To identify the grpah correspondig to the given function find the zeros of the function (use the quadratic formula):
[tex]\begin{gathered} f(x)=\frac{1}{2}x^2+2x-6 \\ \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \\ a=\frac{1}{2} \\ \\ b=2 \\ c=-6 \\ \\ \\ x=\frac{-2\pm\sqrt[]{2^2-4(\frac{1}{2})(-6)}}{2(\frac{1}{2})} \\ \\ x=\frac{-2\pm\sqrt[]{4-(-12)}}{1} \\ \\ x=\frac{-2\pm\sqrt[]{16}}{1} \\ \\ x=\frac{-2\pm4}{1} \\ \\ x_1=-2+4=2 \\ x_2=-2-4=-6 \end{gathered}[/tex]As the zeros are 2 and -6 the graph cross the x-axis in x=2 and x=-6
