# Comparing Graphs of Parametric Equations

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft  per second at an angle of elevation of 52°. Consider the horizontal and vertical positions of the dart at any time t. Neglect air resistance.

Part I: Complete the following steps:

1. Find parametric equations that model the problem situation.
2. Use the GeoGebra tool to graph the parametric equations. (Refer to this tutorial as needed: https://www.geogebra.org/m/XUv5mXTm (Links to an external site.).)
3. Find all possible values for t that represent the situation.
4. State the parametric equations.
5. State the domain and range for each parametric equation.
6. Plot the minimum and maximum heights of the dart.
7. Plot the minimum and maximum horizontal positions of the dart.
8. Find the time at which the dart reaches the maximum height.
9. Save your GeoGebra work as a .pdf file for submission.

Part II: Based on your work in Part I, discuss the following:

1. Discuss why this situation can be modeled using parametric equations.
2. Discuss how you determined all possible values for t that represent the situation.
3. Discuss what the domain and range of the parametric equations mean in the context of this problem.
4. What do maximum and minimum values of the parametric equations represent in this context?
5. Discuss how your answers to Part I would be affected if:
1. The initial velocity is increased.
2. The angle of elevation is decreased.
6. Provide at least two other real-world situations that can be modeled using parametric functions and respond to the following:
1. What common characteristics do the real-world scenarios you chose share?
2. What did you look for in the way that the real-world scenario can be modeled?
3. How can you verify that the real-world scenarios you chose can be modeled by parametric functions?

Discussion may include: The horizontal and vertical positions of the dart as a function of time can serve as parametric equations to model this situation. In this case, t can be any non-negative real number. The domains of the parametric equations are any non-negative real number. The range of the vertical position function is, assuming the ground is at a height of 0, also any non-negative real number while the range of the horizontal position function is any real number assuming 0 is where the dart is first thrown.  Any real-world scenario involving parametric equations should involve an independent parameter such as time and at least two variables that are a function of the independent parameter.

Requirements:

You must submit two files for this assignment.  The first file should contain the computations, graphs, diagrams, etc., associated with the questions in Part I.  This file may be formatted as a numbered list of answers.  Unless stated in the problem, a narrative discussion is not required, but you must provide enough information to show how you arrived at the answer.

The second file should be a 2-3 page narrative paper, written in APA format, associated with the situation described in Part II.  Specific requirements for the paper are provided below:

1. Your paper should be 2-3 pages in length (not counting the title page and references page) and cite and integrate at least two credible outside sources. The CSU-Global Library is a great place to find resources. Your textbook is a credible resource.
2. Include a title page, introduction, body, conclusion, and a reference page.
1. The introduction should describe or summarize the topic or problem. It might discuss the general applications of the topic or it might introduce the unique terminology associated with the topic.
2. The body of your paper should address the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.
3. The conclusion should summarize your thoughts about what you have determined from your analysis in completing the assignment. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.
3. Include any tables of data or calculations, calculated values, and/or graphs referenced in the paper. (Note: The minimum required length excludes any tables, graphs, etc.)
4. Document formatting, citations, and style should conform to the CSU-Global Virtual Library CSU-Global Guide to Writing and APA: Introduction. A short summary containing much that you need to know about paper formatting, citations, and references is contained in the New Sample APA Paper. In addition, information in the CSU-Global Virtual Library under the Writing Center/APA Resources tab has many helpful areas (Writing Center, Writing Tips, Template Examples/Papers Essays, Figures and Tables, and others).