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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 golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
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emma Reply
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
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Maurice
answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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