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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

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)
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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
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what is complex numbers
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ahmed
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Solve 2cos x + 3sin x = 0.5
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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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