Problem 41 - Precalculus with Calculus Previews

Question

Use a graphing utility to obtain the graph of the given set of parametric equations. $$ x=6 \sin 4 t, y=4 \sin t, 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The graph is an ellipse-like figure with horizontal loops, oscillating 4 times within one vertical cycle.

Step 1: Understand the Parametric Equations

We have the parametric equations given as \(x=6 \sin 4t\) and \(y=4 \sin t\). The parameter \(t\) varies from 0 to \(2\pi\). These equations define a relationship between \(x\), \(y\), and \(t\).

Step 2: Set Up the Graphing Utility

Use a graphing utility, such as a graphing calculator or software like Desmos, GeoGebra, or an online tool that supports parametric equations. Input the equations \(x(t) = 6 \sin 4t\) and \(y(t) = 4 \sin t\) into the graphing tool. Set the parameter \(t\) to range from 0 to \(2\pi\).

Step 3: Visualize the Graph

With the parametric equations entered, generate the graph. You should see an ellipse-shaped graph with a repetitive loop or oscillation pattern, indicating the changes in the \(x\) and \(y\) coordinates over the specified range of \(t\).

Step 4: Analyze the Graph

Observe the graph's behavior as \(t\) changes. Note that \(x\) undergoes oscillations at a frequency determined by \(\sin 4t\), which results in 4 cycles within the interval, whereas \(y\) oscillates as \(\sin t\), producing a single cycle. Hence, the resulting graph would resemble a series of loops horizontally stretched across the \(x\)-axis and vertically constrained.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool designed to help visualize mathematical equations effortlessly. Just like a physical graphing calculator, digital graphing tools allow you to input equations and instantly see the corresponding graph. This is extremely useful for parametric equations because it lets you visualize complex relationships between variables in real-time.

To use a graphing utility, begin by selecting a tool such as Desmos, GeoGebra, or another online platform supporting parametric equations. Insert the parametric equations into the tool's input fields. For the given problem, you'd enter \(x(t) = 6 \sin 4t\) and \(y(t) = 4 \sin t\).

Next, define the range for the parameter \(t\). In this case, \(t\) varies from 0 to \(2\pi\). Setting the correct range is crucial, as it determines which parts of the function will be visible on the graph. Once everything is set, the program instantly plots the graph, revealing the shape described by these equations.

Trigonometric Functions

Trigonometric functions like sine and cosine, integral to this exercise, are key components in many parametric equations. They describe how things oscillate and rotate in a circular pattern. Here, the parametric equations involve these functions extensively.

For the function \(x=6 \sin 4t\), the sine function repeats its cycle four times as \(t\) progresses from 0 to \(2\pi\). This is due to the factor of 4 in the angle \(4t\), which makes the curve oscillate at a higher frequency. The coefficient 6 stretches the cycle horizontally, affecting the amplitude of the oscillation.

The function \(y=4 \sin t\) involves a basic sine function completing one full cycle from 0 to \(2\pi\). The coefficient 4 dictates the amplitude, meaning the peaks and troughs of the curve reach 4 units above and below the central axis.

Understanding these trigonometric properties, including cycle frequency and amplitude, is essential in predicting how the graph of parametric equations will look.

Visualization of Graphs

Graphs are an essential part of understanding mathematical concepts, as they turn abstract numbers into visible patterns and shapes. When visualizing parametric equations, the graph can show how two related variables change with respect to a third parameter.

In the problem at hand, the graph produced by the equations \(x=6 \sin 4t\) and \(y=4 \sin t\) creates an elliptical shape that appears to have loops. This particular form stems from how the sine functions control the oscillations of \(x\) and \(y\) with different frequencies.

The changes in \(x\) and \(y\) as \(t\) varies are crucial to noticing. \(x\) changes more rapidly due to the higher frequency sine function, resulting in a horizontally stretched pattern, while \(y\) varies more slowly, confining the graph vertically. Such characteristics give the graph its distinctive geometry.

By seeing the synchronous relationship between \(x\), \(y\), and \(t\), students can grasp the dynamic behavior of these equations, enhancing their understanding of parametric relationships.