Question

How do you find the domain and range of `y = −3x^2 − 3x + 4`?

Answer 1

This is a quadratic funtion and the domain is all the Real `x`. This means that you can give every value of `x` in the real domain and your function devolves a value of `y`.

The range is a little bit tricky.
The graph of a quadratic is a PARABOLA...basically a U shaped curve.

The position of the lowes (or highest) point gives us the possibility to "see" the range!
Your quadratic has `-3` in front of `x^2` so it is a "sad" parábola, a inverted U shaped curve
The highest point is called the Vertex and is given (the coordinates) as (if you have your quadratic in the general form:
`ax^2+bx+c=0`):
`x_v=-b/(2a)`
`y_v=-Delta/(4a)`
with `Delta=b^2-4ac`

in your case:
`x_v=-(-3)/(2*-3)=-1/2`
`y_v=[9-4(-3*4)]/(4*-3)=4.75`

So your range is all the `y` less or equals to `4.75`.

Graphically:
graph{-3x^2-3x+4 [-10.62, 7.16, 1.22, 10.11]}