Answer
Vertex: $\left(-7,-2\right)$
Axis of Symmetry: $y=-2$
Domain: $\{x|x\ge-7\}$
Range: set of all real numbers
Graph of $x=3y^2+12y+5$
Work Step by Step
To find the properties of the given equation, $
x=3y^2+12y+5
,$ 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(3y^2+12y\right)+5
\\\\
x&=3\left(y^2+4y\right)+5
.\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&=3\left(y^2+4y+\left(\dfrac{4}{2}\right)^2\right)+\left[5-3\left(\dfrac{4}{2}\right)^2\right]
\\\\
x&=3\left(y^2+4y+4\right)+\left[5-12\right]
\\
x&=3\left(y+2\right)^2-7
.\end{align*}(Note that $a\left(\dfrac{b}{2}\right)^2\Rightarrow
3\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(-7,-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=3y^2+12y+5 & x=3y^2+12y+5
\\
x=3(-4)^2+12(-4)+5 & x=3(-3)^2+12(-3)+5
\\
x=3(16)-48+5 & x=3(9)-36+5
\\
x=48-48+5 & x=27-36+5
\\
x=5 & x=-4
.\end{array}
Thus the points $
(5,-4)
$ and $
(-4,-3)
$ are points on the parabola. Reflecting these points about the axis of symmetry, then $
(-4,-1)
$ and $
(5,-0)
$ are also points on the parabola.
Using the points $\{
(5,-4),(-4,-3),
\left(-7,-2\right)
(-4,-1),(5,0)
\}$ 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-7\}
$. The range (all $y$-values used in the graph) is the set of all real numbers.
