# What is the range of  y=3x^2+2x+1?

May 12, 2015

The range represents the set of $y$ values that your function can give as output.

In this case you have a quadratic that can be represented, graphically, by a parabola.

By finding the Vertex of your parabola you'll find the lower $y$ value attained by your function (and consequently the range).

I know that this is a parabola of the "U" type because the coefficient ${x}^{2}$ of your equation is $a = 3 > 0$.
Considering your function in the form $y = a {x}^{2} + b x + c$ the coordinates of the Vertex are found as:
${x}_{v} = - \frac{b}{2 a} = - \frac{2}{6} = - \frac{1}{3}$
${y}_{v} = - \frac{\Delta}{4 a} = - \frac{{b}^{2} - 4 a c}{4 a} = - \frac{4 - 4 \left(3 \cdot 1\right)}{12} = \frac{8}{12} = \frac{2}{3}$
Giving:

So Range: $y \ge \frac{2}{3}$