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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

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin

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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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