<< Chapter < Page Chapter >> Page >

Eliminating the parameter

In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as x and y . Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter t from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.

Eliminating the parameter from polynomial, exponential, and logarithmic equations

For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for t . We substitute the resulting expression for t into the second equation. This gives one equation in x and y .

Eliminating the parameter in polynomials

Given x ( t ) = t 2 + 1 and y ( t ) = 2 + t , eliminate the parameter, and write the parametric equations as a Cartesian equation.

We will begin with the equation for y because the linear equation is easier to solve for t .

y = 2 + t y 2 = t

Next, substitute y 2 for t in x ( t ) .

x = t 2 + 1 x = ( y 2 ) 2 + 1 Substitute the expression for  t  into  x . x = y 2 4 y + 4 + 1 x = y 2 4 y + 5 x = y 2 4 y + 5

The Cartesian form is x = y 2 4 y + 5.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given the equations below, eliminate the parameter and write as a rectangular equation for y as a function
of x .

x ( t ) = 2 t 2 + 6 y ( t ) = 5 t

y = 5 1 2 x 3

Got questions? Get instant answers now!

Eliminating the parameter in exponential equations

Eliminate the parameter and write as a Cartesian equation: x ( t ) = e t and y ( t ) = 3 e t , t > 0.

Isolate e t .

x = e t e t = 1 x

Substitute the expression into y ( t ) .

y = 3 e t y = 3 ( 1 x ) y = 3 x

The Cartesian form is y = 3 x .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Eliminating the parameter in logarithmic equations

Eliminate the parameter and write as a Cartesian equation: x ( t ) = t + 2 and y ( t ) = log ( t ) .

Solve the first equation for t .

            x = t + 2      x 2 = t ( x 2 ) 2 = t Square both sides .

Then, substitute the expression for t into the y equation.

y = log ( t ) y = log ( x 2 ) 2

The Cartesian form is y = log ( x 2 ) 2 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Eliminate the parameter and write as a rectangular equation .

x ( t ) = t 2 y ( t ) = ln t t > 0

y = ln x

Got questions? Get instant answers now!

Eliminating the parameter from trigonometric equations

Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.

First, we use the identities:

x ( t ) = a cos t y ( t ) = b sin t

Solving for cos t and sin t , we have

x a = cos t y b = sin t

Then, use the Pythagorean Theorem:

cos 2 t + sin 2 t = 1

Substituting gives

cos 2 t + sin 2 t = ( x a ) 2 + ( y b ) 2 = 1

Eliminating the parameter from a pair of trigonometric parametric equations

Eliminate the parameter from the given pair of trigonometric equations where 0 t 2 π and sketch the graph.

x ( t ) = 4 cos t y ( t ) = 3 sin t

Solving for cos t and sin t , we have

x = 4 cos t x 4 = cos t y = 3 sin t y 3 = sin t

Next, use the Pythagorean identity and make the substitutions.

cos 2 t + sin 2 t = 1 ( x 4 ) 2 + ( y 3 ) 2 = 1 x 2 16 + y 2 9 = 1

The graph for the equation is shown in [link] .

Graph of given ellipse centered at (0,0).
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
modul questions trigonometry
Thamarai Reply
(1+cosA)(1-cosA)=sin^2A
BINCY
differentiate f(t)=1/4t to the power 4 +8
Jessica Reply
I need trigonometry,polynomial
duru Reply
ok
Augustine
Why is 7 on top
Bertha Reply
simplify cot x / csc x
Catherine Reply
👉🌹Solve🌻 Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
what is the period of cos?
SIYAMTHEMBA Reply
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
Patrick
simplify: cot x/csc x
Catherine
sorry i didnt realize you were actually asking someone else to put their question on here. i thought this was where i was supposed to.
Catherine
some to dereve formula for bulky density
kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
questions
Thamarai
ok
AjA
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask