\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Parametric Equations - Graphing

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On this page we discuss how to graph parametric equations.

Resources

Topics You Need To Understand For This Page

precalculus

basics of parametrics

Related Topics and Links

Wikipedia - Parametric Equations

Graphing Parametric Equations

As you learned when graphing functions, you need some experience with the equations and what the graphs look like in order to be able to know what a graph looks like just from the equation. As you are learning, we suggest using Winplot, Geogebra or Maple Learn to graph parametric equations. You can find out more information on the Graphing Utilities page.

You may be asked to plot a set of parametric equations by hand. This is usually done by plotting points to get an idea of the shape of the curve.
Here is a video showing how to plot parametric equations. He includes examples using a graphing utility and plotting by hand.

MIP4U - Introduction to Parametric Equations

video by MIP4U

Okay, let's try these practice problems before going on.

Schaum's Outline of Calculus, 6e: 1,105 Solved Problems + 30 Videos

Practice

Unless otherwise instructed, graph these parametric equations. Indicate the direction of the curve on the graph. If you eliminate the parameter, state restrictions on the domain and range, if any exist.

Graph the parametric equations \(x=1+\sqrt{t}\), \(y=t^2-4t\) on \( 0\leq t\leq4\).

Problem Statement

Graph the parametric equations \(x=1+\sqrt{t}\), \(y=t^2-4t\) on \( 0\leq t\leq4\).

Solution

PatrickJMT - 51 video solution

video by PatrickJMT

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Graph the parametric equations \(x=\sqrt{t}\), \(y=1-t\)

Problem Statement

Graph the parametric equations \(x=\sqrt{t}\), \(y=1-t\)

Solution

PatrickJMT - 52 video solution

video by PatrickJMT

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For the parametric equations \(x=1+\sin(t)\), \(y=-2+\cos(t)\), find the point that corresponds to \(t=\pi/2\), graph the equations and eliminate the parameter.

Problem Statement

For the parametric equations \(x=1+\sin(t)\), \(y=-2+\cos(t)\), find the point that corresponds to \(t=\pi/2\), graph the equations and eliminate the parameter.

Hint

In the video, he eliminates the parameter first and then graphs. Doing so makes it easier to graph and to see what is going on with the equations.

Problem Statement

For the parametric equations \(x=1+\sin(t)\), \(y=-2+\cos(t)\), find the point that corresponds to \(t=\pi/2\), graph the equations and eliminate the parameter.

Hint

In the video, he eliminates the parameter first and then graphs. Doing so makes it easier to graph and to see what is going on with the equations.

Solution

PatrickJMT - 47 video solution

video by PatrickJMT

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Eliminate the parameter for the parametric equations \( x = 4 - 2t \), \( y = 3 + 6t - 4t^2 \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 4 - 2t \), \( y = 3 + 6t - 4t^2 \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = \sqrt{t+1} \), \( y = \dfrac{1}{t+1} \); \( t \gt -1 \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Problem Statement

Eliminate the parameter for the parametric equations \( x = \sqrt{t+1} \), \( y = \dfrac{1}{t+1} \); \( t \gt -1 \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = 3\sin(t) \), \( y = -4\cos(t) \); \( 0 \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 3\sin(t) \), \( y = -4\cos(t) \); \( 0 \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = 3\sin(2t) \), \( y = -4\cos(2t) \); \( 0 \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 3\sin(2t) \), \( y = -4\cos(2t) \); \( 0 \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = 3\sin(t/3) \), \( y = -4\cos(t/3) \); \( 0 \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 3\sin(t/3) \), \( y = -4\cos(t/3) \); \( 0 \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = 3 - 2\cos(3t) \), \( y = 1 + 4\sin(3t) \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Specify a range on \(t\) so that the curve is traced only once.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 3 - 2\cos(3t) \), \( y = 1 + 4\sin(3t) \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Specify a range on \(t\) so that the curve is traced only once.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = 4\sin(t/4) \), \( y = 1 - 2\cos^2(t/4) \); \( -54\pi \lt t \lt 34\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Find the number of times the curve is traced.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 4\sin(t/4) \), \( y = 1 - 2\cos^2(t/4) \); \( -54\pi \lt t \lt 34\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Find the number of times the curve is traced.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = \sqrt{4 + \cos(5t/2)} \), \( y = 1 + (1/3)\cos(5t/2) \); \( -48\pi \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Find the number of times the curve is traced.

Problem Statement

Eliminate the parameter for the parametric equations \( x = \sqrt{4 + \cos(5t/2)} \), \( y = 1 + (1/3)\cos(5t/2) \); \( -48\pi \lt t \lt 2\pi \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Find the number of times the curve is traced.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = 2e^t \), \( y = \cos(1 + e^{3t}) \); \( 0 \lt t \lt 3/4 \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Problem Statement

Eliminate the parameter for the parametric equations \( x = 2e^t \), \( y = \cos(1 + e^{3t}) \); \( 0 \lt t \lt 3/4 \), sketch the graph of the parametric curve and give any limits that might exist on x and y.

Solution

The solution can be found on this page.

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Eliminate the parameter for the parametric equations \( x = (1/2)e^{-3t} \), \( y = e^{-6t} + 2e^{-3t} - 8 \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Find the range on \(t\) for one trace of the curve.

Problem Statement

Eliminate the parameter for the parametric equations \( x = (1/2)e^{-3t} \), \( y = e^{-6t} + 2e^{-3t} - 8 \), sketch the graph of the parametric curve and give any limits that might exist on x and y. Find the range on \(t\) for one trace of the curve.

Solution

The solution can be found on this page.

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Practice Instructions

Unless otherwise instructed, graph these parametric equations. Indicate the direction of the curve on the graph. If you eliminate the parameter, state restrictions on the domain and range, if any exist.

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