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Section exercises

Verbal

What is a system of parametric equations?

A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, x = f ( t ) and y = f ( t ) .

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Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.

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Explain how to eliminate a parameter given a set of parametric equations.

Choose one equation to solve for t , substitute into the other equation and simplify.

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What is a benefit of writing a system of parametric equations as a Cartesian equation?

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What is a benefit of using parametric equations?

Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.

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Why are there many sets of parametric equations to represent on Cartesian function?

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Algebraic

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation.

{ x ( t ) = 5 t y ( t ) = 8 2 t

y = 2 + 2 x

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{ x ( t ) = 6 3 t y ( t ) = 10 t

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{ x ( t ) = 2 t + 1 y ( t ) = 3 t

y = 3 x 1 2

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{ x ( t ) = 3 t 1 y ( t ) = 2 t 2

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{ x ( t ) = 2 e t y ( t ) = 1 5 t

x = 2 e 1 y 5 or y = 1 5 l n ( x 2 )

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{ x ( t ) = e 2 t y ( t ) = 2 e t

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{ x ( t ) = 4 log ( t ) y ( t ) = 3 + 2 t

x = 4 log ( y 3 2 )

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{ x ( t ) = log ( 2 t ) y ( t ) = t 1

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{ x ( t ) = t 3 t y ( t ) = 2 t

x = ( y 2 ) 3 y 2

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{ x ( t ) = t t 4 y ( t ) = t + 2

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{ x ( t ) = e 2 t y ( t ) = e 6 t

y = x 3

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{ x ( t ) = t 5 y ( t ) = t 10

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{ x ( t ) = 4 cos t y ( t ) = 5 sin t  

( x 4 ) 2 + ( y 5 ) 2 = 1

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{ x ( t ) = 3 sin t y ( t ) = 6 cos t

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{ x ( t ) = 2 cos 2 t y ( t ) = sin t  

y 2 = 1 1 2 x

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{ x ( t ) = cos t + 4 y ( t ) = 2 sin 2 t

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{ x ( t ) = t 1 y ( t ) = t 2

y = x 2 + 2 x + 1

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{ x ( t ) = t y ( t ) = t 3 + 1

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{ x ( t ) = 2 t 1 y ( t ) = t 3 2

y = ( x + 1 2 ) 3 2

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For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x - y table.

{ x ( t ) = 2 t 1 y ( t ) = t + 4

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{ x ( t ) = 4 t y ( t ) = 3 t + 2

y = 3 x + 14

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{ x ( t ) = 2 t 1 y ( t ) = 5 t

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{ x ( t ) = 4 t 1 y ( t ) = 4 t + 2

y = x + 3

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For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x ( t ) = t or by setting y ( t ) = t .

y ( x ) = 2 sin x + 1

{ x ( t ) = t y ( t ) = 2 sin t + 1

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x ( y ) = 3 log ( y ) + y

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x ( y ) = y + 2 y

{ x ( t ) = t + 2 t y ( t ) = t

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For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using x ( t ) = a cos t and y ( t ) = b sin t . Identify the curve.

x 2 16 + y 2 36 = 1

{ x ( t ) = 4 cos t y ( t ) = 6 sin t ; Ellipse

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x 2 + y 2 = 10

{ x ( t ) = 10 cos t y ( t ) = 10 sin t ; Circle

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Parameterize the line from ( 3 , 0 ) to ( −2 , −5 ) so that the line is at ( 3 , 0 ) at t = 0 , and at ( −2 , −5 ) at t = 1.

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Parameterize the line from ( −1 , 0 ) to ( 3 , −2 ) so that the line is at ( −1 , 0 ) at t = 0 , and at ( 3 , −2 ) at t = 1.

{ x ( t ) = 1 + 4 t y ( t ) = 2 t

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Parameterize the line from ( −1 , 5 ) to ( 2 , 3 ) so that the line is at ( −1 , 5 ) at t = 0 , and at ( 2 , 3 ) at t = 1.

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Parameterize the line from ( 4 , 1 ) to ( 6 , −2 ) so that the line is at ( 4 , 1 ) at t = 0 , and at ( 6 , −2 ) at t = 1.

{ x ( t ) = 4 + 2 t y ( t ) = 1 3 t

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Technology

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.

{ x 1 ( t ) = 3 t y 1 ( t ) = 2 t 1  and  { x 2 ( t ) = t + 3 y 2 ( t ) = 4 t 4

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{ x 1 ( t ) = t 2 y 1 ( t ) = 2 t 1  and  { x 2 ( t ) = t + 6 y 2 ( t ) = t + 1

yes, at t = 2

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For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

{ x 1 ( t ) = 3 t 2 3 t + 7 y 1 ( t ) = 2 t + 3

t x y
–1
0
1
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{ x 1 ( t ) = t 2 4 y 1 ( t ) = 2 t 2 1

t x y
1
2
3
t x y
1 -3 1
2 0 7
3 5 17
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{ x 1 ( t ) = t 4 y 1 ( t ) = t 3 + 4

t x y
-1
0
1
2
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Extensions

Find two different sets of parametric equations for y = ( x + 1 ) 2 .

answers may vary: { x ( t ) = t 1 y ( t ) = t 2  and  { x ( t ) = t + 1 y ( t ) = ( t + 2 ) 2

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Find two different sets of parametric equations for y = 3 x 2.

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Find two different sets of parametric equations for y = x 2 4 x + 4.

answers may vary: , { x ( t ) = t y ( t ) = t 2 4 t + 4  and  { x ( t ) = t + 2 y ( t ) = t 2

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Questions & Answers

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
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
Practice Key Terms 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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