What is the fundamental characteristic of parametric equations?

Parametric equations define $x$ and $y$ as functions of an independent parameter, typically $t$, where $x = f(t)$ and $y = g(t)$, meaning $x$ and $y$ are not directly related.

What is the significance of the parameter $t$ in defining motion along the curve?

The parameter $t$ is often treated as time, defining the path taken by an object from an initial point (at the start of the interval $i$) to a terminal point (at the end of the interval $i$).

How is the algebraic equation derived from parametric equations?

The algebraic equation is found by eliminating the parameter $t$ from the two parametric equations, resulting in an equation solely in terms of $x$ and $y$.

What is the resulting algebraic equation when $x = A \cos t$ and $y = A \sin t$ are given?

Using the identity $\cos^2 t + \sin^2 t = 1$, the resulting algebraic equation is $x^2 + y^2 = A^2$, which is the equation for a circle centered at $(0,0)$ with radius $A$.

In the example $x = \sin t$ and $y = \sin 2t$, how was $\cos t$ expressed in terms of $x$ and $y$ to eliminate $t$?

By using the double-angle identity $y = 2 \sin t \cos t$, and knowing $x = \sin t$, $\cos t$ was isolated as $y / (2x)$ before applying the Pythagorean identity.

Menggambar Kurva Parametrik | Kalkulus 2

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

Menggambar Kurva Parametrik | Kalkulus 2

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