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### Chapter 10: Parametric Equations and Polar Coordinates

#### Learning module LM 10.1: Parametrized Curves:

3 kinds of functions, 3 kinds of curves
The cycloid
Visualizing parametrized curves
Tracing curves and ellipses
Lissajous figures

# Tracing curves and ellipses

Tracing Curves and Ellipses

We already saw that $x=\cos(t)$, $y=\sin(t)$ gives a circle traced counter-clockwise. In this demonstration, we'll look at something slightly more complicated:$$x(t) = a\cos(t) + h, \qquad y(t)=b\sin(t)+k,$$where $a$, $b$, $h$ and $k$ are numbers that you specify.

 Investigate the interactive figure to the right. (i) Try changing $a$, $b$, $h$ and $k$, one at a time, to understand what each of them does. (ii) How would you make a circle of radius 2, centered at the origin? (iii) How would you make an ellipse centered at the origin that is twice as wide as it is high? Or half as wide as it is high? (iv) How would you make a circle of radius 3, centered at the point (-1,4)?