Parametric equations calculus examples12/3/2023 ![]() For example, consider the graph of a circle, given as r 2 = x 2 + y 2. There are a number of shapes that cannot be represented in the form y = f ( x ), y = f ( x ), meaning that they are not functions. When we graph parametric equations, we can observe the individual behaviors of x x and of y. 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 parameterize a curve, we are translating a single equation in two variables, such as x x and y, y, into an equivalent pair of equations in three variables, x, y, x, y, and t. Parametric equations primarily describe motion and direction. Together, x ( t ) x ( t ) and y ( t ) y ( t ) are called parametric equations, and generate an ordered pair ( x ( t ), y ( t ) ). The x x position of the moon at time, t, t, is represented as the function x ( t ), x ( t ), and the y y position of the moon at time, t, t, is represented as the function y ( t ). ![]() In the example in the section opener, the parameter is time, t. ![]() For this reason, we add another variable, the parameter, upon which both x x and y y are dependent functions. Vary over time and so are functions of time. 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. 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. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. 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. As we trace out successive values of t, t, the orientation of the curve becomes clear. 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. In this section, we will consider sets of equations given by x ( t ) x ( t ) and y ( t ) y ( t ) where t t is the independent variable of time.
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