1. The problem is to analyze the function $$y = -|x| + \frac{1}{2}$$ and understand its shape and key features. 2. The function involves the absolute value of $x$, which normally creates a V-shaped graph with vertex at the origin. 3. The negative sign in front of $|x|$ reflects the graph vertically, turning the V shape upside down. 4. The constant term $\frac{1}{2}$ shifts the entire graph upward by $\frac{1}{2}$ units. 5. Therefore, the vertex of the graph is at the point $(0, \frac{1}{2})$. 6. The graph decreases linearly on both sides of the vertex with slope $-1$ for $x > 0$ and slope $1$ for $x < 0$ (due to the reflection). 7. The $y$-intercept is at $y = \frac{1}{2}$ when $x=0$. 8. There are no $x$-intercepts because $- |x| + \frac{1}{2} = 0$ implies $|x| = \frac{1}{2}$, so $x = \pm \frac{1}{2}$ are the $x$-intercepts. Final answer: The graph is an upside-down V with vertex at $(0, \frac{1}{2})$ and $x$-intercepts at $\pm \frac{1}{2}$.