Question

Analytically determine what type(s) of symmetry, if any, the graph of the equation would possess. is the graph odd or even?Show your work.22) x^2 + y^2 =3|x|

Answer 1

$\begin{gathered} \text{solve for y:} \\ y^2=3|x|-x^2 \\ y=\pm\sqrt[]{3|x|-x^2} \end{gathered}$

Rewrite the equation as:

$f(x)=\pm\sqrt[]{3|x|-x^2}$

Even and odd functions:

$\begin{gathered} \text{If f(x) is even:} \\ f(x)=f(-x) \\ \text{If f(x) is odd:} \\ f(-x)=-f(x) \end{gathered}$

Let's check:

$\begin{gathered} x=1 \\ f(1)=\sqrt[]{3|1|-1^2}=\sqrt[]{3-2}=\sqrt[]{2} \\ x=-1 \\ f(-1)=\sqrt[]{3|-1|-(-1)^2}=\sqrt[]{3-1}=\sqrt[]{2} \\ \text{ Since:} \\ f(1)=f(-1) \end{gathered}$

We can conclude t