The x-axis of the image intersection of the quadratic function intersects the Y-axis of the point a (- 1,0) B (3,0) and the point C (0, - 3 / 2) to find the analytical formula of the parabola The x-axis of image intersection of quadratic function is at the intersection of point a (- 1,0) B (3,0) Y-axis and point C (0,3 / 2) 1. Find the analytical formula of parabola 2. P is a moving point on the parabola above the x-axis, and the maximum area of △ PAB is calculated

The x-axis of the image intersection of the quadratic function intersects the Y-axis of the point a (- 1,0) B (3,0) and the point C (0, - 3 / 2) to find the analytical formula of the parabola The x-axis of image intersection of quadratic function is at the intersection of point a (- 1,0) B (3,0) Y-axis and point C (0,3 / 2) 1. Find the analytical formula of parabola 2. P is a moving point on the parabola above the x-axis, and the maximum area of △ PAB is calculated

∵ a (- 1,0), B (3,0) ∵ let the analytic formula be y = a (x + 1) (x-3) and C be substituted into the analytic formula, then 3 / 2 = a (0 + 1) (0-3), then a = - 1 / 2, then the parabola analytic formula is y = - 1 / 2 (x + 1) (x-3), and the symmetry axis is x = 1, then the vertex coordinate is (1,2). Let P point coordinate be (a, b) △ PAB area = (1 / 2) ×| ab | × B = 2B ∵ when p is the vertex

The image of quadratic function is a parabola passing through a (4,0) B (0, - 4) C (2,4)
(1) Finding the analytic expression of function
(2) Find its vertex coordinates

Y = ax ^ 2 + BX + C0 = 16A + 4B + C-4 = C4 = 4A + 2B + C: 16A + 4B = 44a + 2B = 8, that is 16A + 8b = 32, B = 7, a = 8-14) / 4 = - 3 / 2Y = - 3 / 2x ^ 2 + 7x-4 vertex (- B / 2a, (4ac-b ^ 2) / (4a)), that is: (- 7 / (2 * (- 3 / 2)), (4 * (- 3 / 2) * (- 4) - 7 ^ 2) / 4 * (- 3 / 2)), that is: (- 7 / (- 3)

Find the analytic expression of quadratic function that meets the following conditions: 1. The image of quadratic function passes through three points a (0,0) B (1, - 3) C (2,8) 2. The vertex of parabola is (- 1,1)
Find the analytic expression of quadratic function that meets the following conditions: 1. The image of quadratic function passes through three points a (0,0) B (1, - 3) C (2,8)
2. The vertex of the parabola is (- 1,1), and its intersection with the Y axis is a (0, - 1)
3. The image passes through (1,0), (- 1,8), and has the same opening direction and shape as the square of parabola y = 2x
4. The intersection coordinates of parabola and X-axis are (- 1,0), (1,0) and pass through point (2,3)

Description: this topic investigates various expressions of quadratic function. It is more comprehensive and best to collect!
(1) (Note: this question tests the general formula and the students' computing ability)
Let y = ax ^ 2 + BX + C (a ≠ 0, a, B, C are constants)
From the title, there is
c=0;
-3=a+b;
8=2a+b.
To solve a system of quadratic equations, we have
y=7x^2-10x.
(2) (Note: this question tests the vertex form and the students' computing ability)
Let the analytic formula be y = a (x + m) ^ 2 + K (a ≠ 0, a, m, K are constants), and the vertex coordinates be (- m, K)
From the title, there is
y=a(x+1)^2+1
And if the function is over (0, - 1), then it is substituted with, and there is
y=-2(x+1)^2+1.
(3) (Note: the note of this question is "the same opening direction and shape as the square of parabola y = 2x
”Therefore, a = 2)
Let the analytic formula be y = 2aX ^ 2 + BX + C, then the problem, image (1,0), (- 1,8) form a system of equations, and the solution is obtained
y=2x^2-4x+2.
(4) (Note: this question tests the intersection formula and students' computing ability)
Let y = a (x-x1) (x-x2)
From the title, substitute, have
y=a(x+1)(x-1)
In addition, if the image is over (2,3), then it is substituted, and a = 1 is obtained
∴ y=x^2-1