# Parametrics: Definition & Equations

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?

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).

## 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.

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.

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.

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.

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