For the quadratic function $f(x) = x^2 + 6x + 11$:

1. Complete the square to write the equation in graphing form.

   Hint (a):

   Use the Math Notes box.

   Example 3: Solve $x^2+5x+4=0$ by completing the square.

   Solution: This method works most efficiently when the coefficient of $x^2$ is $1$.
   Rewrite the equation as $x^2+5x=-4$. Rewrite the left side as an incomplete square.

   $2.5$ $+$ $x$	$2.5x$		$=-4$
   	$x^2$	$2.5x$
   	$x+2.5$

   Complete the square and rewrite as
   $(x+2.5)^2-6.25=-4$ or $(x+2.5)^2=2.25$

   Take the square root of both sides, $x+2.5=\underline{+}1.5$. Solving for $x$ reveals that $x+-1$ or $x=-4$.

2. State the vertex and sketch a graph of the parabola.

   Hint (b):

   In the standard equation, the vertex is $(h, k)$.

   Partial Answer (b):

   $(-3, 2)$

3. What does the graph tell you about the solutions to the equation $x^2 + 6x + 11 = 0$? Explain.

   Hint (c):

   Are there any $x$-intercepts?