Section 1.1 - Parametric Equations

Section 1.1 - Parametric Equations

Introduction to Parametric Equations

Parametric equations provide a powerful way to describe curves in the plane by expressing the coordinates as functions of a third variable, called a parameter. This approach is especially useful for representing curves that cannot be described as functions in the form $y = f(x)$.

• Parametric Equations: A pair of equations where both $x$ and $y$ are given as functions of a parameter $t$:

$x = f(t)$ $y = g(t)$

• Each value of $t$ determines a point $(x, y)$ in the coordinate plane.

• As $t$ varies, the point $(x, y)$ traces out a curve.

Definition and Key Concepts

• Parameter: A third variable (often $t$) that determines the position on the curve.

• Parametric Curve: The set of all points $(x, y)$ as $t$ varies over an interval.

• Eliminating the Parameter: The process of rewriting parametric equations as a single equation in $x$ and $y$ by removing $t$.

Why Use Parametric Equations?

• Some curves, such as loops or vertical lines, cannot be described by a single function $y = f(x)$.

• Parametric equations allow for more flexibility in describing motion and orientation.

• They are useful in physics and engineering to describe the path of moving objects.

General Properties

• The direction in which the curve is traced depends on how $t$ increases.

• Points may be traced at non-uniform speeds as $t$ changes.

• The same curve can have different parametric representations.

Examples of Parametric Equations

• Example 1: Parabola $x = t^2 - 2t$ $y = t^2 + 1$ $0 \\leq t \\leq 4$ Interpretation: As $t$ increases from 0 to 4, the point $(x, y)$ traces a portion of a parabola from $(0, 1)$ to $(8, 5)$.

• Example 2: Circle (Centered at the Origin) $x = \\sin t$ $y = \\cos t$ $0 \\leq t \\leq 2\\pi$ Interpretation: As $t$ varies from $0$ to $2\\pi$, the point $(x, y)$ traces a unit circle.

• Example 3: Circle (Centered at $(h, k)$, Radius $r$) $x = h + r \\cos t$ $y = k + r \\sin t$ $0 \\leq t \\leq 2\\pi$

• Example 4: Line through Two Points $(x_1, y_1)$ and $(x_2, y_2)$ $x = x_1 + (x_2 - x_1)t$ $y = y_1 + (y_2 - y_1)t$ $0 \\leq t \\leq 1$

• Example 5: Cycloid (Path of a Point on a Rolling Wheel of Radius $a$) $x(t) = a t - a \\sin t$ $y(t) = a - a \\cos t$ Application: The cycloid describes the path traced by a point on the rim of a rolling wheel.

Eliminating the Parameter

To convert parametric equations to a single equation in $x$ and $y$, solve one equation for $t$ and substitute into the other.

• Example: Given $x = \\sin t$, $y = \\sin^2 t$

• Solve for $t$ in terms of $x$: $t = \\arcsin x$

• Substitute into $y$: $y = (\\sin t)^2 = x^2$

• Result: The curve is $y = x^2$

Comparison: Cartesian vs. Parametric Equations

Cartesian Equation	Parametric Equations
Describes $y$ as a function of $x$ (or vice versa)	Describes $x$ and $y$ as functions of a parameter $t$
Not all curves can be represented (e.g., vertical lines, loops)	Can represent a wider variety of curves, including those with multiple $y$ for a single $x$
Single equation	System of two equations

Key Takeaways

• Parametric equations are essential for describing complex curves and motion.

• They provide flexibility and are widely used in calculus, physics, and engineering.

• Eliminating the parameter can help relate parametric and Cartesian forms.

• The same curve can have multiple parametric representations.

Additional info: The notes above include standard definitions and examples of parametric equations, as well as a comparison table for clarity. The cycloid example is a classic application in physics and engineering.