or (1 + 22, - 4) Let s △ PAB = s △ cab

The coordinates of the intersection of the parabola y = 4x2-1 and the X axis are
When y = 0, I get 1 / 2 or - 1 / 2, right

y=4x²-1
If y = 0 is substituted, then 4x & # 178; - 1 = 0
That is (2x + 1) (2x-1) = 0
Or 2x + 1 = 0, or 2x-1 = 0
The solution is x = - 1 / 2, or x = 1 / 2
Then the intersection coordinates are (- 1 / 2,0) and (1 / 2,0)
When we talk about the coordinates of the intersection, we should write the X and Y coordinates in perfect pairs

The parabola passes through points (2, - 3), and the abscissa of its intersection with X axis is - 1 and 3
(1) Find the analytical formula of parabola (2) find the symmetry axis and vertex coordinates of parabola by matching method; (3) draw a sketch; (4) observe the image, when x takes what value, the function value is less than zero? When x takes what value, y decreases with the increase of X?

Let y = a (x + 1) (x-3) bring in point (2, - 3) a = 1, y = x & sup2; - 2x-3 (2) y = x & sup2; - 2x + 1-4 = (x-1) & sup2; - 4 axis of symmetry x = 1 vertex coordinates (1, - 4) (3) slightly (4) when x is greater than - 1 and less than 3, function value is less than zero, when x is less than 1, y decreases with the increase of X