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Access the following online resource for additional instruction and practice with graphs of parametric equations.

Key concepts

  • When there is a third variable, a third parameter on which x and y depend, parametric equations can be used.
  • To graph parametric equations by plotting points, make a table with three columns labeled t , x ( t ) , and y ( t ) . Choose values for t in increasing order. Plot the last two columns for x and y . See [link] and [link] .
  • When graphing a parametric curve by plotting points, note the associated t -values and show arrows on the graph indicating the orientation of the curve. See [link] and [link] .
  • Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See [link] .
  • Projectile motion depends on two parametric equations: x = ( v 0 cos θ ) t and y = 16 t 2 + ( v 0 sin θ ) t + h . Initial velocity is symbolized as v 0 . θ represents the initial angle of the object when thrown, and h represents the height at which the object is propelled.

Section exercises

Verbal

What are two methods used to graph parametric equations?

plotting points with the orientation arrow and a graphing calculator

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What is one difference in point-plotting parametric equations compared to Cartesian equations?

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Why are some graphs drawn with arrows?

The arrows show the orientation, the direction of motion according to increasing values of t .

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Name a few common types of graphs of parametric equations.

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Why are parametric graphs important in understanding projectile motion?

The parametric equations show the different vertical and horizontal motions over time.

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Graphical

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

{ x ( t ) = t y ( t ) = t 2 1

t x y
3
2
1
0
1
2
3
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{ x ( t ) = t 1 y ( t ) = t 2

t 3 2 1 0 1 2
x
y
Graph of the given equations - looks like an upward opening parabola.
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{ x ( t ) = 2 + t y ( t ) = 3 2 t

t 2 1 0 1 2 3
x
y
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{ x ( t ) = 2 2 t y ( t ) = 3 + t

t 3 2 1 0 1
x
y
Graph of the given equations - a line, negative slope.
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{ x ( t ) = t 3 y ( t ) = t + 2

t 2 1 0 1 2
x
y
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{ x ( t ) = t 2 y ( t ) = t + 3

t 2 1 0 1 2
x
y
Graph of the given equations - looks like a sideways parabola, opening to the right.
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For the following exercises, sketch the curve and include the orientation.

{ x ( t ) = t y ( t ) = t

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{ x ( t ) = t y ( t ) = t

Graph of the given equations - looks like the left half of an upward opening parabola.
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{ x ( t ) = 5 | t | y ( t ) = t + 2

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{ x ( t ) = t + 2 y ( t ) = 5 | t |

Graph of the given equations - looks like a downward opening absolute value function.
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{ x ( t ) = 4 sin t y ( t ) = 2 cos t

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{ x ( t ) = 2 sin t y ( t ) = 4 cos t

Graph of the given equations - a vertical ellipse.
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{ x ( t ) = 3 cos 2 t y ( t ) = −3 sin t

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{ x ( t ) = 3 cos 2 t y ( t ) = −3 sin 2 t

Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.
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{ x ( t ) = sec t y ( t ) = tan t

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{ x ( t ) = sec t y ( t ) = tan 2 t

Graph of the given equations- looks like an upward opening parabola.
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{ x ( t ) = 1 e 2 t y ( t ) = e t

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For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.

{ x ( t ) = t 1 y ( t ) = t 2

Graph of the given equations- looks like a downward opening parabola.
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{ x ( t ) = t 3 y ( t ) = t + 3

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{ x ( t ) = 2 cos t y ( t ) = sin t

Graph of the given equations- horizontal ellipse.

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{ x ( t ) = 7 cos t y ( t ) = 7 sin t

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{ x ( t ) = e 2 t y ( t ) = e t

Graph of the given equations- looks like the lower half of a sideways parabola opening to the right
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For the following exercises, graph the equation and include the orientation.

x = t 2 , y = 3 t , 0 t 5

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x = 2 t , y = t 2 , 5 t 5

Graph of the given equations- looks like an upwards opening parabola
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x = t , y = 25 t 2 , 0 < t 5

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x ( t ) = t , y ( t ) = t , t 0

Graph of the given equations- looks like the upper half of a sideways parabola opening to the left
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x = 2 cos t , y = 6 sin t , 0 t π

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x = sec t , y = tan t , π 2 < t < π 2

Graph of the given equations- the left half of a hyperbola with diagonal asymptotes
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For the following exercises, use the parametric equations for integers a and b :

x ( t ) = a cos ( ( a + b ) t ) y ( t ) = a cos ( ( a b ) t )

Graph on the domain [ π , 0 ] , where a = 2 and b = 1 , and include the orientation.

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Graph on the domain [ π , 0 ] , where a = 3 and b = 2 , and include the orientation.

Graph of the given equations - vertical periodic trajectory
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Graph on the domain [ π , 0 ] , where a = 4 and b = 3 , and include the orientation.

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Questions & Answers

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim

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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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