Answer
The x-intercepts are (-16,0) and (-8,0)
There aren't any y-intercepts.
The equation has symmetry only with respect to the x-axis.
Work Step by Step
To find the x-intercept(s), we set y to 0 and solve for x:
$(x+12)^2+0^2=16$
$(x+12)^12=16$
$\sqrt{(x+12)^2}=\sqrt{16}$
$x_1+12=-4\rightarrow x_1=-16$
$x_2+12=4\rightarrow x_2=-8$
To find the y-intercept(s), we set x to 0 and solve for y:
$(0+12)^2+y^2=16$
$12^2+y^2=16$
$144+y^2=16$
$y^2=-128$
$\sqrt{y^2}=\sqrt{-128}$
Taking the square root of a negative number is invalid in the real realm. That means that this equation doesn't have y-intercepts.
To test for symmetry with respect to the x-axis, we substitute y for -y and check if it equals the original equation:
$(x+12)^2+(-y)^2=16$
$(x+12)^2+y^2=16 \checkmark$
To test for symmetry with respect to the y-axis, we substitute x for -x and check if it equals the original equation:
$((-x)+12)^2+y^2=16$
$(-x+12)^2+y^2=16$ nope
To test for symmetry with respect to the origin, we substitute x for -x, substitute y for -y and check if it equals the original equation:
$((-x)+12)^2+(-y)^2=16$
$(x+12)^2+y^2=16$ nope
