Question

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

Short answer

Graph moves from (1, 1) to (0, -1) to (-1, 1), showing the orientation.

Step-by-step solution

Understand the Range and Interval

The parameter \( t \) ranges from \( 0 \) to \( \pi \). This means we'll consider the graph of the parametric equations from \( t = 0 \) to \( t = \pi \).

Analyze the Parametric Equations

The equations given are \( x = \cos t \) and \( y = \cos(2t) \). The \( x \)-coordinate is determined by \( \cos t \), and the \( y \)-coordinate is determined by \( \cos(2t) \). Note how \( \cos(t) \) and \( \cos(2t) \) behave over the interval \( [0, \pi] \).

Calculate Key Points

Identify key values of \( t \) to understand the behavior of the graph. For example:- At \( t = 0 \), \( x = \cos(0) = 1 \) and \( y = \cos(0) = 1 \).- At \( t = \frac{\pi}{2} \), \( x = \cos(\frac{\pi}{2}) = 0 \) and \( y = \cos(\pi) = -1 \).- At \( t = \pi \), \( x = \cos(\pi) = -1 \) and \( y = \cos(2\pi) = 1 \).

Determine the Graph Orientation

Consider how the graph moves as \( t \) increases from \( 0 \) to \( \pi \).- The graph starts at point (1, 1) when \( t = 0 \).- It moves to (0, -1) at \( t = \frac{\pi}{2} \).- Finally, it moves to (-1, 1) when \( t = \pi \).This identifies the orientation of the curve.

Sketch and Label the Graph

Using these key points and the orientation, sketch the curve from (1, 1) to (0, -1), and finally to (-1, 1). Label these points and indicate the direction with arrows showing the orientation.

Key concepts

Graph Orientation in Parametric Equations

When working with parametric equations like those provided in the exercise, graph orientation is crucial because it tells us the direction in which the graph is traced as the parameter changes. In this instance, we observe the parameter \( t \) progressing from 0 to \( \pi \).
Knowing the graph orientation can help us accurately sketch the curve and understand its path. For instance, the curve begins at the point \((1, 1)\) when \(t = 0\) and ends at \((-1, 1)\) when \(t = \pi\). This movement provides an understanding of the structure, behavior, and direction of the graph.

• Draw the initial point, usually when \( t=0 \) to see where the graph starts.
• Follow the progression of \( t \) to see if the curve turns left or right, goes up or down.
• Use arrows to show directionality on your sketch, ensuring clarity on how the graph progresses over the interval.

Graph orientation gives insight into the dynamic nature of parametric equations and is a foundational element for sketching them correctly.

Trigonometric Functions and Their Role

Trigonometric functions are fundamental to understanding parametric equations such as \( x = \cos t \) and \( y = \cos(2t) \). Recognizing how these functions behave is essential for analyzing and graphing the equations accurately.

In the given context:

• The function \( \cos(t) \) dictates the \( x \)-values, providing a smooth oscillation from 1 through 0 to -1 as \( t \) goes from 0 to \( \pi \).
• The function \( \cos(2t) \) affects the \( y \)-values, creating a faster oscillation because of its double frequency compared to \( \cos t \).

This doubling of the frequency means that \( \cos(2t) \) completes an entire cycle from 1 to -1 and back to 1 over the interval \( 0 \leq t \leq \pi \). This leads to various intersections and a complex interplay on the graph. Understanding these patterns enables you to predict the behavior of the curve and accurately determine the graph's form.

Interval Analysis of Parametric Equations

Interval analysis involves examining how parametric equations perform over a specific range of the parameter \( t \). In the exercise, \( t \) varies from 0 to \( \pi \), providing an opportunity to analyze the equations thoroughly as they describe a segment of a curve. This analysis entails understanding how each independent variable behaves through increasing \( t \).

For \( x = \cos(t) \):

• When \( t = 0 \), \( x \) starts at its peak value, 1.
• It smoothly decreases to 0 at \( t = \frac{\pi}{2} \) and further to -1 at \( t = \pi \).

For \( y = \cos(2t) \):

• Starting at 1 when \( t = 0 \).
• It reaches -1 at \( t = \frac{\pi}{2} \), illustrating that it cycles faster than \( x \).
• Finally returns back to 1 when \( t = \pi \).

By carefully tracing how \( t \) transforms \( x \) and \( y \), you can visually discern the path traced out and understand the nuances of the graph. Conducting interval analysis provides the essential checkpoints necessary for sketching and verifying the graph correctly.