91Ó°ÊÓ

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. This means, instead of directly relating x and y, we have separate equations for each:
\[ x = t - 3 \] and \[ y = \frac{1}{t} \]
To graph these, we substitute various values of t into both equations to generate points (x, y). These points are then plotted on a graph and connected smoothly. It's crucial to choose a range of t values that reveal the curve's behavior accurately. In this exercise, t was chosen from the set: \(-\text{infinity}, \text{infinity}\). This range ensures we capture the entire behavior of the graph.
Building a table of values helps to systematically organize this process. Here's a simplified approach:

• Select different t values: -2, -1, -0.5, 0.5, 1, and 2.
• Calculate corresponding x and y values using the parametric equations.
• Plot the calculated points on a graph.

Finally, connect the plotted points smoothly to reveal the curve, keeping in mind the trend as t approaches critical values such as 0. This will help in accurately sketching the curve.

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by the intersection of two perpendicular lines: the horizontal x-axis and the vertical y-axis. Each point on this plane is described by a pair of numerical coordinates \((x, y)\), determining its exact position.
In this plane:

• The x-coordinate indicates horizontal position: positive values to the right of the origin and negative values to the left.
• The y-coordinate indicates vertical position: positive values above the origin and negative values below.

To plot the parametric equations \(x = t - 3\) and \(y = \frac{1}{t}\), we:

• Calculate the x and y values by substituting chosen t values.
• Mark each point corresponding to \((x, y)\) values on this plane.
• Pay attention to the overall shape and behavior of the curve as points are plotted.

This step-by-step plotting provides a clear and visual understanding of the relationship between x and y as functions of the parameter t.

A hyperbola is a type of conic section that emerged from slicing a double-cone with a plane. Hyperbolas have two symmetrical curves, each reflecting the shape of the other. These curves are displayed in a mirror-image structure on the coordinate system.
In the context of our parametric equations, \(x = t - 3\) and \(y = \frac{1}{t}\), the resulting graph does show a hyperbola-like structure. Here's why:

• As t approaches 0, \(y\) tends to infinity, causing the curves to shoot outwards.
• The symmetry in \(1/t\) throughout positive and negative values of \(t\) creates mirroring curves.

Understanding hyperbolas helps in predicting and verifying the shape of our parametric equation graph. It's important to note:

• The asymptotes - lines which the curves approach but never actually reach - help guide the curve's shape.
• For large absolute values of \(t\), the \(x\) and \(y\) values settle into predictable patterns, aiding in sketching the entire curve smoothly.

Learning these foundational concepts ensures you can approach and solve parametric equations accurately on a rectangular coordinate system.