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Graph on the domain [ π , 0 ] , where a = 5 and b = 4 , and include the orientation.

Graph of the given equations - vertical periodic trajectory
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If a is 1 more than b , describe the effect the values of a and b have on the graph of the parametric equations.

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Describe the graph if a = 100 and b = 99.

There will be 100 back-and-forth motions.

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What happens if b is 1 more than a ? Describe the graph.

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If the parametric equations x ( t ) = t 2 and y ( t ) = 6 3 t have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

Take the opposite of the x ( t ) equation.

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For the following exercises, describe the graph of the set of parametric equations.

x ( t ) = t 2 and y ( t ) is linear

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y ( t ) = t 2 and x ( t ) is linear

The parabola opens up.

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y ( t ) = t 2 and x ( t ) is linear

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Write the parametric equations of a circle with center ( 0 , 0 ) , radius 5, and a counterclockwise orientation.

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

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Write the parametric equations of an ellipse with center ( 0 , 0 ) , major axis of length 10, minor axis of length 6, and a counterclockwise orientation.

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For the following exercises, use a graphing utility to graph on the window [ 3 , 3 ] by [ 3 , 3 ] on the domain [ 0 , 2 π ) for the following values of a and b , and include the orientation.

{ x ( t ) = sin ( a t ) y ( t ) = sin ( b t )

Technology

For the following exercises, look at the graphs that were created by parametric equations of the form { x ( t ) = a cos ( b t ) y ( t ) = c sin ( d t ) . Use the parametric mode on the graphing calculator to find the values of a , b , c , and d to achieve each graph.

Graph of the given equations

a = 4 , b = 3 , c = 6 , d = 1

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Graph of the given equations

a = 4 , b = 2 , c = 3 , d = 3

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For the following exercises, use a graphing utility to graph the given parametric equations.

  1. { x ( t ) = cos t 1 y ( t ) = sin t + t
  2. { x ( t ) = cos t + t y ( t ) = sin t 1
  3. { x ( t ) = t sin t y ( t ) = cos t 1

Graph all three sets of parametric equations on the domain [ 0 , 2 π ] .

Graph of the given equations

Graph of the given equations

Graph of the given equations

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Graph all three sets of parametric equations on the domain [ 0 , 4 π ] .

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Graph all three sets of parametric equations on the domain [ 4 π , 6 π ] .

Graph of the given equations

Graph of the given equations

Graph of the given equations

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The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along?

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Explain the effect on the graph of the parametric equation when we switched sin t and cos t .

The y -intercept changes.

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Explain the effect on the graph of the parametric equation when we changed the domain.

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Extensions

An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation y ( t ) = 16 t 2 + 20 t , while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

y ( x ) = 16 ( x 15 ) 2 + 20 ( x 15 )

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A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation y ( t ) = 16 t 2 + 10 t + 5 . Write parametric equations for the ball’s position, and then eliminate time to write height as a function of horizontal position.

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For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time t . Neglect air resistance.

Find parametric equations that model the problem situation.

{ x ( t ) = 64 t cos ( 52 ° ) y ( t ) = 16 t 2 + 64 t sin ( 52 ° )

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Find all possible values of x that represent the situation.

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When will the dart hit the ground?

approximately 3.2 seconds

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Find the maximum height of the dart.

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At what time will the dart reach maximum height?

1.6 seconds

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For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.

An epicycloid: { x ( t ) = 14 cos t cos ( 14 t ) y ( t ) = 14 sin t + sin ( 14 t ) on the domain [ 0 , 2 π ] .

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A hypocycloid: { x ( t ) = 6 sin t + 2 sin ( 6 t ) y ( t ) = 6 cos t 2 cos ( 6 t ) on the domain [ 0 , 2 π ] .

Graph of the given equations - a hypocycloid
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A hypotrochoid: { x ( t ) = 2 sin t + 5 cos ( 6 t ) y ( t ) = 5 cos t 2 sin ( 6 t ) on the domain [ 0 , 2 π ] .

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A rose: { x ( t ) = 5 sin ( 2 t ) sin t y ( t ) = 5 sin ( 2 t ) cos t on the domain [ 0 , 2 π ] .

Graph of the given equations - a four petal rose
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Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply

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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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