A [b][color=#ff0000]parameterized curve[/color][/b] is a function from a subset of [math]\mathbb{R}[/math] into Euclidean space ([math]\mathbb{R}^n[/math]). For the time being we are going to stick to [b][color=#ff0000]plane[/color][/b] curves, meaning our codomain is [math]\mathbb{R}^2[/math].[br][br]Notation:[br][br][math]\vec{c}:\left[a,b\right]\to\mathbb{R}^2[/math] where [math]\vec{c}\left(t\right)=\left(x\left(t\right),y\left(t\right)\right)[/math].[br][br]The input variable [math]t[/math] is often called the [b][color=#ff0000]parameter[/color][/b]. The functions [math]x\left(t\right)[/math] and [math]y\left(t\right)[/math] are called the [b][color=#ff0000]component functions[/color][/b] of [math]\vec{c}[/math]. The function itself is called a [b][color=#ff0000]path[/color] in [math]\mathbb{R}^2[/math] [/b]and the image of the function is called the [color=#ff0000][b]image curve[/b] [/color]of the path. The points [math]\vec{c}\left(a\right)[/math] and [math]\vec{c}\left(b\right)[/math] are the [b][color=#ff0000]endpoints[/color][/b] of the curve.[br][br]The first thing I want you to do is just play. In the GeoGebra applet below you can type in two component functions and a domain of definition to see the resulting parameterized curve. Take some time to experiment. Try fixing a couple component functions and changing the domain to see how the resulting curve changes. Then hold the domain steady and edit the component functions creating different curves. Can you parameterize a line segment? A circle? A spiral? Can you create a curve whose endpoints are the same? How about a curve with a loop-de-loop?