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

1. Use the idea of completing the square to write it in graphing form.

   Hint (a):

   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:  

   Generic rectangle left edge$2.5$ + $x$	interior top left$2.5x$
   	interior bottom left$x^2$	interior bottom right$2.5x$	$=-4$
   	bottom edge$x$    +    $2.5$

   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, $x+2.5=\pm1.5$.  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. Use the graph to explain why the equation $x^2+6x+11=0$ has no real solutions.

   Hint (c):

   Are there any $x$-intercepts?