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?
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
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)
Thx Steve. Are there any handheld graphic calculators (eg TI-84 Plus CE) that can graph this?
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.
To graph the equation (x^2 + y^2 - 1)^3 = x^2 * y^3, you can start by rearranging the equation into a more suitable form for graphing.
First, expand the left side of the equation using the cube of a binomial:
(x^2 + y^2 - 1)(x^2 + y^2 - 1)(x^2 + y^2 - 1) = x^2 * y^3
This simplifies to:
(x^2 + y^2 - 1)(x^2 + y^2 - 1)(x^2 + y^2 - 1) - x^2 * y^3 = 0
Now that the equation is in terms of x and y, you can plot the graph using a graphing tool or software. Here's how you can proceed using an online graphing software:
1. Open a graphing tool or a graphing software website.
2. Enter the equation (x^2 + y^2 - 1)(x^2 + y^2 - 1)(x^2 + y^2 - 1) - x^2 * y^3 = 0 into the equation field.
3. Adjust the graphing window to include the desired range for both x and y axes.
4. Click on the "Graph" or "Plot" button to generate the graph of the equation.
As for the polar form of the equation, you can convert it by using the following polar coordinate transformations:
1. Convert x^2 + y^2 - 1 = 0 to r = 1, which represents a circle with radius 1 centered at the origin.
2. Convert x^2 * y^3 = 0 to r^2 * cos(theta)^2 * r * sin(theta)^3 = 0.
However, graphing complex equations in polar coordinates can be quite challenging due to the more intricate expressions involved. It is often easier to graph such equations in Cartesian coordinates (x, y) rather than polar coordinates (r, theta).