Let [math]P_0=\left(x_0,y_0\right)[/math] and [math]P_1=\left(x_1,y_1\right)[/math] be two distinct points in the plane. We will call the standard parameterization of the directed line segment from [math]P_0[/math] to [math]P_1[/math] the path:[br][math]\vec{r}\left(t\right)=\left(x_0+t\left(x_1-x_0\right),y_0+t\left(y_1-y_0\right)\right)=\left(x_0+t\Delta x,y_0+t\Delta y\right),t\in\left[0,1\right][/math][br][br]In the applet below adjust the values of [math]a[/math] and [math]b[/math] to see the corresponding changes in the image curve [math]\vec{p}\left(t\right)=\left(x_0+\left(at+b\right)\Delta x,y_0+\left(at+b\right)\Delta y\right)[/math][br]Jot down your observations.

Again we find the the underlying rectangular equation of the line does not change, but the portion of this line that lies on the image curve changes quite a bit. Changing the value of [math]a[/math] changes both the length of the directed line segment and the time it takes to be "drawn", while changing the value of [math]b[/math] shifts the starting point along the line.