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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

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit 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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