Answer
Vertex: $\left(-5,-2\right)$
Axis of Symmetry: $y=-2$
Domain: $\{x|x\ge-5\}$
Range: set of all real numbers
Graph of $x=4y^2+16y+11$
Work Step by Step
To find the properties of the given equation, $
x=4y^2+16y+11
,$ convert the equation in the form $x=a(y-k)^2+h$.
Grouping the $y$-variables together and making the coefficient of $y^2$ equal to $1$, the given equation is equivalent to
\begin{align*}
x&=\left(4y^2+16y\right)+11
\\
x&=4\left(y^2+4y\right)+11
.\end{align*}
Completing the square of the right-side expression by adding $\left(\dfrac{b}{2}\right)^2,$ the equation above is equivalent to
\begin{align*}
x&=4\left(y^2+4y+\left(\dfrac{4}{2}\right)^2\right)+\left[11-4\left(\dfrac{4}{2}\right)^2\right]
\\\\
x&=4\left(y^2+4y+4\right)+\left[11-16\right]
\\
x&=4\left(y+2\right)^2-5
.\end{align*}(Note that $a\left(\dfrac{b}{2}\right)^2\Rightarrow
4\left(\dfrac{4}{2}\right)^2
$ should be subtracted as well to cancel out the term that was added to complete the square.)
Since the vertex of a parabola defined by $x=(y-k)^2+h$ is given by $(h,k),$ then the vertex of of the equation above is $
\left(-5,-2\right)
$.
The axis of symmetry is given by $y=k$. Hence, the axis of symmetry of the parabola with the given equation is $
y=-2
$.
To graph the parabola, find points on the parabola by substituting values of of $y$ and then solving for $x$. That is,
\begin{array}{l|r}
\text{If }y=-4: & \text{If }y=-3:
\\\\
x=4y^2+16y+11 & x=4y^2+16y+11
\\
x=4(-4)^2+16(-4)+11 & x=4(-3)^2+16(-3)+11
\\
x=4(16)-64+11 & x=4(9)-48+11
\\
x=64-64+11 & x=36-48+11
\\
x=11 & x=-1
.\end{array}
Thus the points $
(11,-4)
$ and $
(-1,-3)
$ are points on the parabola. Reflecting these points about the axis of symmetry, then $
(-1,-1)
$ and $
(11,0)
$ are also points on the parabola.
Using the points $\{
(11,-4),(-1,-3),
\left(-5,-2\right)
(-1,-1),(11,0)
\}$ the graph of the given equation is derived (see graph above).
Using the graph above, the domain (all $x$-values used in the graph) is $
\{x|x\ge-5\}
$. The range (all $y$-values used in the graph) is the set of all real numbers.
