Copyright

Parametrics: Definition & Equations

Instructor: Emily Cadic

Emily has a master's degree in engineering and currently teaches middle and high school science.

Parametrics are a special type of mathematical expression that build on traditional rectangular equations. Learn how parametric equations can be used to simplify problems, create graphs, and enhance our understanding of the behavior of those graphs.

Defining Parametrics

Parametrics involve the use of a third variable, t, to rewrite a single function, y = f(x), into two separate equations in terms of t, x(t) and y(t). In parametric problems, t is known as the parameter, while x(t) and y(t) are known as parametric equations.

Why Parametrics?

Parametric equations are advantageous when you are working with x and y variables that do not have a direct relationship. For example, in physics you may be asked to determine how far a ball will go if it is kicked at 15 m/s with an angle of 45°. The maximum vertical and horizontal distances come out to 25 and 17 meters, respectively. In other words, the ball travels about 1.5 times more vertical than horizontal distance. Wouldn't it therefore be correct to state that y = 1.5x?

null

Concluding that y = 1.5x implies that the reason the ball reaches vertical 25 meters is because it traveled 17 horizontal meters, which is incorrect. The horizontal and vertical distances traveled are not directly related. They are indirectly related in that they both depend on 1) how the ball was kicked, and 2) how long the ball spent in the air (a third variable known as t).

Time-dependent physics problem are such a common application of parametrics, the letter t seemed like the most natural choice of variable.

Types of Parametric Curves

The three types of parametric curves that you are most likely to encounter are in your studies are:

  • Ellipses
  • Circles
  • Parabolas

The parametric equations associated with circles and ellipses are very similar; however, parabolas vary too much to generalize (see Table 1).

Table 1
null

Graphing Parametric Equations

In this example, we will see how parametric equations provide us with an opportunity to discover important details about a graph.

Example: Graphing an Ellipse

Graph the parametric curve associated with the following equations:

x = -4cos(2t)

y = 3sin(2t)

The first important detail we can glean from our parametric expressions is the range of values needed to create a complete curve. An ellipse, like a circle, spans 360 degrees. Inputs greater than 360 degrees would only lead to a retracing of the same path. If our input ranges from 0 to 360 degrees, then our parameter t in this problem, which is being multiplied by 2, needs to encompass 0 to 180 degrees (360 degrees divided by 2).

The second important detail you can find by examining the parametric equations of a curve is something called the direction of motion. To determine the direction of motion of this ellipse, solve each parametric equation for the appropriate range of t (Table 2), and then plot. Note that the parameter used to compute x and y will not actually appear in the graph.

Table 2
null

As you plot the results of Table 2, notice that each subsequent point requires a clockwise movement as you progress towards completing the ellipse. The direction of motion is therefore clockwise, which has been indicated with a directional arrow in Graph 1.

Graph 1
null

Another interesting point about this graph is that it could have been generated in any number of different ways with a slight tweak to our original equations (let's call them V1). To demonstrate, we can compare V1 with similar equations given in Table 3.

Table 3
null

If you were to graph equations V2, you would get the same graph as V1; however, it would require double the range. If you think of the parameter as time, V2 traces the elliptical path in half the rate as V1. You can confirm that all three graphs (V1, V2, and V3) appear the same using GraphSketch, a great online utility for parametric equations.

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support