Definition of Parametric Equations
📌 Parametric equations define $x$ and $y$ as functions of a third, independent variable called the parameter, denoted as $t$: $x = f(t)$ and $y = g(t)$ over an interval $I$.
📐 The set of points $(x, y) = (f(t), g(t))$ defined by these equations constitutes a parametric curve.
🧭 The parameter $t$ often represents time, defining the movement from an initial point ($t=A$) to a terminal point ($t=B$).

Eliminating the Parameter (Finding Algebraic Equations)
📌 To find the algebraic equation (in terms of $x$ and $y$ only), the parameter $t$ must be eliminated using algebraic manipulation or trigonometric identities.
🔄 Example 1 ($\boldsymbol{x = t^2 - 4}$, $\boldsymbol{y = 2t}$): Since $t = y/2$, substituting this into the $x$ equation yields the algebraic form: $x = (y/2)^2 - 4$ or $x = y^2/4 - 4$.
⭕ Example 4 ($\boldsymbol{x = a \cos t}$, $\boldsymbol{y = a \sin t}$): Using the identity $\cos^2 t + \sin^2 t = 1$, the parametric equations transform into the equation of a circle centered at $(0,0)$ with radius $A$: $x^2 + y^2 = A^2$.

Graphing and Describing Motion
📌 Sketching the curve requires evaluating $(x, y)$ pairs for various values of $t$ within the defined interval $I=[A, B]$.
➡️ The initial point is $(f(A), g(A))$ and the terminal point is $(f(B), g(B))$.
🏹 The direction of motion along the curve must be indicated with arrows on the graph corresponding to increasing values of the parameter $t$.

Specific Curve Examples
🔷 Example 5 (Ellipse): For $x = 4 \cos t$ and $y = 3 \sin t$ over $0 \le t \le 2\pi$, the algebraic form is $\frac{x^2}{16} + \frac{y^2}{9} = 1$, representing an ellipse centered at $(0,0)$.
➰ Example 6 (Lissajous Figure): The equations $x = \sin t$ and $y = \sin 2t$ yield the algebraic equation $x^2(1 - x^2) = \frac{y^2}{4}$, which traces a complex closed curve (Lissajous figure) when $t$ ranges from $0$ to $4\pi$.

Key Points & Insights
➡️ Parametric equations are considered easier than series and sequences and must be understood for exams due to their simplicity.
✅ When graphing, always clearly label the initial point and the terminal point based on the parameter interval $[A, B]$.
✍️ Ensure that the direction of motion is shown on the final curve sketch using arrows to represent the particle's path over time $t$.

📸 Video summarized with SummaryTube.com on Nov 19, 2025, 05:46 UTC