Question

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$x=\frac{3 t}{1+t^{3}}, y=\frac{3 t^{2}}{1+t^{3}}$$

Short answer

Question: Use a graphing utility to graph the curve represented by the parametric equations $$x = \frac{3t}{1+t^3}$$ and $$y = \frac{3t^2}{1+t^3}$$ over the interval $$t \in [-10, 10]$$ and describe any notable features observed in the graph. Answer: After graphing the curve with the parametric equations, we observe that it has an asymptote at $$x = 0$$, a maximum point near $$(1, 1)$$, and a minimum point near $$(-1,1)$$, as well as exhibiting symmetry about the y-axis.

Step-by-step solution

Choose an appropriate interval for the parameter 't'

First, we need to choose an appropriate interval for the parameter 't'. Since we're looking to generate all features of interest, we should look for a reasonably large range for 't'. One option could be a range from the real numbers, though you may need to adjust the interval if it doesn't generate all features of interest. For this exercise, the interval we will use for the parameter 't' is: $$t \in [-10, 10]$$

Plug in the parametric equations into a graphing utility

In step 2, we will input the given parametric equations into a graphing utility. There are many free graphing utilities available online. For this exercise, we will use the popular graphing utility called Desmos (https://www.desmos.com/calculator). Once on the site, input the following parametric equations: $$x = \frac{3t}{1+t^3}$$ $$y = \frac{3t^2}{1+t^3}$$ Set the parameter 't' to go from -10 to 10 to cover the interval we chose in step 1.

Graph the curve and observe the features of interest

Now that we've input the parametric equations and set the interval for 't', we can look at the graph generated by the graphing utility. Observing the graph, take note of any interesting features like asymptotes, maximum, minimum, or points of inflection. After following these three steps, we will have successfully graphed the given parametric equations and analyzed the features of interest over the specified interval.

Key concepts

graphing utility

Graphing utilities are tools that help us visualize mathematical equations by generating their graphical representation. These tools are incredibly useful, especially when dealing with complex equations, such as parametric equations.

• They can plot points, lines, curves, and even surfaces in a coordinate plane.
• Most graphing utilities can handle a variety of functions, including linear, quadratic, and even parametric equations like the ones in this exercise.
• These tools may also have features that allow you to adjust the view, zoom in and out, and manipulate the graph to better understand its properties.

By using a graphing utility, students can visually understand the behavior of equations, aiding in their overall comprehension and problem-solving skills.

Desmos

Desmos is a popular online graphing utility that is free and particularly user-friendly. It allows users to plot and analyze mathematical equations with ease.

• Desmos supports a wide array of mathematical expressions, including parametric equations, which are the focus of this exercise.
• The platform's grid system is intuitive, enabling users to input equations and immediately see their results.
• One advantage of Desmos is that it runs on most web browsers, eliminating the need for software installation.

This makes Desmos an accessible tool for students everywhere, providing an educational space to experiment with a range of mathematical concepts.

parameter interval

When working with parametric equations, choosing the correct parameter interval is crucial. The parameter interval determines the start and end values of the parameter (in this case, 't'), thus affecting the part of the curve that will be graphed.

• A well-chosen interval will allow you to capture all the significant features of the graph such as loops, endpoints, or any peculiar behaviors.
• For our exercise, the interval \([-10, 10]\) was chosen for 't', providing a wide range to ensure all features are visible.
• Adjusting the interval can help focus on specific parts of the graph or reveal intricate behaviors that might otherwise be missed.

By experimenting with different parameter intervals, you can develop a deeper understanding of how the parametric equations describe the curve.

curve features

When you graph parametric equations, you can observe various features of the curve that can provide insights into its behavior. Here are some key features to look for:

• **Asymptotes**: Lines that the curve approaches but never crosses. These can indicate undefined behavior or infinity.
• **Maxima and Minima**: Points where the curve reaches a highest or lowest point, respectively, can indicate turning points or significant changes in direction.
• **Inflection Points**: Points where the curve changes concavity, providing insights into the graph's curvature.

Observing these features can help you understand the dynamics of the curves produced by the given parametric equations. This can provide deeper comprehension when analyzing real-world data represented by these mathematical models.