Parametric Equations: Concepts, Graphs, and Applications

Parametric Equations and Plane Curves

Definition and Basic Properties

Parametric equations are a powerful way to describe curves in the plane by expressing both x and y as functions of a third variable, called the parameter (usually t). This approach is especially useful for representing curves that cannot be described easily by a single function y = f(x).

• Plane Curve: A set of points (x, y) such that $x = f(t)$ and $y = g(t)$, where f and g are continuous on an interval I.

• Parametric Equations: The equations $x = f(t)$ and $y = g(t)$ are called parametric equations with parameter t.

Graphing Parametric Equations

Graphing a Parabola Parametrically

To graph a curve defined by parametric equations, create a table of values for t, then compute corresponding x and y values and plot the points.

• Example: $x = t^2$, $y = 2t + 3$, for $t$ in $[-3, 3]$

• Table of values: $t = -3 ightarrow (x, y) = (9, -3)$ $t = 0 ightarrow (x, y) = (0, 3)$ $t = 3 ightarrow (x, y) = (9, 9)$

• The direction of the curve is indicated by increasing t.

Eliminating the Parameter

To find the equivalent rectangular (Cartesian) equation, solve one parametric equation for t and substitute into the other.

• Given $y = 2t + 3$, solve for $t$: $t = rac{y - 3}{2}$

• Substitute into $x = t^2$: $x = igg( rac{y - 3}{2}igg)^2 = rac{1}{4}(y - 3)^2$

• Rectangular Equation: $x = rac{1}{4}(y - 3)^2$, for $x$ in $[0, 9]$

Graphing Calculator Solution

Parametric mode on a graphing calculator allows direct input of $x = f(t)$ and $y = g(t)$ to visualize the curve and its direction as t increases.

Graphing Other Plane Curves Parametrically

Ellipse Example

• Example: $x = 2t + 5$, $y = igg( rac{4 - t^2}{1}igg)^{1/2}$, for $t$ in $[-2, 2]$

• Eliminate $t$: $t = rac{x - 5}{2}$, substitute into $y$: $y = igg(4 - igg( rac{x - 5}{2}igg)^2igg)^{1/2}$

• Rectangular form: $4y^2 + (x - 5)^2 = 16$

• This is the equation of an ellipse. Since $y ext{ is defined as } igg(4 - t^2igg)^{1/2}$, only the upper half is graphed.

Graphing a Line Parametrically

• Example: $x = t^2$, $y = t^2$, for $t$ in $[0, ext{infinity})$

• Rectangular form: $y = x$, $x ext{ and } y ext{ are both } ext{greater than or equal to } 0$

• This represents half the line $y = x$ for $x ext{ and } y ext{ } ext{greater than or equal to } 0$.

Alternative Forms of Parametric Equations

Non-uniqueness of Parametric Representations

Parametric equations for a curve are not unique; different parameterizations can describe the same curve.

• For $y = f(x)$, a simple parameterization is $x = t$, $y = f(t)$, for $t$ in the domain of $f$.

• Example: For $y = (x - 2)^2 + 1$:

  • First form: $x = t$, $y = (t - 2)^2 + 1$

  • Second form: $x = t + 2$, $y = t^2 + 1$

Applications: Projectile Motion

Parametric Equations in Physics

Parametric equations are used to model the position of moving objects, such as projectiles, as functions of time.

• General Form: $x = f(t)$, $y = g(t)$, where $t$ is time.

Projectile Motion Example

• For a projectile launched at $heta = 45^ ext{o}$ with initial speed $v_0$ (feet/sec): $x = v_0 rac{ oot 2 elax}{2} t$ $y = v_0 rac{ oot 2 elax}{2} t - 16t^2$ for $t$ in $[0, k]$

• Eliminate $t$ to get the rectangular form: $x = v_0 rac{ oot 2 elax}{2} t ightarrow t = rac{2x}{v_0 oot 2 elax}$ Substitute into $y$: $y = v_0 rac{ oot 2 elax}{2} igg( rac{2x}{v_0 oot 2 elax}igg) - 16 igg( rac{2x}{v_0 oot 2 elax}igg)^2$ $y = x - rac{32}{v_0^2} x^2$

• This is a downward-opening parabola describing the projectile's path.

Summary Table: Parametric vs. Rectangular Forms

Curve Type	Parametric Form	Rectangular Form
Parabola	$x = t^2$, $y = 2t + 3$	$x = rac{1}{4}(y - 3)^2$
Ellipse (upper half)	$x = 2t + 5$, $y = igg(4 - t^2igg)^{1/2}$	$4y^2 + (x - 5)^2 = 16$, $y ext{ } ext{greater than or equal to } 0$
Line (ray)	$x = t^2$, $y = t^2$, $t ext{ } ext{greater than or equal to } 0$	$y = x$, $x ext{ } ext{greater than or equal to } 0$
Projectile Path	$x = v_0 rac{ oot 2 elax}{2} t$, $y = v_0 rac{ oot 2 elax}{2} t - 16t^2$	$y = x - rac{32}{v_0^2} x^2$

Key Takeaways

• Parametric equations provide a flexible way to describe curves and motion in the plane.

• Eliminating the parameter allows conversion to rectangular form for further analysis.

• Applications include modeling physical phenomena such as projectile motion.

• Graphing calculators can visualize parametric curves and their direction.

Additional info: These notes expand on the provided slides and text, adding definitions, step-by-step examples, and a summary table for clarity and completeness.