Answer
Intercepts
$x$ intercepts: $(-16,0)$ and $(-8,0)$
Symmetrical in $y$ axis
Work Step by Step
$\left(x+12\right)^{2}+\ y^{2}=16$
$x$-intercept:
$\left(x+12\right)^{2}+\ y^{2}=16$
$\left(x+12\right)^{2}+\ (0)^{2}=16$
$x+12 = \pm\sqrt {16}$
$x=-12\pm4$
$x=-16, -8$
$y$-intercept:
$\left(x+12\right)^{2}+\ y^{2}=16$
$\left(0+12\right)^{2}+\ y^{2}=16$
$144 +y^2=16$
$y=\pm\sqrt {16-144}$
$y=\pm\sqrt {-128}$
There is no $y$ intercept
Therefore, the $x$ intercepts are $(-16,0)$ and $(-8,0)$
Test for symmetry
Symmetry for $x$ axis:
$\left(x+12\right)^{2}+\ y^{2}=16$
$x^2 +24x +144 + y ^2 = 16$
$\left(-x+12\right)^{2}+\ y^{2}=16$
$x^2-24x+144+y^2=16$
Not symmetric about $x$ axis
Symmetry for $y$ axis:
$\left(x+12\right)^{2}+\ y^{2}=16$
$x^2 +24x +144 + y ^2 = 16$
$\left(x+12\right)^{2}+\ y^{2}=16$
$\left(x+12\right)^{2}+\ (-y)^{2}=16$
$\left(x+12\right)^{2}+\ y^{2}=16$
Symmetric about $y$ axis
Symmetry for origin:
$\left(x+12\right)^{2}+\ y^{2}=16$
$x^2 +24x +144 + y ^2 = 16$
$\left(x+12\right)^{2}+\ y^{2}=16$
$\left(-x+12\right)^{2}+\ (-y)^{2}=16$
$x^2 -24x +144 + y ^2 = 16$
Not symmetric in the origin
Therefore, it is symmetrical in $y$ axis.
