Parametric Curves and Equations

These flashcards cover key concepts related to parametric curves and equations, focusing on definitions, examples, and methods for sketching and parametrization.

What are parametric equations used for in describing curves?

Parametric equations are used to describe curves that cannot be expressed as a function, particularly when dealing with entire circles.

How can parameters often be eliminated from parametric equations?

Parameters can be eliminated by solving one equation for the parameter and substituting it into the other equation.

What is the direction of motion in parametric curves indicated by?

The direction of motion in parametric curves is indicated by arrows as the parameter increases.

What is the form of a parametric equation for a circle centered at the origin?

The parametric equations for a circle centered at the origin are given by x = cos(t) and y = sin(t).

What defines a parabola in parametric equations?

A parabola can be represented in parametric equations, for example, x = t^2 and y = 2t.

What determines the number of times a parametric curve goes around before completing a cycle?

The number of times a curve goes around is determined by the parameter limits, such as 0 to 2π for one full rotation.

What is a common method for parametrizing a line segment?

A common method for parametrizing a line segment is using linear functions, such as x(t) = x0 + (x1-x0)t and y(t) = y0 + (y1-y0)t.

In a parametric curve defined by x = t^2, y = 2t, what is the shape of the curve?

The parametric equations define a parabola that opens upwards.

What do you need to sketch a curve defined by parametric equations?

To sketch a curve defined by parametric equations, you plot points for various values of the parameter and consider the direction of motion.

What relationship describes the parametric equations for a circle with center not at the origin?

The parametric equations are adjusted to x = h + r cos(t) and y = k + r sin(t), where (h,k) is the center of the circle.