Parametric functions Knowledge-Based Study

Study notes

1. In standard Cartesian representation, a curve in the plane is expressed by a single equation relating two variables, such as y = f(x) or F(x, y) = 0. However, to describe the motion of an object or curves that fail the vertical line test, it is often more effective to express both x and y coordinates as independent functions of a third variable, called a parameter, typically denoted by $t$.

   Definition

   A parametric curve in the coordinate plane is defined by a pair of parametric equations:

   $$x = f(t), \quad y = g(t)$$

   where $t$ varies over a specified interval $I$. Each real number $t \in I$ yields an ordered pair $(x, y) = (f(t), g(t))$. As $t$ increases through the interval, the point $(x, y)$ traces out the curve in a specific direction, referred to as the orientation of the curve.

   To identify the geometric path of a parametric curve, we can often eliminate the parameter $t$ to produce a familiar Cartesian equation in terms of $x$ and $y$.

   Method of Substitution

   When one parametric equation can be solved explicitly for $t$, we substitute that expression into the other equation.

   Example: Consider the parametric system:

   $$x = 2t - 1, \quad y = t^2 + 3, \quad \text{for } t \in \mathbb{R}$$

   Solving the first equation for $t$ yields $t = \frac{x + 1}{2}$. Substituting this into the second equation produces:

   $$y = \left(\frac{x + 1}{2}\right)^2 + 3 = \frac{(x + 1)^2}{4} + 3$$

   This represents a parabola with vertex at $(-1, 3)$.

   Domain and Range Restrictions

   A Cartesian equation obtained by eliminating the parameter may describe an entire curve, while the parametric equations may trace only a subset of it. The domain and range of the parametric curve are strictly determined by the domain of the parameter $t$:

   $$\text{Domain: } \{x \mid x = f(t), \, t \in I\}, \quad \text{Range: } \{y \mid y = g(t), \, t \in I\}$$

   Curves exhibiting rotational symmetry, such as circles and ellipses, are naturally described using trigonometric functions where the parameter represents an angle.

   Circles and Ellipses

   A circle of radius $r$ centered at the origin is parameterized by:

   $$x = r \cos(t), \quad y = r \sin(t), \quad 0 \le t < 2\pi$$

   Using the Pythagorean identity $\cos^2(t) + \sin^2(t) = 1$, we confirm the Cartesian form:

   $$\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1 \implies x^2 + y^2 = r^2$$

   For an ellipse centered at $(h, k)$ with horizontal semi-axis $a$ and vertical semi-axis $b$, the parametric equations are:

   $$x = h + a \cos(t), \quad y = k + b \sin(t), \quad 0 \le t < 2\pi$$

   Eliminating $t$ produces the standard form:

   $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$

   Unlike a static Cartesian equation, parametric equations convey kinematic information: how, when, and in what direction the curve is traversed.

   Comparing Trajectories

   Different sets of parametric equations can represent the exact same geometric path while differing in orientation and traversal speed:

   • $x = \cos(t), \, y = \sin(t)$ for $0 \le t \le 2\pi$ traces the unit circle counterclockwise starting at $(1, 0)$.
   • $x = \cos(t), \, y = -\sin(t)$ for $0 \le t \le 2\pi$ traces the same unit circle clockwise starting at $(1, 0)$.
   • $x = \cos(2t), \, y = \sin(2t)$ for $0 \le t \le \pi$ completes the entire circle in half the parameter interval, representing double the tracing speed.

   Self-Intersection

   A parametric curve may intersect itself if there exist distinct parameter values $t_1 \ne t_2$ such that $f(t_1) = f(t_2)$ and $g(t_1) = g(t_2)$. This property enables parametric systems to model complex loops, knots, and orbital trajectories that cannot be represented as single-valued functions $y = f(x)$.