1. **State the problem:** Analyze the quadratic function $f(x) = x^2 - 4x - 8$. 2. **Formula and rules:** A quadratic function is generally written as $f(x) = ax^2 + bx + c$. - The graph is a parabola. - If $a > 0$, the parabola opens upward (concave up). - The vertex formula is $\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$. - The y-intercept is at $(0, c)$. 3. **Identify coefficients:** Here, $a=1$, $b=-4$, and $c=-8$. 4. **Concavity:** Since $a=1 > 0$, the parabola opens upward (concave up). 5. **Find vertex:** $$x_{vertex} = -\frac{b}{2a} = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2$$ Evaluate $f(2)$: $$f(2) = (2)^2 - 4(2) - 8 = 4 - 8 - 8 = -12$$ So, the vertex is at $(2, -12)$. 6. **Find y-intercept:** At $x=0$: $$f(0) = 0^2 - 4(0) - 8 = -8$$ So, the y-intercept is at $(0, -8)$. 7. **Summary:** - The parabola opens upward. - Vertex at $(2, -12)$. - Y-intercept at $(0, -8)$. This matches the given information and confirms the analysis.