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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

what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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