when x=2 and y=9
would mean that at a distance of 2 km form the town centre the population density is 9 thousand per km^2
at the town centre, x = 0, so
y = 0+0+9 or 9000 per km^2
The max density would be at the vertex of your parabola.
A quick way to find the vertex:
for y = ax^2 + bx + c, the x value of the vertex is -b/(2a)
so for yours, x = -6/-6 = 1
when x=1, y = -3+6+9 = 12
so 1 km from the centre, the density is 12000/km^2
Since y is your density, and density is obviously a positive number, we would use your parabola only when it is above the x-axis
so let's find the x-intercepts
-3x^2 + 6x + 9 = 0
x^2 - 2x - 3 = 0
(x-3)(x+1) = 0
x = 3 or x = -1
but x = -1 makes little sense here, since clearly x > 0
so your model would only be useful for
x between 0 and 3, and the range of density would be from 0 to 12.
Brilliant, thanks very much for that
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