Parametric Equations and Curves

These flashcards cover key concepts related to parametric equations and curves, helping students prepare for their exam.

What are curves defined by parametric equations?

Curves that are represented in terms of parameters rather than as functions using y = f(x) or x = g(y).

What is the significance of parametric equations for a circle?

Parametric equations are used to describe a whole circle, not just a portion of it.

What happens when parameters are eliminated from parametric equations?

It allows finding a relationship between x and y without explicit dependence on t.

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

x = r * cos(t), y = r * sin(t), where r is the radius.

How do we determine the direction of motion in a parametric curve?

By examining the increasing or decreasing behavior of the parameter t, often indicated by arrows on a sketch.

What is the relationship between parameter t and x and y in terms of sine and cosine?

For a parametric curve using trigonometric functions, often x = r * cos(t) and y = r * sin(t).

How many cycles are completed when t ranges from 0 to 2π in parametric equations involving trigonometric functions?

One complete cycle.

How do we sketch a curve defined by parametric equations?

By plotting the points corresponding to various values of the parameter t.

What transformations are needed for parametric equations when the circle's center is at (h, k)?

The equations become x = h + r * cos(t) and y = k + r * sin(t).

What is the parabola equation in parametric form?

x = t, y = at^2 + bt + c, where a, b, and c determine the shape and position of the parabola.