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

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1. Write the function in standard form:
   
   $$
   f(x) = -x^2 - 6x + 27
   $$

2. Factor out $$-1$$ from the quadratic and linear terms:

$$
   f(x) = -\left(x^2 + 6x\right) + 27
   $$

3. Complete the square for the expression inside the parentheses. Recall that:

$$
   x^2 + 6x = \left(x+3\right)^2 - 9
   $$

Substitute this back into the function:

$$
   f(x) = -\left[\left(x+3\right)^2 - 9\right] + 27
   $$

4. Distribute the negative sign:

$$
   f(x) = -\left(x+3\right)^2 + 9 + 27
   $$

5. Combine like terms:

$$
   f(x) = -\left(x+3\right)^2 + 36
   $$

6. Identify the vertex form of the parabola:

$$
   f(x) = -\left(x+3\right)^2 + 36
   $$
   
   From this form, the vertex is $$(-3, 36)$$.

7. Determine the orientation of the parabola:
   
   Since the coefficient of $$\left(x+3\right)^2$$ is negative ($$-1$$), the parabola opens downward.

8. Find the $$x$$-intercepts by solving $$ f(x) = 0 $$:

$$
   -x^2 - 6x + 27 = 0
   $$
   
   Multiply both sides by $$-1$$:

$$
   x^2 + 6x - 27 = 0
   $$
   
   Apply the quadratic formula:

$$
   x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-27)}}{2(1)} = \frac{-6 \pm \sqrt{36 + 108}}{2} = \frac{-6 \pm \sqrt{144}}{2}
   $$
   
   Simplify:

$$
   x = \frac{-6 \pm 12}{2}
   $$
   
   Thus, the $$x$$-intercepts are:

$$
   x = \frac{6}{2} = 3 \quad \text{and} \quad x = \frac{-18}{2} = -9
   $$

9. Find the $$y$$-intercept by substituting $$ x = 0 $$:

$$
   f(0) = -0^2 - 6(0) + 27 = 27
   $$

10. Summary of key features for the graph of $$ f(x) = -x^{2}-6 x+27 $$:
    
    - Vertex: $$(-3, 36)$$
    - Parabola opens downward.
    - $$x$$-intercepts: $$ x = -9 $$ and $$ x = 3 $$
    - $$y$$-intercept: $$ y = 27 $$

The correct graph will show a downward-opening parabola with a vertex at $$(-3, 36)$$, $$x$$-intercepts at $$-9$$ and $$3$$, and passing through the point $$(0, 27)$$.
The graph of $$ -x^{2}-6x+27 $$ is a downward-opening parabola with a vertex at $$(-3, 36)$$, $$x$$-intercepts at $$-9$$ and $$3$$, and a $$y$$-intercept at $$27$$.

Select the graph that represents the function

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