<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Why are some graphs drawn with arrows?

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

Got questions? Get instant answers now!

Name a few common types of graphs of parametric equations.

Got questions? Get instant answers now!

Why are parametric graphs important in understanding projectile motion?

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

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

{ 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.
Got questions? Get instant answers now!

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

t 2 1 0 1 2 3
x
y
Got questions? Get instant answers now!

{ 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.
Got questions? Get instant answers now!

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

t 2 1 0 1 2
x
y
Got questions? Get instant answers now!

{ 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.
Got questions? Get instant answers now!

For the following exercises, sketch the curve and include the orientation.

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

Got questions? Get instant answers now!

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

Graph of the given equations - looks like the left half of an upward opening parabola.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Graph of the given equations - looks like a downward opening absolute value function.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Graph of the given equations - a vertical ellipse.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

{ 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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Graph of the given equations- looks like an upward opening parabola.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

{ x ( t ) = t 3 y ( t ) = t + 3

Got questions? Get instant answers now!

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

Graph of the given equations- horizontal ellipse.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

{ 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
Got questions? Get instant answers now!

For the following exercises, graph the equation and include the orientation.

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

Got questions? Get instant answers now!

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

Graph of the given equations- looks like an upwards opening parabola
Got questions? Get instant answers now!

x = t , y = 25 t 2 , 0 < t 5

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

x = 2 cos t , y = 6 sin t , 0 t π

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Graph of the given equations - vertical periodic trajectory
Got questions? Get instant answers now!

Graph on the domain [ π , 0 ] , where a = 4 and b = 3 , and include the orientation.

Got questions? Get instant answers now!

Questions & Answers

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask