Write and equation of the translated or rotated graph in general form (picture below)

The answer is ellipse; 3x² + y² + 6x - 6y + 3 = 0

Step-by-step explanation:

* At first lets talk about the general form of the conic equation

- Ax² + Bxy + Cy²  + Dx + Ey + F = 0

∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.

∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.

∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.

* Now we will study our equation:

* 3x² + y² = 9

∵ A = 3 , B = 0 , C = 1

∴ B² - 4 AC = (0)² - 4(3)(1) = -12

∴ B² - 4AC < 0

∴ The graph is ellipse or circle

* If A and C are nonzero, have the same sign, and are not

 equal to each other, then the graph is an ellipse.

* If A and C are equal and nonzero and have the same

 sign, then the graph is a circle.

∵ A and C have same signs with different values

∴ It is an ellipse

* Now lets study T(-1 , 3), that means the graph will translate

 1 unit to the left and 3 units up

∴ x will be (x - -1) = (x + 1) and y will be (y - 3)

* Lets substitute the x by ( x + 1) and y by (y - 3) in the equation

∴ 3(x + 1)² + (y - 3)² = 9

* Use the foil method

∴ 3(x² + 2x + 1) + (y² - 6y + 9) = 9

* Open the brackets

∴ 3x² + 6x + 3 + y² - 6y + 9 = 9

* Collect the like terms

∴ 3x² + y² + 6x - 6y + 12 = 9

∴ 3x² + y² + 6x - 6y + 12 - 9 = 0

∴ 3x² + y² + 6x - 6y + 3 = 0

* The answer is ellipse of equation 3x² + y² + 6x - 6y + 3 = 0

the probability is 0