Given the parabolas:
[tex]x=-2\mleft(y+3\mright)^2-4[/tex][tex]x=2\mleft(y+3\mright)^2+4[/tex]
You can identify that they are horizontal parabolas written in Vertex Form:
[tex]x=(y-k)^2+h[/tex]
Where the Vertex is:
[tex](h,k)[/tex]
In order to match each attribute to its corresponding equation, you need to find the Vertex, the Focus, and the Directrix of each parabola:
1. For the first parabola:
[tex]x=-2\mleft(y+3\mright)^2-4[/tex]
• You can identify that its Vertex is:
[tex](-4,-3)[/tex]
• By definition, the Focus of a horizontal parabola is:
[tex](h+\frac{1}{4a},k)[/tex]
In this case:
[tex]a=-2[/tex]
Then, you get that the Focus is:
[tex]=(-4+\frac{1}{4(-2)},-3)=(-4-\frac{1}{8},-3)=(-\frac{33}{8},-3)[/tex]
• By definition, the Directrix is:
[tex]undefined[/tex]
