To solve this problem, we need to identify which of the given equations matches the graph of the function $f(x)$ shown.

Step 1: Identify key features of the graph

• Vertex: The graph has its vertex at the point $(1, -9)$.
• Direction: The graph opens upwards, which indicates that the leading coefficient of $x^2$ is positive.
• X-Intercepts: The graph crosses the x-axis at $x = -2$ and $x = 4$.

Step 2: Analyze the given options

1. Option A: $f(x) = x^2 - 2x - 8$

   • This is in standard form $ax^2 + bx + c$.
   • To find the vertex, use the vertex formula $x = -\frac{b}{2a} = -\frac{-2}{2 \times 1} = 1$.
   • Plugging $x = 1$ into the equation: $f(1) = 1^2 - 2(1) - 8 = 1 - 2 - 8 = -9$.
   • The vertex is $(1, -9)$, which matches the graph.
   • To check the x-intercepts, set $f(x) = 0$: $x^2 - 2x - 8 = 0$ Factor the equation: $(x - 4)(x + 2) = 0$ So the x-intercepts are $x = 4$ and $x = -2$, which match the graph.

2. Option B: $f(x) = -x^2 + 2x - 8$

   • This opens downwards due to the negative sign in front of $x^2$, which does not match the graph.

3. Option C: $f(x) = (x - 2)(x + 4)$

   • This factored form has roots at $x = 2$ and $x = -4$, which do not match the graph's intercepts.

4. Option D: $f(x) = -(x - 1)^2 - 9$

   • This is in vertex form with the vertex at $(1, -9)$, but it opens downwards (due to the negative sign in front), so it does not match the graph.

Conclusion

The correct equation that matches the graph is Option A: $f(x) = x^2 - 2x - 8$.

Would you like more details on any specific part of the solution? Here are some related questions you might find helpful:

1. How do you convert a quadratic equation from standard form to vertex form?
2. What are the properties of parabolas based on the sign of the leading coefficient?
3. How do you find the x-intercepts of a quadratic function?
4. What is the significance of the vertex in the graph of a quadratic function?
5. How can you determine if a parabola opens upwards or downwards?

Tip: To quickly determine the vertex of a parabola in standard form, use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.