<< 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

what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Practice Key Terms 1

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask