Answer
$(a)$ Symmetry with respect to $y$-axis
$(b)$ $x$-intercept $B(0,0)$
$y$-intercept $A(0,2)$ and $B(0,0)$
Work Step by Step
$(a)$ As we can see from the image of the graph above, it is symmetrical with respect to the $y$-axis.
Algebraically, if we replace $x$ by $−x$ equation will be still equivalent to the original one.
$(b)$
$x$-intercept happens when $y=0$
$x^2+(0-1)^2=1$
$x^2+1=1$
$x^2=0$
$x=0$
$B(0,0)$
$y$-intercept happens when $x=0$
$0^2+(y-1)^2=1$
$y^2-2y+1=1$
$y^2-2y=0$
$y(y-2)=0$
$y_1=0$
$y_2=2$
$A(0,2)$ and $B(0,0)$