To determine which graph represents the function $f(x) = -x^2 - 4x + 5$, we can analyze the given quadratic function step by step.

Step 1: Analyze the function

1. Form: The function is quadratic, in the standard form $f(x) = ax^2 + bx + c$.

   • $a = -1$ (negative, so the parabola opens downward).
   • $b = -4$, $c = 5$.

2. Vertex: The x-coordinate of the vertex is found using: $x = \frac{-b}{2a} = \frac{-(-4)}{2(-1)} = \frac{4}{-2} = -2.$ Substitute $x = -2$ into the equation to find $y$: $f(-2) = -(-2)^2 - 4(-2) + 5 = -4 + 8 + 5 = 9.$ So, the vertex is at $(-2, 9)$.

3. Y-intercept: When $x = 0$: $f(0) = -(0)^2 - 4(0) + 5 = 5.$ The y-intercept is at $(0, 5)$.

4. Axis of symmetry: The vertical line $x = -2$ is the axis of symmetry.

5. Opening: Since $a = -1$, the parabola opens downward.

Step 2: Match with the graphs

• The correct graph must have a vertex at $(-2, 9)$, open downward, and cross the y-axis at $(0, 5)$.
• The top-left graph matches these conditions.

Final Answer:

The top-left graph represents the function $f(x) = -x^2 - 4x + 5$.

Would you like a detailed explanation of any specific step?

Here are 5 related questions to deepen your understanding:

1. How can you determine if a parabola opens upward or downward just by looking at the equation?
2. What role does the vertex play in understanding quadratic functions?
3. How can you find the roots of this function $f(x) = -x^2 - 4x + 5$?
4. How does the value of $c$ in the quadratic equation relate to the graph?
5. What transformations occur to the parent function $y = x^2$ to form $y = -x^2 - 4x + 5$?

Tip: Always sketch the vertex and axis of symmetry first when graphing a parabola—it helps to structure your graph effectively!