Practice Parametric Graphs in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Plotting and analyzing curves defined by parametric equations x = f(t), y = g(t), including eliminating the parameter, determining direction of motion, and finding tangent lines.
To sketch a parametric curve, make a table of t, x, and y values, then plot the (x, y) points and connect them in order of increasing t. Arrows on the curve show the direction of travel. Alternatively, you can sometimes eliminate t to get a familiar Cartesian equation—but you may lose information about direction and speed.
Example 1
easySketch the direction of motion for the parametric curve x = t, y = t^2 as t increases from -2 to 2.
Example 2
mediumDescribe the graph of x = 2\cos(t), y = 5\sin(t) for 0 \le t \le 2\pi, including shape, direction, and starting point.
Example 3
mediumWhat is the difference between the graphs of (a) x = t, y = t^2 and (b) x = \sin(t), y = \sin^2(t)?
Example 4
hardFind the slope of the tangent line to the curve x = t^2 + 1, y = t^3 - 3t at the point where t = 2.