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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

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
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
7.5 and 37.5
Nando
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
the 28th term is 175
Nando
192
Kenneth
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
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write down the polynomial function with root 1/3,2,-3 with solution
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
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write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
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Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
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In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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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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