1. **State the problem:** We are given the function $$g(x) = -(x + 5)^2 + 8$$ and need to find: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 2. **Formula and rules:** - The function is a quadratic in vertex form $$g(x) = a(x-h)^2 + k$$ where vertex is at $$(h, k)$$. - Since $$a = -1 < 0$$, the parabola opens downward. - The y-intercept is found by evaluating $$g(0)$$. - The x-intercepts are found by solving $$g(x) = 0$$. 3. **Find the vertex:** From the given function, vertex is at $$(-5, 8)$$. 4. **Find the y-intercept:** Evaluate $$g(0) = -(0 + 5)^2 + 8 = -25 + 8 = -17$$. So, y-intercept is $$(0, -17)$$. 5. **Find the x-intercepts:** Set $$g(x) = 0$$: $$0 = -(x + 5)^2 + 8$$ $$ (x + 5)^2 = 8$$ $$x + 5 = \pm \sqrt{8} = \pm 2\sqrt{2}$$ $$x = -5 \pm 2\sqrt{2}$$ 6. **Maximum or minimum:** Since $$a = -1 < 0$$, the parabola opens downward, so the vertex is a maximum point. Maximum value is $$8$$ at $$x = -5$$. 7. **Range:** Because the parabola opens downward with maximum $$8$$, the range is: $$(-\infty, 8]$$. **Final answers:** (A) y-intercept: $$(0, -17)$$ (B) Vertex: $$(-5, 8)$$ (C) Maximum: $$8$$ at $$x = -5$$ (D) Range: $$(-\infty, 8]$$