1. **State the problem:** We need to identify which graph corresponds to the system of equations: $$y = (x - 2)^2 + 6$$ $$y + \frac{1}{2}x = -5$$ 2. **Analyze the first equation:** The first equation is a parabola in vertex form: $$y = (x - 2)^2 + 6$$ - The parabola opens upwards because the coefficient of the squared term is positive. - The vertex is at $(2, 6)$. 3. **Analyze the second equation:** Rewrite the second equation to slope-intercept form: $$y + \frac{1}{2}x = -5 \implies y = -5 - \frac{1}{2}x$$ - The slope is $-\frac{1}{2}$ (negative slope). - The y-intercept is $-5$ (below zero). 4. **Check the points given for the line:** - At $x = -4$: $$y = -5 - \frac{1}{2}(-4) = -5 + 2 = -3$$ - At $x = 4$: $$y = -5 - \frac{1}{2}(4) = -5 - 2 = -7$$ The line passes through $(-4, -3)$ and $(4, -7)$, which is consistent with a negative slope and y-intercept below zero. 5. **Compare with graph descriptions:** - The parabola opens upwards with vertex at $(2,6)$. - The line has a negative slope and crosses the y-axis below zero. Only **Graph A** matches the parabola opening upwards with vertex at $(2,6)$ and a line with negative slope crossing y-axis below zero. **Final answer:** Graph A