(a) Consider the equation y = x² - 2x. What curve does this represent in R²? Make a sketch of the curve. (b) What surface does the equation in (a) above represent in R³? Make a sketch of the surface. (c) If we now had the equation z = y²-2y, describe the features of this surface in R³, and describe how we could use the answer to (b) above to answer this question. Sketch what the surface looks like.

Epicycloid If the circle C of Exercise 63 rolls on the outside of the larger circle, the curve traced out by P is called an epicycloid. Find parametric equations for the epicycloid. Hypocycloid A circle C of radius b rolls on the inside of a larger circle of radius a centered at the origin. Let P be a fixed point on the smaller circle, with the initial position at the point (a,0) as shown in the figure. The curve traced out by P is called a hypocycloid. a Show that parametric equations of hypocycloid are x=(ab)cos+bcos(abb) y=(ab)sinbsin(abb) b If a=4b, the hypocycloid is called an asteroid. Show that in this case parametric equations can be reduced to x=acos3y=asin3 Sketch the curve. Eliminate the parameter to obtain an equation for the asteroid in rectangular coordinates.

1. Consider the equation E : z² = x − 3. (a) Sketch the graph of E as a cylinder in R³. (b) Find the equation of the surface generated when the parabola with equation E on the xz plane is revolved about the z-axis. Sketch the surface of revolution.

Match the equations of the surface with the graphs below.