Plot the curve $y^2 = (x^2 +12x +36)/(x^4 -4x^3 - 12x^2 - 32x 64)$

Question

Can you help me find the following to plot this curve,

$$y^2 = \frac{x^2 +12x +36}{x^4 -4x^3 - 12x^2 - 32x 64}$$

1. In Explicit Form
2. Find the Symmetry
3. X & Y intercepts
4. Vertical and Horizontal Assymptotes
5. Plot

I've been trying to figure this out but the professor only gave an example in the $y^2 =P(x)$ form but this is $y^2 = P(x) /Q(x)$ so I'm a little confused how to go about plotting this, thanks in advance to those that can help.

Answer 1

I assume you have a typo, and you meant $x^4-4 x^3-12 x^2+32 x+64$. In that case, the denominator is just: $$(x-4)^2(x+2)$$ And the numerator is $(x+6)^2$.

Now can you solve the rest yourself?