17calculus
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Gradients
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

Parametrics

on this page: ► graphing     ► eliminating the parameter

Parametrics is a way to plot a graph by specifying the x, y (and sometimes z) values separately. We do so using a parameter, another variable used to link equations together. It is simple but deceptively powerful.

Search 17Calculus

For starters we will use the parameter t. Here is an example.

\( x = \cos(t) \)

\( y = \sin(t) \)

Notice that, although we have two separate equations, the x and y are linked by the value of t. In this example, we take a value for the parameter t and plug that value into both equations, to get a corresponding point \((x,y)\). When \(t=0\), we get \(x=\cos(0)=1\) and \(y=\sin(0)=0\). Writing this as an ordered pair, we have \((x,y)=(1,0)\). So, the point \((1,0)\) corresponds to the value \(t=0\).

One of the advantages of using parametric equations is that we can describe many more graphs than we could when we had only x and y. Also, we are not going to require the graphs to be functions now, i.e. they may not pass the vertical line test.

Another piece of information we get from parametric equation is direction. The use of the variable t as a parameter is not random. Often, we assign a meaning to the parameter and sometimes that meaning is time. When you graph a set of parametric equations, the graph is swept out in a certain direction. This is an inherent feature of the parametric equations. We will often start at \( t=0 \) and increase t, giving the idea that time is passing. By adjusting the parametric equations, we can reverse the direction that the graph is swept.

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, I suggest using winplot to graph parametric equations. It is easy to use, has a very short learning curve and, best of all, it's free. You can find out more information on the Tools page.

Eliminating The Parameter

Sometimes we are given the set of parametric equations and we are asked to write the equation without the parameter by eliminating the parameter. This not always possible but with some equations there are ways to do it.

The easiest technique to try is to solve one of the equations for the parameter and then substitute the result in the other equation. Here is an example.
\( x = 2t \) and \( y = t^2 \)
Solve the first equation for t giving \( t=x/2\) and substitute into the second equation.
\( y = (x/2)^2 = x^2/4 \)

A second technique involves the use of trig identities. For example, given the parametric equations \( x=\cos(t)\) and \(y=\sin(t)\), we know that \( \cos^2(x) + \sin^2(x) = 1 \). So we can write \( x^2 + y^2 = 1 \) which eliminates the parameter.

A third way is by inspection. Sometimes it is obvious what a substitution might be. For example, if we have \(x=e^t\) and \(y=e^{3t}+1\) then we can rewrite \(y=(e^t)^3+1\). Then we can replace \(e^t\) with \(x\) in the last equation (because our first parametric equation was \(x=e^t\)) to get \(y=x^3+1\). Of course, the first technique would have worked too by solving \(x=e^t\) for \(t\) and then substituting and simplifying but by standing back and looking at the equations more carefully, the solution was much easier.

Okay, time for some practice problems.

Practice Problems

Level A - Basic

Practice A01

Eliminate the parameter t from \(x=3t\) and \(y=t^2\).

answer

solution

Practice A02

Eliminate the parameter t from \(x=3\cos(t)\) and \(y=2\sin(t)\).

answer

solution

Practice A03

Eliminate the parameter t from \(x=5\cos(t)\) and \(y=3\sin(t)\).

answer

solution

Practice A04

Eliminate the parameter from the parametric equations \(x=t-1\), \(y=4t+1\).

solution

Practice A05

Eliminate the parameter from the parametric equations \(x=\sqrt{t+1}\), \(y=3t+2\).

solution

Practice A06

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

solution

Practice A07

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

solution


Level B - Intermediate

Practice B01

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.

solution

Practice B02

Eliminate the parameter from the parametric equations \(x=4e^{t/4}\), \(y=3e^t\).

solution

Practice B03

Find all points of intersection of the parametric curves \(C_1: x=t+1; y=t^2\) and \(C_2: x=3t+1; y=t^2+1\).

solution

Real Time Web Analytics
menu top search practice problems
17
menu top search practice problems 17