# 21 Parametric Equations

### Learning Objectives

In this section, you will:

- Parameterize a curve.
- Eliminate the parameter.
- Find a rectangular equation for a curve defined parametrically.
- Find parametric equations for curves defined by rectangular equations.

Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in (Figure). At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.

In this section, we will consider sets of equations given by and where is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values ofthe orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.

### Parameterizing a Curve

When an object moves along a curve—or curvilinear path—in a given direction and in a given amount of time, the position of the object in the plane is given by the *x-*coordinate and the *y-*coordinate. However, bothand

vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which bothandare dependent functions. In the example in the section opener, the parameter is time,Theposition of the moon at time,is represented as the functionand theposition of the moon at time,is represented as the functionTogether, and are called parametric equations, and generate an ordered pairParametric equations primarily describe motion and direction.

When we parameterize a curve, we are translating a single equation in two variables, such asandinto an equivalent pair of equations in three variables,andOne of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time.

When we graph parametric equations, we can observe the individual behaviors ofand ofThere are a number of shapes that cannot be represented in the formmeaning that they are not functions. For example, consider the graph of a circle, given asSolving forgivesor two equations:andIf we graphandtogether, the graph will not pass the vertical line test, as shown in (Figure). Thus, the equation for the graph of a circle is not a function.

However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

### Parametric Equations

Suppose is a number on an interval, The set of ordered pairs, (x,y), with and forms a plane curve based on the parameter Both x and y are dependent on t, so the equations and are the parametric equations with parameter t.

### Parameterizing a Curve

Parameterize the curvelettingGraph both equations.

## Show Solution

Ifthen to findwe replace the variablewith the expression given inIn other words, Make a table of values similar to (Figure), and sketch the graph.

See the graphs in (Figure). It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated asincreases.

#### Analysis

The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of

### Try It

Construct a table of values and plot the parametric equations:

## Show Solution

### Finding a Pair of Parametric Equations

Find a pair of parametric equations that models the graph ofusing the parameterPlot some points and sketch the graph.

## Show Solution

Ifand we substituteforinto theequation, thenOur pair of parametric equations is

To graph the equations, first we construct a table of values like that in (Figure). We can choose values aroundfromtoThe values in thecolumn will be the same as those in thecolumn becauseCalculate values for the column

The graph ofis a parabola facing downward, as shown in (Figure). We have mapped the curve over the interval shown as a solid line with arrows indicating the orientation of the curve according toOrientation refers to the path traced along the curve in terms of increasing values ofAs this parabola is symmetric with respect to the linethe values ofare reflected across the *y*-axis.

### Try It

Parameterize the curve given by

## Show Solution

### Finding Parametric Equations That Model Given Criteria

An object travels at a steady rate along a straight path toin the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.

## Show Solution

The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The *x*-value of the object starts atmeters and goes to 3 meters. This means the distance *x* has changed by 8 meters in 4 seconds, which is a rate of orWe can write the *x*-coordinate as a linear function with respect to time asIn the linear function templateand

Similarly, the *y*-value of the object starts at 3 and goes towhich is a change in the distance *y* of −4 meters in 4 seconds, which is a rate of orWe can also write the *y*-coordinate as the linear functionTogether, these are the parametric equations for the position of the object, where

and

are expressed in meters and

represents time:

Using these equations, we can build a table of values for and (see (Figure)). In this example, we limited values ofto non-negative numbers. In general, any value ofcan be used.

From this table, we can create three graphs, as shown in (Figure).

#### Analysis

Again, we see that, in (Figure)(c), when the parameter represents time, we can indicate the movement of the object along the path with arrows.