In this video I go over an example graphing parametric equations x = t2 - 2t and y = t + 1. Plotting several points from these equations shows a shape that looks like a parabola, but we can in fact confirm this is actually this case. If we eliminate the t variable from both x and y functions we do in fact get a parabola that opens up horizontally.
Also in this video I show that sometimes we restrict the t variable to be within a given interval, in which the resulting curve has a initial point, (f(a), g(a)), and a terminal point, (f(b), g(b)). Overall this is a very interesting and useful video to show how common functions such as the parabola can be written in parametric equation form; so make sure to watch this video!
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Example:
Sketch and identify the curve defined by the parametric equations:
Solution:
Each value of t gives a point on the curve, as shown in the table and curve below:
A particle whose position is given by the parametric equations moves along the curve in the direction of the arrows as t increases.
Notice that at consecutive points marked on the curve appear at equal time intervals but not at equal distances.
That is because the particle slows down and then speeds up as t increases.
It appears that the curve maybe be a parabola.
This can be confirmed by eliminating the parameter t as follows:
Note that no restriction was placed on the parameter t in this example, so we assumed that t could by any real number.
But sometimes we restrict t to lie in a finite interval.
For instance, consider the parametric curve:
This curve is part of the parabola that starts at (0, 1) and ends at the point (8, 5).
The arrowhead indicates the direction in which the curve is traced as t increases from 0 to 4.
In general, the curve with parametric equations:
