answer:
To write the equation in standard form, we need to complete the square for both the x and y terms. Let's start with the x terms:
-x^2 - 2x + y^2 + 6y - 73 = 0
To complete the square for the x terms, we need to add (2/2)^2 = 1 to both sides of the equation:
-x^2 - 2x + 1 + y^2 + 6y - 73 + 1 = 0 + 1
Simplifying, we have:
-(x^2 + 2x + 1) + y^2 + 6y - 72 = 1
Now, let's do the same for the y terms:
-(x^2 + 2x + 1) + (y^2 + 6y + 9) - 72 = 1 + 9
Simplifying further:
-(x^2 + 2x + 1) + (y^2 + 6y + 9) = 82
Factoring the x and y terms:
-(x + 1)^2 + (y + 3)^2 = 82
This is the standard form of the conic.
To plot the graph, you can use a graphing calculator like Desmos. Set the equation to -(x + 1)^2 + (y + 3)^2 = 82 and plot it.
The vertices of the graph can be found by changing the signs in the equation:
(x + 1)^2 - (y + 3)^2 = 82
The x and y coordinates of the vertices can be found by setting the expressions inside the parentheses to zero:
x + 1 = 0 => x = -1
y + 3 = 0 => y = -3
So, the vertices of the graph are (-1, -3).
real