However, This Equation Does Not Tell the Whole Story. Although It Tells You Where

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AP CALCULUS BC / LECTURE NOTES / MR. RECORD
Section Number:
10.2 / Topics: Plane Curves and Parametric Equations / Day: 1 of 2

Until now, you have been representing a graph by a single equation involving two variables. In this section, you will study situations in which three variables are used to represent a curve in a plane.

Consider the path followed by an object that is propelled into the air at an angle of 45°. If the initial velocity of the object is 48 feet per second, the object travels the parabolic path given by Rectangular Equation

However, this equation does not tell the whole story. Although it tells you where

the object has been, it doesn’t tell you when the object was at a given point .

To determine the time, we can introduce a third variable, t, called a parameter. By

Writing both x and y as functions of t, you obtain the parametric equations.

Parametric Equation for x Parametric Equation for y

From this information, you can determine that at time t = 0, the object is at the point (0, 0). Similarly at time t = 1, the object is at the point and so on. For this problem, x and y are continuous functions of t, and the resulting path is called the plane curve.

Example 1: Sketching a Curve.

Sketch the curve described by the parametric equations

Note: It sometimes happens that two different sets of parametric equations will have the same graph.

The set of equations below illustrate this. It’s significant to note that in the case below, the graph would be traced more rapidly.

Eliminating the Parameter

Example 2: Adjusting the Domain After Eliminating the Parameter.

Sketch the curve described by the parametric equations

by eliminating the parameter and adjusting the domain of the resulting rectangular equation.

Angles as Parameters

Example 3: Using Trigonometry to Eliminate a Paramter.

Sketch the curve represented by

by eliminating the parameter and finding the corresponding rectangular equation.

Using the Techniques shown in Example 3, we can conclude that the graph of the parametric equations

is the ellipse (traced counterclockwise) given by .

The graph of the parametric equations

is the ellipse (traced clockwise) given by

Finding Parametric Equations

Example 4: Finding Parametric Equations for a Given Graph.

Find a set of parametric equations that represents the graph of using each of the following parameters.

a. b.

AP CALCULUS BC / LECTURE NOTES / MR. RECORD
Section Number:
10.2 / Topics: Plane Curves and Parametric Equations
-  Cycloids / Day: 2 of 2

Cycloids

Example 5: Parametric Equations for a Cycloid.

Determine the curve traced by a point P on the circumference of a circle of radius a rolling along a straight line in a plane. Such a curve is called a cycloid. (See animation.)

Let parameter be the measure of the circle’s rotation.

Let the point begin at the origin.

a. Using the picture above, write an expression for the measure of .

b. Find four ways to express .

c. Find three ways to express .

Because the circle rolls along the x-axis, you know that

d. Using your results from parts b and c above and the fact that , write parametric equations

for x and y.

e. What are the values of the derivatives of the two parametric equations, at the points