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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 )

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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

can you solve it step b step
Ching Reply
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth 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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