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Table 1 Actual data collected x and y

Date
t
x (longitude)
(East/West)
y (latitude)
(North/South)
September 26

0
18.7
-82.4
September 27

1
21.3
-83.4
September 28

2
24.4
-83.0
September 29

3
27.3
-81.6
September 30

4
30.2
-79.3

Step 3: Create a Mathematical Model

Work through the following steps to create two parametric equations where x is a function of t and y is a function of t. Remember t is just a parametric variable. You are creating two functions x(t) and y(t) ***If you use a linear regression for this portfolio the highest grade you are able to earn is a 70***
Plot x (longitude is the vertical axis) versus t (horizontal axis) (2 point)

Plot y (latitude is the vertical axis) versus t (horizontal axis). These should be two separate graphs. Make sure to submit the 2 graphs for your instructor to view. Label your axes and choose appropriate scales and ranges for your axis. Include a title for each graph. (2 point)

What type of function or regression model do you think would best fit the data based on your graphs? (2 points)

What type of function will you be using for x (longitude versus t) ___________________

What type of function will you be using for y (latitude versus t) ___________________

Use your calculator to create a formula for the model you have chosen. Enter the ordered pairs into lists and have the calculator create the best fit function for your model. For example, if your path appears to be exponential, you will have a model of the form y = abt using the ExpReg feature on the calculator. If you think the function is quadratic your model will have the form y = at2 + bt + c using the QuadReg feature on the calculator. You will then do the same for x. You do not have to use the same model type for both x and y. Pick the model that fits each one best! Remember do not use a linear function!

Directions to create this model on the TI84Plus or Desmos Calculator are at end of portfolio.

d. Write your final equations: (3 points)

x(t) =

y(t) =

1 answer

To create parametric equations for the given data, we will analyze the data points for both longitude (x) and latitude (y) over time (t).

Step 1: Data Overview

Based on the provided data, we have:

| Date | t | x (longitude) | y (latitude) | |---------------|---|----------------|---------------| | September 26 | 0 | 18.7 | -82.4 | | September 27 | 1 | 21.3 | -83.4 | | September 28 | 2 | 24.4 | -83.0 | | September 29 | 3 | 27.3 | -81.6 | | September 30 | 4 | 30.2 | -79.3 |

Step 2: Plotting the Data

You would need to create graphs using a graphing calculator or plotting software (like Desmos). However, I will guide you on descriptions for effective plots.

  1. Graph for x (longitude) vs. t (time):

    • Title: "Longitude vs. Time"
    • x-axis: Time (t), range from 0 to 4
    • y-axis: Longitude (x), range from 18 to 31 (to accommodate data points)
    • Plot points: (0, 18.7), (1, 21.3), (2, 24.4), (3, 27.3), (4, 30.2)

  2. Graph for y (latitude) vs. t (time):

    • Title: "Latitude vs. Time"
    • x-axis: Time (t), range from 0 to 4
    • y-axis: Latitude (y), range from -84 to -79
    • Plot points: (0, -82.4), (1, -83.4), (2, -83.0), (3, -81.6), (4, -79.3)

Step 3: Choosing the Model Type

Based on the graphs:

  • For x (longitude), the relationship appears to be quadratic as the values increase more rapidly as t increases.
  • For y (latitude), it appears that the values fluctuate more and do not follow a clear pattern, but a quadratic fit could accommodate the variability.

Final Model Types

  • For x (longitude): Quadratic
  • For y (latitude): Quadratic

Step 4: Create the Quadratic Models

You will input your data into your calculator's regression feature to find the best fit quadratic equations.

Using a quadratic regression function, the general form for a quadratic equation is:

  • \( y = at^2 + bt + c \)

Once you perform the regression analysis on your calculator, you would get equations of the form:

  1. Final Equations:
    • x(t) = \( a_x t^2 + b_x t + c_x \)
    • y(t) = \( a_y t^2 + b_y t + c_y \)

Example Final Equations: (These are placeholders; you must compute the actual coefficients using your calculator.)

  • x(t) = \( 2.95t^2 + 15.42t + 18.7 \) (Example)
  • y(t) = \( -0.83t^2 - 82.67t - 82.4 \) (Example)

You will want to substitute the actual coefficients obtained from your regression analysis in place of the placeholders.

Now, submit your graphs along with these equations to your instructor.