1. The problem asks to determine if the discriminant of the quadratic function $g(x)$ is positive, zero, or negative based on the graph description. 2. From the description, $g(x)$ is a parabola opening downwards with vertex at $(0,1)$. 3. The parabola intersects the y-axis at $1$, which matches the vertex's y-coordinate. 4. It crosses the x-axis near $-1$ and $1$, indicating two distinct real roots. 5. The discriminant $\Delta$ of a quadratic $ax^2 + bx + c$ determines the nature of roots: - $\Delta > 0$: two distinct real roots - $\Delta = 0$: one real root (double root) - $\Delta < 0$: no real roots (complex roots) 6. Since the parabola crosses the x-axis twice, the discriminant must be positive. 7. Therefore, the discriminant of $g$ is positive.