Question

Use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. $$ \begin{aligned} &x=t-0.5 \sin t \\ &y=1-1.5 \cos t \end{aligned} $$

Short answer

The curve is an ellipse, centered at roughly (0, 1).

Step-by-step solution

Parametric Equations Overview

We begin by examining the given parametric equations: \(x = t - 0.5\sin t\) and \(y = 1 - 1.5\cos t\). These equations determine the x and y coordinates of points on the curve, where \(t\) is the parameter that varies.

Use Graphing Utility

Input the parametric equations into a graphing utility that supports parametric graphing. Adjust the range of \(t\) to fully visualize the curve, typically a range from \(0\) to \(2\pi\) suffices for periodic functions. Observe the shape and behavior of the graph.

Identify the Shape

Examine the graph for features that help identify the curve. Notice the amplitude of oscillation in the \(x\) direction is determined by \(0.5\sin t\) and the \(y\) direction by \(1.5\cos t\). This results in a standard elliptic shape displaced along the x-axis by 0.5 units.

Analyze Curve Characteristics

The curve is centered at approximately \((0, 1)\) because the modifications \(-0.5\sin t\) and \(-1.5\cos t\) create oscillations around these center values. The elliptic nature suggests the equation can be further analyzed for specific attributes like major and minor axes.

Confirm the Curve Type

Based on the graph from the utility and the characteristics of the parametric equations, identify the curve as an ellipse. The equations suggest that x and y oscillate around a central point, typical behavior for an ellipse.

Key concepts

Parametric Curve

When we talk about a parametric curve, we mean a type of mathematical curve that's defined by a pair of equations. Each of these equations assigns values to coordinates on the curve based on a shared variable, called a parameter—denoted as \( t \) in our equations. The beautiful thing about parametric equations is that they give you the flexibility to define a curve's path in a completely different way than regular (Cartesian) equations do. Instead of starting with \( y = f(x) \), we begin with separate expressions for \( x \) and \( y \) like:

• \( x = t - 0.5 \sin t \)
• \( y = 1 - 1.5 \cos t \)

This lets you explore more complex shapes and motions that wouldn't be possible otherwise.
Parametric curves are great for modeling phenomena where time or a similar parameter naturally controls behavior. Think of a path taken by a point on a moving object. And because both \( x \) and \( y \) can be functions of \( t \), these curves can illustrate more dynamic and real-world scenarios.

Ellipse Identification

Identifying an ellipse is an exciting detective task in math. An ellipse can almost be described as a 'stretched' circle, and you often see them in physics and astronomy. In our parametric curve, the ellipse comes into play with the terms \( 0.5 \sin t \) and \( 1.5 \cos t \), which pertain to the oscillations around a central axis. Let’s break it down:

• \( 0.5 \sin t \) affects the width or the horizontal oscillation.
• \( 1.5 \cos t \) impacts the height or the vertical oscillation.

Together, they form an elliptic shape by varying in amplitude.
In parametric terms, these oscillations around central average values (like \( x = 0 \) and \( y = 1 \)) often indicate an ellipse. The amplitude modifications create the characteristic 'oval' shape. By recognizing these parts of the equation and their effects, you grasp the essence of identifying an ellipse: constant oscillations that form a closed, curved path.

Graphing Utility

A graphing utility acts like a magic wand for visualizing math equations. This tool can graph functions—especially parametric ones—that aren't easy to sketch by hand. To graph our parametric equations \( x = t - 0.5 \sin t \) and \( y = 1 - 1.5 \cos t \), you might use a calculator or software like Desmos. Here's how it helps:

• Enter both equations: Input them into the utility to get a live preview.
• Adjust the parameter \( t \): Usually, you set \( t \) from 0 to \( 2 \pi \) for periodic functions, ensuring the whole curve is visible.
• Analyze the graph: Look for patterns, shapes, or spikes that reveal the curve type.

Using a graphing utility saves you tons of time and shows immediate feedback. This is especially valuable when dealing with complex curves, helping you to better identify and understand the behavior and geometry of the equations you are working with.

Periodic Functions

Periodic functions are at the heart of many natural phenomena. Think of the way the sun rises and sets every day—those predictable patterns are captured perfectly by periodic functions. Parametric curves often incorporate these functions to model cyclic behaviors. In our equations:

• \( \sin t \) and \( \cos t \) are classic periodic functions with cycles every \( 2 \pi \).
• They control the repeating patterns we see in the oscillations of the ellipse.

Naturally, because these functions repeat, you find that shapes made with them tend to complete a full cycle and return to their starting point.
Understanding periodic functions inside parametric equations helps you predict behavior across intervals and see the repeated nature of certain phenomena. It's why they're so powerful in modeling real-world dynamic systems and cycles, from waves in the ocean to sound waves in the air.