Given that the vertex of the parabola y = (x + a) square + 2A square + 3a-5 is on the coordinate axis, find a and vertex coordinates

Given that the vertex of the parabola y = (x + a) square + 2A square + 3a-5 is on the coordinate axis, find a and vertex coordinates

Given the vertex coordinates of the parabola y = (x + a) + 2A + 3a-5 on the coordinate axis, find the value of the letter A, and point out the vertex coordinates,

Given that the vertex of the parabola y = (x + a) square + a square + 3a-4 is on the coordinate axis, find the value of the letter A, and point out the vertex coordinates?

y=(x+a) ²+a²+3a-4
Vertex (- A, a & # 178; + 3a-4)
If the vertex is on the axis, then
-A = 0 or a & # 178; + 3a-4 = 0
a²+3a-4＝0
（a-1）（a+4）＝0
A = 1 or a = - 4
When a = 0
a²+3a-4＝-4
Then when a = 0, vertex (0, - 4)
When a = 1, vertex (- 1,0)
When a = - 4, vertex (4,0)

The square of the parabola y = x is first translated down two unit lengths, and then rotated 180 degrees around its vertex

The parabola y = - (x-1) ^ 2 + 1 is rotated 180 ° around its vertex and then translated up and down to make it intersect with the y-axis at the point (0,5)

When the parabola y = - (x-1) & # 178; + 1 rotates 180 ° around its vertex, the opening direction of the vertex is still (1,1), which is just opposite to the original. At this time, the equation is y = (x-1) & # 178; + 1 moves downward, and the ordinate of the vertex will change. If it is set to t, then the equation is y = (x-1) & # 178; + T. substituting (0,5) into 5 = 1 + T, the solution is obtained

The parabola y = - (x-1) & sup2; + 1 is rotated 180 degrees around the vertex and then translated up and down to make it intersect with the straight line y = 2x-3 at a point on the y-axis to find a new intersection between the parabola and the straight line

The parabola y = - (x-1) & sup2; + 1 changes to y = (x-1) & sup2; + 1 after 180 degrees around the vertex. Let y = (x-1) & sup2; + 1 translate B units up and down, B can be positive or negative, positive number means up and negative number means down, then the new parabola is y = (x-1) & sup2; + 1 + B. the line y = 2x-3 intersects Y axis

The ellipse equation with the center at the origin, the line 3x + 4y-12 = 0 and the focus of the two axes as the vertex and focus respectively is ()

The two intersections are (0,3), (4,0), 3

It is known that the parabola y is equal to the square of MX - 3 (M + 2) x + 3M + 12
(1) When m is the value, there are two intersections between the parabola and the x-axis?
(2) If the x-axis of the parabola intersects with a and B, they are on both sides of the origin and ab is equal to 2 √ 3, the analytic expression of the quadratic function is obtained
Please tell me the process and the answer, thank you!

(1) Let MX ^ 2-3 (M + 2) x + 3M + 12 = 0
When there are two intersections of parabola and x-axis, the quadratic equation of one variable has two unequal heels
△=b^2-4ac=-m^2-4m+12>0
Get - 6

The parabola y = ax & # 178; + BX + C passes through points (0,1) and (2, - 3). Write out two analytic expressions of functions satisfying the above two points!
I want to know the process

First, the point (0,1) is taken into the parabolic equation to obtain C = 1,
Then (2, - 3) is substituted into the equation to get 2A + B = - 2, and a is not equal to zero. There are many such equations, as long as a and B satisfy the above conditions. For example, if a takes 1, B = - 4, then the equation is y = x & # 178; - 4x + 1. You can also make a = - 1, then B = 0, then y = - X & # 178; + 1

If the abscissa of the intersection of the parabola y = ax & # 178; + BX + C and X axis is - 1,5 respectively, then when x satisfies, the value of the function is y

When x = - 1, A-B + C = 0
&When x = 5, 25A + 5B + C = 0
B = - 4A, C = - 5A are obtained from the above two formulas
So y = ax2-4ax-5a
Since the original problem is a parabola, it is impossible for a to be 0
The solution is y = ax2-4ax-5a & lt; 0
When a & gt; 0, x2-4x-5 & lt; 0, (X-5) (x + 1) & lt; 0
-1<x<5,
When a & lt; 0, x2-4x-5 & lt; 0, (X-5) (x + 1) & gt; 0
X & lt; - 1 or X & gt; 5
To sum up, when a & gt; 0, - 1 & lt; X & lt; 5,
X & lt; - 1 or X & gt; 5

The function y = MX2 + x-2m (M is a constant), and the intersection of the image and X-axis has the following properties______ One

When m = 0, y = x, the intersection of image and X-axis has one