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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 golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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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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