1. **State the problem:** We are given the quadratic function $$y = x^2 - 8x - 48$$ and want to analyze its graph. 2. **Formula and rules:** A quadratic function is generally written as $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants. The graph is a parabola. 3. **Find the vertex:** The vertex of a parabola given by $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $a=1$, $b=-8$, so $$x = -\frac{-8}{2 \times 1} = \frac{8}{2} = 4.$$ 4. **Calculate the vertex's y-coordinate:** Substitute $x=4$ into the function: $$y = 4^2 - 8 \times 4 - 48 = 16 - 32 - 48 = -64.$$ So the vertex is at $(4, -64)$. 5. **Find the roots (x-intercepts):** Solve $$x^2 - 8x - 48 = 0$$. Factor the quadratic: $$x^2 - 8x - 48 = (x - 12)(x + 4) = 0.$$ 6. **Roots:** Set each factor to zero: $$x - 12 = 0 \Rightarrow x = 12,$$ $$x + 4 = 0 \Rightarrow x = -4.$$ 7. **Summary:** The parabola opens upwards (since $a=1 > 0$), has vertex at $(4, -64)$, and crosses the x-axis at $x = -4$ and $x = 12$. This gives a complete picture of the graph's shape and key points.