which is the graph of the function $f(x) = \\frac{1}{2}x^2 + 2x - 6$?

Explanation:

Step1: Find vertex of parabola

For $f(x)=ax^2+bx+c$, vertex $x=-\frac{b}{2a}$. Here $a=\frac{1}{2}$, $b=2$, so $x=-\frac{2}{2\times\frac{1}{2}}=-2$. Calculate $f(-2)=\frac{1}{2}(-2)^2+2(-2)-6=\frac{1}{2}(4)-4-6=2-10=-8$. Vertex: $(-2,-8)$

Step2: Find x-intercepts

Set $f(x)=0$: $\frac{1}{2}x^2+2x-6=0$ Multiply by 2: $x^2+4x-12=0$ Factor: $(x+6)(x-2)=0$ Solutions: $x=-6$ and $x=2$ Intercepts: $(-6,0)$ and $(2,0)$

Step3: Match to graph

The top graph has vertex $(-2,-8)$ and intercepts $(-6,0),(2,0)$.

Answer:

The topmost provided graph (the first one with vertex at $(-2,-8)$ and x-intercepts at $x=-6$ and $x=2$) is the graph of $f(x)=\frac{1}{2}x^2+2x-6$.