Analyze the graph of `y=3x^(2)+2x+5`

Explanation

First observe that `a=3,b=2,c=5`. Since a`gt0` , the graph opens up, and the vertex is a mnimum. Since `h=-(2)/(6)=-(1)/(3) and k=3(-(1)/(3))^(2)+2(-(1)/(3))+5=(14)/(3)`, the vertex is `(-(1)/(3),(14)/(3))`, the minimum is `(14)/(3)`, and the range is `((14)/(3),oo)`. the axis of symmetry is `x=-(1)/(3)`. the figure below shows the graph of `y=3x^(2)+2x+5`. the graph does not cross the x-axis. this means that the solutions to the equation `3x^(2)+2x+5=0` are imaginary numbers.