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

Sep 4, 2015

The domain is R (all real numbers) or $\left(- \infty , + \infty\right)$ and range is ($\frac{5}{3} , \infty$)

#### Explanation:

It is a quadratic function, hence first it can be written vertex form as following:

y= $3 \left({x}^{2} + \frac{4}{3} x\right) + 3$

= $3 \left({x}^{2} + \frac{4}{3} x + \frac{4}{9}\right) - \frac{4}{3} + 3$

=$3 {\left(x + \frac{2}{3}\right)}^{2} + \frac{5}{3}$

The function therefore represents a vertical parabola, opening up with vertex at $\left(- \frac{2}{3} , \frac{5}{3}\right)$

The domain is therefore R (all real numbers) or $\left(- \infty , + \infty\right)$ and range is ($\frac{5}{3} , \infty$)