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

Mar 26, 2015

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:
$a {x}^{2} + b x + c = 0$):
${x}_{v} = - \frac{b}{2 a}$
${y}_{v} = - \frac{\Delta}{4 a}$
with $\Delta = {b}^{2} - 4 a c$

${x}_{v} = - \frac{- 3}{2 \cdot - 3} = - \frac{1}{2}$
${y}_{v} = \frac{9 - 4 \left(- 3 \cdot 4\right)}{4 \cdot - 3} = 4.75$
So your range is all the $y$ less or equals to $4.75$.