<< 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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
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