Parametric Equations
Learning Objectives
By the end of this section, you will be able to:- Parameterize a curve.
- Eliminate the parameter.
- Find a rectangular equation for a curve defined parametrically.
- Find parametric equations for curves defined by rectangular equations.
Figure 1
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 of , the 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, both and vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both and are dependent functions. In the example in the section opener, the parameter is time, . The position of the moon at time, , is represented as the function , and the position of the moon at time, , is represented as the function . Together, and are called parametric equations, and generate an ordered pair . Parametric equations primarily describe motion and direction. When we parameterize a curve, we are translating a single equation in two variables, such as and , into an equivalent pair of equations in three variables, , and . One 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 of and of . There are a number of shapes that cannot be represented in the form , meaning that they are not functions. For example, consider the graph of a circle, given as . Solving for gives , or two equations: and . If we graph and together, the graph will not pass the vertical line test, as shown in Figure 2. Thus, the equation for the graph of a circle is not a function. Figure 2
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.
A General Note: Parametric Equations
Suppose is a number on an interval, . The set of ordered pairs, , where and , forms a plane curve based on the parameter . The equations and are the parametric equations.Example 1: Parameterizing a Curve
Parameterize the curve letting . Graph both equations.Solution
If , then to find we replace the variable with the expression given in . In other words, . Make a table of values similar to the table below, and sketch the graph. Figure 3. (a) Parametric (b) Rectangular
Analysis of the Solution
The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of .Example 2: Finding a Pair of Parametric Equations
Find a pair of parametric equations that models the graph of , using the parameter . Plot some points and sketch the graph.Solution
If and we substitute for into the equation, then . Our pair of parametric equations is
To graph the equations, first we construct a table of values like that in the table below. We can choose values around , from to . The values in the column will be the same as those in the column because . Calculate values for the column .
Figure 4
Example 3: Finding Parametric Equations That Model Given Criteria
An object travels at a steady rate along a straight path to in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.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 at meters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of , or . We can write the x-coordinate as a linear function with respect to time as . In the linear function template and . Similarly, the y-value of the object starts at 3 and goes to , which is a change in the distance y of −4 meters in 4 seconds, which is a rate of , or . We can also write the y-coordinate as the linear function . Together, 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 . In this example, we limited values of to non-negative numbers. In general, any value of can be used.
Figure 5. (a) A graph of vs. , representing the horizontal position over time. (b) A graph of vs. , representing the vertical position over time. (c) A graph of vs. , representing the position of the object in the plane at time .
Analysis of the Solution
Again, we see that, in Figure 5(c), when the parameter represents time, we can indicate the movement of the object along the path with arrows.Methods for Finding Cartesian and Polar Equations from Curves
In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as and . Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations
For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for . We substitute the resulting expression for into the second equation. This gives one equation in and .Example 4: Eliminating the Parameter in Polynomials
Given and , eliminate the parameter, and write the parametric equations as a Cartesian equation.Solution
We will begin with the equation for because the linear equation is easier to solve for .
Next, substitute for in .
The Cartesian form is .
Analysis of the Solution
This is an equation for a parabola in which, in rectangular terms, is dependent on . From the curve’s vertex at , the graph sweeps out to the right. See Figure 6. In this section, we consider sets of equations given by the functions and , where is the independent variable of time. Notice, both and are functions of time; so in general is not a function of . Figure 6
Try It 3
Given the equations below, eliminate the parameter and write as a rectangular equation for as a function of .Example 5: Eliminating the Parameter in Exponential Equations
Eliminate the parameter and write as a Cartesian equation: and .Solution
Isolate .
Substitute the expression into .
The Cartesian form is .
Analysis of the Solution
The graph of the parametric equation is shown in Figure 7(a). The domain is restricted to . The Cartesian equation, is shown in Figure 7(b) and has only one restriction on the domain, .Example 6: Eliminating the Parameter in Logarithmic Equations
Eliminate the parameter and write as a Cartesian equation: and .Solution
Solve the first equation for .
Then, substitute the expression for into the equation.
The Cartesian form is .
Analysis of the Solution
To be sure that the parametric equations are equivalent to the Cartesian equation, check the domains. The parametric equations restrict the domain on to ; we restrict the domain on to . The domain for the parametric equation is restricted to ; we limit the domain on to .Eliminating the Parameter from Trigonometric Equations
Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem. First, we use the identities:
Solving for and , we have
Then, use the Pythagorean Theorem:
Substituting gives
Example 7: Eliminating the Parameter from a Pair of Trigonometric Parametric Equations
Eliminate the parameter from the given pair of trigonometric equations where and sketch the graph.Solution
Solving for and , we have
Next, use the Pythagorean identity and make the substitutions.
The graph for the equation is shown in Figure 8.
Figure 8
Analysis of the Solution
Applying the general equations for conic sections, we can identify as an ellipse centered at . Notice that when the coordinates are , and when the coordinates are . This shows the orientation of the curve with increasing values of .Try It 5
Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation:and .
Solution
Rewriting this set of parametric equations is a matter of substituting for . Thus, the Cartesian equation is .
Example 8: Finding a Cartesian Equation Using Alternate Methods
Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.Solution
Method 1. First, let’s solve the equation for . Then we can substitute the result into the equation.
Now substitute the expression for into the equation.
Method 2. Solve the equation for and substitute this expression in the equation.
Make the substitution and then solve for .
Finding Parametric Equations for Curves Defined by Rectangular Equations
Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent , and then substitute it into the equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for as the domain of the rectangular equation, then the graphs will be different. The following video shows examples of how to find Cartesian representations of parametric equations of different kinds. https://youtu.be/tW6N7DFTvrMExample 9: Finding a Set of Parametric Equations for Curves Defined by Rectangular Equations
Find a set of equivalent parametric equations for .Solution
An obvious choice would be to let . Then . But let’s try something more interesting. What if we let Then we have
The set of parametric equations is
Figure 9
Key Concepts
- Parameterizing a curve involves translating a rectangular equation in two variables, and , into two equations in three variables, x, y, and t. Often, more information is obtained from a set of parametric equations.
- Sometimes equations are simpler to graph when written in rectangular form. By eliminating , an equation in and is the result.
- To eliminate , solve one of the equations for , and substitute the expression into the second equation.
- Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for in one of the equations, and substitute the expression into the second equation.
- There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.
- Find an expression for such that the domain of the set of parametric equations remains the same as the original rectangular equation.
Glossary
- parameter
- a variable, often representing time, upon which and are both dependent
Section Exercises
1. What is a system of parametric equations? 2. Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter. 3. Explain how to eliminate a parameter given a set of parametric equations. 4. What is a benefit of writing a system of parametric equations as a Cartesian equation? 5. What is a benefit of using parametric equations? 6. Why are there many sets of parametric equations to represent on Cartesian function? For the following exercises, eliminate the parameter to rewrite the parametric equation as a Cartesian equation. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. For the following exercises, rewrite the parametric equation as a Cartesian equation by building an table. 26. 27. 28. 29. For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting or by setting . 30. 31. 32. 33. For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using and . Identify the curve. 34. 35. 36. 37. 38. Parameterize the line from to so that the line is at at , and at at . 39. Parameterize the line from to so that the line is at at , and at at . 40. Parameterize the line from to so that the line is at at , and at at . 41. Parameterize the line from to so that the line is at at , and at at . For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect. 42. 43. For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations. 44.–1 | ||
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