Introduction

Understanding motion through curves and lines is a fundamental part of math, especially in precalculus. Parametric equations step in to make this concept more digestible. They express paths through equations that depend on an independent variable, wonderfully illustrating motion in both curved and linear paths. Here, we’ll dive into the parametric equation for a circle and the parametric equation of a line.

What Are Parametric Equations?

Parametric equations are unique because they express a set of quantities through functions of one or more parameters (independent variables). Unlike regular equations that relate x and y directly, parametric equations involve another variable, typically t.

Example: Comparing Equations

• Traditional Circle Equation: x^2 + y^2 = r^2
• Parametric Circle Equation: (x(t), y(t)) = (r \cdot \cos t, r \cdot \sin t)

Notice how the parametric version uses a parameter, t, to independently define positions of x and y.

Parametric Equation for Circle

Let’s work through a circle’s parametric equation using the concept of the unit circle. A unit circle has a radius of 1 and is centered at the origin, (0, 0).

• Key Features: Center at (0, 0) and radius 1.
• Parametric Formula: (x(t), y(t)) = (\cos t, \sin t)
• Domain: 0 \leq t \leq 2\pi

Example 1: Motion Around a Circle

Problem: Describe a point’s position moving counterclockwise around the unit circle from t = 0 to t = \pi/2.

1. Substitute Values:

• At t = 0, x(0) = \cos(0) = 1, y(0) = \sin(0) = 0
• At t = \pi/2, x\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0, y\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1

The point moves from (1, 0) to (0, 1) along the circle’s path.

Transformations of the Parametric Function

Circles can be moved around the plane and resized by tweaking their parametric equations.

• Transformations: Adjust the center to (h, k) and modify the radius to r.
• New Parametric Form: (x(t), y(t)) = (h + r \cdot \cos t, k + r \cdot \sin t)

Example 2: Transforming the Circle

Problem: Find the parametric equations for a circle centered at (2, 3) with a radius of 4.

1. Apply Transformations:

• Center (h, k) = (2, 3)
• Radius r = 4

New Parametric Equations: (x(t), y(t)) = (2 + 4 \cdot \cos t, 3 + 4 \cdot \sin t)

Parametric Equations for Line Segments

A line can also be expressed using parametric equations. Instead of motion around, consider linear motion.

• Definition: Expressed as functions of a parameter for x(t) and y(t).
• Example: Line from (x_1, y_1) to (x_2, y_2). The image to the right shows the line from (-1,1) to (1,4)

Example 3: Parametric Equation of a Line Segment

Problem: Create a parametric equation for a line segment from (1, 2) to (5, 6).

Determine Changes:

• Change in x = 5 - 1 = 4
• Change in y = 6 - 2 = 4

Write Parametric Equations:

• Start point: (1, 2)
• x(t) = 1 + 4t, \quad y(t) = 2 + 4t, where 0 \leq t \leq 1

The final parametric equation is (1+4t, 2+4t) where 0 \leq t \leq 1.

Quick Reference Vocabulary

Conclusion

Understanding parametric equations unlocks the fascinating world of describing motion in both circles and lines. These equations render curvy paths into manageable data points through parameters. Whether illustrating a circle’s geometric beauty or a linear path’s straightforwardness, parametric equations shine in capturing motion’s elegance in precalculus.

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