Select the graph that represents the function $$ -x^{2}-6 x+27 $$

Question

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1. **Write the function:**  
   We start with the function  
   $$
   -x^2 - 6x + 27.
   $$

2. **Complete the square to find the vertex:**  
   Factor out $$-1$$ from the quadratic and linear terms:  
   $$
   -\left(x^2 + 6x\right) + 27.
   $$  
   Complete the square inside the parentheses:  
   - The expression $$x^2+6x$$ can be rewritten as  
     $$
     x^2+6x+9-9 = (x+3)^2 - 9.
     $$  
   Substitute back:  
   $$
   -\left[(x+3)^2 - 9\right] + 27 = -(x+3)^2 + 9 + 27 = -(x+3)^2 + 36.
   $$

3. **Identify the vertex:**  
   The vertex form is  
   $$
   -(x+3)^2 + 36,
   $$  
   which shows that the vertex is at  
   $$
   (-3, 36).
   $$

4. **Determine the parabola’s direction:**  
   Since the coefficient of $$(x+3)^2$$ is negative, the parabola opens downward.

5. **Find the x-intercepts (optional):**  
   Set the function equal to 0:  
   $$
   -x^2-6x+27=0.
   $$  
   Multiply by $$-1$$:  
   $$
   x^2+6x-27=0.
   $$  
   Use the quadratic formula:  
   $$
   x = \frac{-6 \pm \sqrt{6^2-4\cdot1\cdot(-27)}}{2\cdot1} = \frac{-6 \pm \sqrt{36+108}}{2} = \frac{-6 \pm \sqrt{144}}{2}.
   $$  
   Thus,  
   $$
   x = \frac{-6 \pm 12}{2}.
   $$  
   This gives:  
   $$
   x = 3 \quad \text{or} \quad x = -9.
   $$  
   The x-intercepts are at $$(3, 0)$$ and $$(-9, 0)$$.

6. **Find the y-intercept (optional):**  
   Substitute $$x = 0$$ into the function:  
   $$
   f(0) = -0^2-6\cdot0+27 = 27.
   $$  
   So the y-intercept is at $$(0, 27)$$.

7. **Graph description summary:**  
   The graph of  
   $$
   -x^2-6x+27
   $$  
   is a downward-opening parabola with:  
   - Vertex at $$(-3, 36)$$  
   - X-intercepts at $$(-9, 0)$$ and $$(3, 0)$$  
   - Y-intercept at $$(0, 27)$$  
   - Axis of symmetry given by $$x = -3$$.

8. **Conclusion – Selecting the graph:**  
   The graph that represents the function is the one displaying a parabola that opens downward, with the vertex at $$(-3, 36)$$, crossing the x-axis at $$(-9, 0)$$ and $$(3, 0)$$, and crossing the y-axis at $$(0, 27)$$.
The graph of $$ -x^2 - 6x + 27 $$ is a downward-opening parabola with its vertex at $$(-3, 36)$$, x-intercepts at $$(-9, 0)$$ and $$(3, 0)$$, and y-intercept at $$(0, 27)$$.

Select the graph that represents the function

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