a cardiod microphone pattern is given by the following equation:

$\displaystyle \left{({x}^{2}+{y}^{2}-x\right)}^{2}={x}^{2}+{y}^{2} $

the expanded form of this equation is

$\displaystyle x^4-2 x^3+2 x^2 y^2+x^2-2 x y^2+y^4 = x^2+y^2$

(a) Find the intercepts of the graph of the equation

to find $\displaystyle {x}_{intercepts}$ I set $\displaystyle y = 0$

thus resulting in

$\displaystyle x^4-2x^3=0\Rightarrow x^3\left( x-2 \right) $

so $\displaystyle {x}_{intercepts} = 0, 2$

to find $\displaystyle {y}_{intercepts}$ I set $\displaystyle x = 0$

thus resulting in

$\displaystyle Y^2=1$ so $\displaystyle {y}_{intercepts}$ are $\displaystyle \pm 1$

(b) Test for symmetry with respect to the x-axis

my questions is that i do not how the symmetry is found except by observation from the graph. so symmetry is only on x-axis not on y-axis.

the books answers were

**(a)**$\displaystyle \left( 0,0 \right),\left( 2,0 \right) \left( 0,1\right) \left(0,-1 \right)$

**(b)**x=axis symmetry

but wolfframalpha gave something else

btw like the upgrades to latex