Finding the Minimum Value of a Parabola: Tips and Techniques

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What is the smallest value in the image set of the function:

f(x)=3x^2+10x+219
 
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Welcome to the forum!

The vertex (minimum or maximum) of a parabola $f(x)=ax^2+bx+c$ has $x$-coordinate $x=-\dfrac{b}{2a}$.

For the future, please read the forum http://mathhelpboards.com/rules/, especially rule #11. In this case, you could write what facts about parabolas you know and what exactly your difficulty is in applying them.