Question

How do you find the domain and range of `f(x) = -7(x - 2)^2 - 9`?

Answer 1

Domain: `(-oo, oo)`
Range: (-00, -9]

Explanation:

Right now, this equation is in vertex form. That is really useful, because it tells us the vertex, which is either the maximnum or the minimum, depending on whether the graph is flipped or not.

`f(x) = color(red)(-)7 (x color(orange)(-2))^2color(blue)(-9)`

gives us a vertex of

`(- color(orange)(-2), color(blue)(-9)`

or `(2, -9)`.

That's our vertex, and thanks to the negative (`color(red)(-)`) in front of `7`, we can tell our graph is flipped, so instead of `(2, -9)` being
the minimum, it is our maximum

So, the range is easy to figure out from here:

The highest point it will be is at `-9`, so the range is `(-oo, -9]`

The domain is not restricted. So, from left to right, the graph extends forever. Thus, the domain is `(-oo, oo)`