Question

Sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph. \(x=\cos t, \quad y=\sin (2 t), \quad 0 \leq t \leq 2 \pi\)

Short answer

The graph is an elongated figure-eight with loops above and below the x-axis, oriented left to right as \( t \) increases.

Step-by-step solution

Understand the Equations

The given parametric equations are: \( x = \cos t \) and \( y = \sin (2t) \). Here, \( t \) is a parameter that varies within the interval \( 0 \leq t \leq 2\pi \). These equations define the position of a point \((x, y)\) as \( t \) changes.

Identify Key Features

For the given parametric equations, notice that \( x = \cos t \) produces values for \( x \) between -1 and 1 as \( t \) varies. Similarly, \( y = \sin(2t) \) also produces values between -1 and 1. The parameter \( t \) affects the orientation and shape of the curve.

Consider the Range of t

The parameter \( t \) ranges from \( 0 \) to \( 2\pi \). This means the graph completes the full periodic cycle allowed by \( \cos t \) and \( \sin (2t) \). The orientation of the curve, how it progresses from start to end, is determined by increasing \( t \).

Sketch the Curve without Graphing Utility

To visualize without a graphing utility, plot some points. For example, \( t = 0 \) gives \( x = 1 \), \( y = 0 \); \( t = \frac{\pi}{2} \), gives \( x = 0 \), \( y = 1 \); \( t = \pi \), gives \( x = -1 \), \( y = 0 \); and so forth. Connect these points considering the cyclic nature of sine and cosine.

Determine the Orientation

As \( t \) increases from 0 to \( 2\pi \), the graph will trace a figure starting from \( (1, 0) \), moving to \( (0, 1) \), then \( (-1, 0) \), and back, completing its shape within the first cycle of \( t \), and then repeating.

Conclusions from Graphing Utility

Using a graphing utility provides confirmation of the shape and orientation. Enter the equations \( x = \cos t \) and \( y = \sin (2t) \) into a graphing tool to visualize. The expected graph will resemble an elongated figure-eight shape with one loop above and one below the x-axis, showing the orientation from left to right as \( t \) increases.

Key concepts

Graphing Techniques

When it comes to graphing parametric equations, it's essential to understand how to extract the geometric nature of a curve just from its equations. The parametric form consists of separate equations for the x and y coordinates, typically depending on a third parameter, often denoted as \( t \). This method is a powerful technique for visualizing curves that might be difficult to describe otherwise.

Start by considering values of \( t \) within the given interval. In our exercise, \( t \) ranges from 0 to \( 2\pi \). As you substitute these values, you'll be sketching portions or even full cycles of the function described by these equations. Values of \( x = \cos t \) will lie between -1 and 1, while \( y = \sin(2t) \), with double the frequency of sine's normal cycles, also oscillates between -1 and 1.

Pointers include:

• Use a step-wise approach to "connect the dots"—this means calculate \( x \) and \( y \) for selected values of \( t \) within the specified range.
• Plot each of those points on a coordinate plane to get the skeleton of your curve.
• Consider using a graphing utility for a more precise and complete curve visualization.

Effectively, this process allows complex curves to be dissected into comprehensible parts, showing notable shapes and features in a manageable manner.

Orientation of Curves

Orientation describes the direction in which a parametric curve is traversed as the parameter value changes. For the parametric equations \( x = \cos t \) and \( y = \sin(2t) \), orientation reveals itself in the progression of \( t \) from 0 to \( 2\pi \). This means tracking the movement of the point \((x, y)\) on the plane as \( t \) increases.

Determining orientation involves:

• Identifying starting point: \( t = 0 \) yields \((1, 0)\), a starting point on the curve.
• Monitoring subsequent points, like \( t = \frac{\pi}{2} \), gives \((0, 1)\), and \( t = \pi \) yields \((-1, 0)\).
• Observing the direction along these points tells us how the curve moves. Here, it moves from right to left and then returns, giving a complete periodic cycle.

The curve's path will look like a loop or a figure-eight, starting from the defined point and moving symmetrically along the path. This understanding of orientation is key to faithfully reconstructing the motion implied by the parametric equations.

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, play pivotal roles in parametric equations. They dictate how the curve behaves, and interactions between these functions generate fascinating shapes. In our context, we deal with functions \( x = \cos t \) and \( y = \sin(2t) \).

Important considerations include:

• Both sine and cosine functions are periodic, oscillating between -1 and 1.
• The \( \cos t \) represents typical cosine wave patterns, important in the x-component of the curve.
• The \( \sin(2t) \) indicates a sine wave that completes two full cycles over the same interval of \( t \), reflecting higher frequency compared to the cosine component.

Interestingly, as \( t \) switches, these functions control the oscillation and give the curve its form.
Understanding these trig functions enables you to predict the waveforms involved, thus making it easier to sketch the resulting curve with the given equations.