Question

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

Answer 1

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=ax^2+bx+c` the coordinates of the Vertex are found as:
`x_v=-b/(2a)=-2/6=-1/3`
`y_v=-Delta/(4a)=-(b^2-4ac)/(4a)=-(4-4(3*1))/12=8/12=2/3`
Giving:

So Range: `y>=2/3`