Practice Parametric Graphs in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

Plotting and analyzing curves defined by parametric equations x=f(t)x = f(t), y=g(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 tt, xx, and yy values, then plot the (x,y)(x, y) points and connect them in order of increasing tt. Arrows on the curve show the direction of travel. Alternatively, you can sometimes eliminate tt to get a familiar Cartesian equationβ€”but you may lose information about direction and speed.

Showing a random 20 of 50 problems.

Example 1

medium
For x=t2x = t^2, y=t3y = t^3, where is the curve's cusp?

Example 2

easy
At what parameter value does x=t2x = t^2, y=tβˆ’1y = t - 1 reach the yy-axis?

Example 3

medium
A particle follows x=3βˆ’tx = 3 - t, y=2tβˆ’1y = 2t - 1. Find the slope of its straight-line path.

Example 4

easy
For x=tx = t, y=ty = t, what graph results?

Example 5

medium
For x=sin⁑tx = \sin t, y=cos⁑ty = \cos t at t=0t = 0, find the starting point and initial direction.

Example 6

easy
For x=tx = t, y=2ty = 2t, list the points at t=0,1,2t = 0, 1, 2.

Example 7

easy
Eliminate the parameter to graph x=tx = t, y=βˆ’t+3y = -t + 3.

Example 8

medium
For x=t2x = t^2, y=t3y = t^3, find where the tangent is vertical.

Example 9

hard
For x=t2x = t^2, y=t3βˆ’ty = t^3 - t, find the slope of the tangent at t=βˆ’1t = -1.

Example 10

challenge
For x=tβˆ’sin⁑tx = t - \sin t, y=1βˆ’cos⁑ty = 1 - \cos t (a cycloid), find the points where the tangent is horizontal on [0,2Ο€][0, 2\pi].

Example 11

medium
Find the tangent slope of x=cos⁑tx = \cos t, y=sin⁑ty = \sin t at t=Ο€/4t = \pi/4.

Example 12

challenge
For x=2cos⁑tx = 2\cos t, y=3sin⁑ty = 3\sin t, eliminate the parameter and name the curve and its intercepts.

Example 13

challenge
Find the concavity of x=t2x = t^2, y=t3y = t^3 at t=1t = 1.

Example 14

challenge
For x=t3βˆ’4tx = t^3 - 4t, y=t2βˆ’4y = t^2 - 4, find the point of self-intersection.

Example 15

medium
For x=1+2cos⁑tx = 1 + 2\cos t, y=βˆ’3+2sin⁑ty = -3 + 2\sin t, describe the Cartesian curve.

Example 16

easy
Find where x=tβˆ’1x = t - 1, y=t+1y = t + 1 crosses the yy-axis.

Example 17

medium
For x=3cos⁑tx = 3\cos t, y=2sin⁑ty = 2\sin t, sketch direction and starting point.

Example 18

hard
Find the slope of the tangent line to the curve x=t2+1x = t^2 + 1, y=t3βˆ’3ty = t^3 - 3t at the point where t=2t = 2.

Example 19

medium
For x=sin⁑tx = \sin t, y=sin⁑ty = \sin t, describe the graph including any restriction.

Example 20

medium
For x=cos⁑tx = \cos t, y=sin⁑2ty = \sin 2t, what is the maximum yy-value of the graph?
← Back to Parametric Graphs Worked Examples Formula Explained