Question

How can I graph this equation: (x-squared + y-squared - 1)cubed = x-squared times y-cubed ?

Is there a polar form of the equation?

Answer 1

Cannot graph here. However, here is an easier way to write your equation online. Online "^" is used to indicate an exponent, e.g., x^2 = x squared.

(x^2 + y^2 -1)^3 = x^2 * y^3

Answer 2

first step: use real math notation

(x^2+y^2-1)^3 = x^2 y^3

As with any function, just start plugging in values for x (or y) and solving to get the other coordinate. Not easy, in this case, since high powers are difficult to solve except in special cases.

The other, easier, way is to use one of the many online graphing utilities. I like this one:

http://www.wolframalpha.com/input/?i=(x%5E2%2By%5E2-1)%5E3+%3D+x%5E2+y%5E3

As for the polar form, the graph indicates there ought to be one, but it will be messy.

(x^2+y^2-1)^3 = x^2 y^3
(r^2-1)^3 = r^5 cos^2θ sin^3θ
(r^2-1)^3 = r^5 (sin^3θ - sin^5θ)
or
8cosθ (r^2-1)^3 = sin^3(2θ)

wow. The symmetries an periodic nature of the second form make an interesting graph.

http://www.wolframalpha.com/input/?i=(r%5E2-1)%5E3+%3D+r%5E5+(sin%5E3%CE%B8+-+sin%5E5%CE%B8)

http://www.wolframalpha.com/input/?i=8cos%CE%B8+(r%5E2-1)%5E3+%3D+sin%5E3(2%CE%B8)

Answer 3

Thx Steve. Are there any handheld graphic calculators (eg TI-84 Plus CE) that can graph this?

Answer 4

don't know. some probably can do polar coordinates, but the implicit definition makes it tricky. They wouldn't look as nice as the online ones, that's for sure.