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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
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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