8.6 Parametric equations (Page 3/6)

Precalculus

Page 3 / 6

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 $t .$

\begin{array}{l} y = 2 + t \\ y - 2 = t \end{array}

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

\begin{array}{l} 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 \end{array}

The Cartesian form is $x = y^{2} - 4 y + 5.$

Got questions? Get instant answers now!

Try It

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

\begin{array}{l} \begin{array}{l} x (t) = 2 t^{2} + 6 \\ y (t) = 5 - t \end{array} \end{array}

$y = 5 - \sqrt{\frac{1}{2} x - 3}$

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 $e^{t} .$

\begin{array}{l} x = e^{- t} \\ e^{t} = \frac{1}{x} \end{array}

Substitute the expression into $y (t) .$

\begin{array}{l} y = 3 e^{t} \\ y = 3 (\frac{1}{x}) \\ y = \frac{3}{x} \end{array}

The Cartesian form is $y = \frac{3}{x} .$

Got questions? Get instant answers now!

Eliminating the parameter in logarithmic equations

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

Solve the first equation for $t .$

\begin{array}{l} x = \sqrt{t} + 2 \\ x - 2 = \sqrt{t} \\ {(x - 2)}^{2} = t & Square both sides . \end{array}

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

\begin{array}{l} y = \log (t) \\ y = \log {(x - 2)}^{2} \end{array}

The Cartesian form is $y = \log {(x - 2)}^{2} .$

Got questions? Get instant answers now!

Try It

Eliminate the parameter and write as a rectangular equation .

\begin{array}{l} \begin{array}{l} x (t) = t^{2} \\ y (t) = \ln t t > 0 \end{array} \end{array}

$y = \ln \sqrt{x}$

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:

\begin{array}{l} x (t) = a \cos t \\ y (t) = b \sin t \end{array}

Solving for $\cos t$ and $\sin t,$ we have

\begin{array}{l} \frac{x}{a} = \cos t \\ \frac{y}{b} = \sin t \end{array}

Then, use the Pythagorean Theorem:

\cos^{2} t + \sin^{2} t = 1

Substituting gives

\cos^{2} t + \sin^{2} t = {(\frac{x}{a})}^{2} + {(\frac{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 $0 \leq t \leq 2 π$ and sketch the graph.

\begin{array}{l} x (t) = 4 \cos t \\ y (t) = 3 \sin t \end{array}

Solving for $\cos t$ and $\sin t,$ we have

\begin{array}{l} x = 4 \cos t \\ \frac{x}{4} = \cos t \\ y = 3 \sin t \\ \frac{y}{3} = \sin t \end{array}

Next, use the Pythagorean identity and make the substitutions.

\begin{array}{r} \cos^{2} t + \sin^{2} t = 1 \\ {(\frac{x}{4})}^{2} + {(\frac{y}{3})}^{2} = 1 \\ \frac{x^{2}}{16} + \frac{y^{2}}{9} = 1 \end{array}

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

Graph of given ellipse centered at (0,0).

Got questions? Get instant answers now!

Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?

ADNAN Reply

linear speed of an object

Melissa Reply

an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object

Melissa

test

Matrix

how to find domain

Mohamed Reply

like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.

Dan

define the term of domain

Moha

if a>0 then the graph is concave

Angel Reply

if a<0 then the graph is concave blank

Angel

what's a domain

Kamogelo Reply

The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.

Spiro

how fast can i understand functions without much difficulty

Joe Reply

what is inequalities

Nathaniel

functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2

Dan

what is set?

Kelvin Reply

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size

Divya Reply

I got 300 minutes. is it right?

Patience

no. should be about 150 minutes.

Jason

It should be 158.5 minutes.

ok, thanks

Patience

100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes

Thomas

158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49

Dan

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

yeah, it does. why do we attempt to gain all of them one side or the other?

Melissa

how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2

Dan

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

Spiro; thanks for putting it out there like that, 😁

Melissa

foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.

Churlene Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

	3 Eliminating the parameter from polynomial, exponential, and By OpenStax Read Online Course
	1 Parameterizing a curve By OpenStax Read Online Course
	Principles of macroeconomics for ap® courses By OpenStax Read Online Course
©flickr: Rod	Medieval Africa By Jesenia Wofford Start Quiz
	Anthropology Biology Culture By Richley Crapo Start Assignment
	Social Dances 2 By Marion Cabalfin Start Quiz
	6 Physiotherapy Flashcards Set 6 By Rhodes Start Flashcards
	39 Biology 39 The Respiratory System MCQ By OpenStax Start Quiz
	3 Microeconomics 03 Demand Supply By OpenStax Start Flashcards
	English Vocabulary By Jordon Humphreys Start Quiz
	4 Psychology MCQ 2009 Final Exam By John Gabrieli Start Exam
	3 Lec:3 Cohort Studies By Janet Forrester Start Quiz