The equation of a parabola with vertex (0, 0), that opens downward and has a focal width of 6 is x²=-24y.
The general equation of a parabola is given by the formula,
[tex](x-h)^2=4p(y-k)[/tex]
where p is the focal length of the parabola, while (h,k) is the coordinate of the centre of the parabola.
Given that the vertex of the parabola is at (0,0), while the focal width is 6. Therefore, the equation of the parabola can be written as,
[tex](x-h)^2=4p(y-k)\\\\(x-0)^2=4(6)(y-0)\\\\x^2= 24y[/tex]
Now, since it is required that the parabola opens downwards, therefore, the equation should be written as,
[tex]x^2 = -24 y[/tex]
Hence, the equation of a parabola with vertex (0, 0), that opens downward and has a focal width of 6 is x²=-24y.
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