Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems.
(a) Graph the curve with equation
Y(y2 - 1) (y - 2) = x(x - 1) (x - 2)
At how many points does this curve have horizontal tangents? Estimate the -coordinates of these points.
(b) Find equations of the tangent lines at the points (0, 1) and (0, 2).
(c) Find the exact -coordinates of the points in part (a).
(d) Create even more fanciful curves by modifying the equation in part (a).