PRECALCULUS: PARAMETRIC FUNCTIONS Directions: Meteorologists use sophisticated models to predict the occurrence, duration, and trajectory of weather events. They build their models based on observations that they have made in the past. By understanding how previous weather events evolved, meteorologists can apply that knowledge to future weather events. Parametric equations can be used to graph the path of an object in space. For example, they can be used to describe the path of a storm moving through an area. In this portfolio, you will use historical storm data to trace the path of a hurricane. From this data, you will use parametric equations to model the path of the storm. Step 1: Analyze Hurricane George of 1980 The table below shows the latitude and longitude of Hurricane George at different times on each day from September 1st to September 8th 1980. Date Longitude Latitude September 1 -38.0 15.6 September 1 -40.8 16.3 September 1 -42.1 16.8 September 2 -43.7 17.3 September 2 -45.7 17.5 September 2 -48.1 17.7 September 2 -50.3 17.8 September 3 -54.5 18.0 September 3 -56.9 18.6 September 3 -59.0 19.7 September 4 -61.0 21 September 4 -62.3 22.1 September 4 -63.6 23.4 September 5 -65.8 26.1 September 5 -68.6 28.5 September 5 -69.4 29 September 5 -70.0 30.6 September 6 -69.6 31.7 September 6 -69.1 32.9Step 4: Use your Parametric Model to nd the predicted x and y at each time point. Using your equations you found in step 3 to nd the predicted it and y coordinates. Plug in the values t = U, 1, 2, 3, and 4 into your parametric equations and enter your values forx and y in the table below. Table 2 Predicted x a y from your model's equation {2 points} x (longitude) redicted from model y (latitude) predicted from model a. Plot your predicted x {table 2) and your actual x [table 1} versus time on one graph. The vertical axis is the longitude, and the horizontal axis is t. (1 pt} b. Plot predicted y (table 2) and your actual y [table 1) versus time on one graph. The vertical axis is latitude and the horizontal axis is t. [1 pt}. Make sure to submit the 2 graphs and label your axes and chose appropriate scales and ranges. Have a key indicating which data are predicted and which are actual. For example use one color for predicted and one color for actual. c. Visually compare the actual and predicted data on the plots? Do they seem to be similar? If not consider using a different model and repeating part 3d and Step 4. Step 5: Check your Parametric Model Now graph the actual x- and y-coordinates from Table 1 using one color and graph the predicted or modeled x- and y-coordinates from Table 2 onto the same graph using a different color. Your )1 coordinates are the horizontal axis and your y coordinates are the vertical axis. Be sure to label which color is which dataset. You may either copy and paste your graph here or upload it along with this worksheet. 1. How does your model compare to the actual path? Be specic and write at least 2 sentences supporting your conclusion [1 pt) 2. Why did you choose the graph family that you did? Did you choose well? Why or why not? {1 pt] 3. Is it possible to write this in rectangular form, eliminating the parameter t. In other words express y In terms of x (eliminate the parameter t and have one function, y =f{x)) Why or why not? (1 pt) Turn it in: - Upload this completed worksheet into the Drop Box. You should have 5 clearly labeled graphs included: Actual 3: versus t Actual y versus t Dverlaid graph of Modeled and Actual 3:: versus t Dverlaid graph of Modeled and Actual y versus t Dverlaid graph of Modeled and Actual x and y coordinatea For all graphs, make sure to label your axis and use an appropriate scale and for overlaid graphs include a key indicating what is each graph. - If you did not paste a copies of your graphs into the worksheet, be sure to also upload the graphs into the Drop Box. Clearly label what is in each le. Step 2: Plot the Hurricane Path Use the data from step 1 to make a table of the storm's horizontal and vertical movement wi'i respect to time. Start with a data point provided in the table in step 1 from September 1 and make this date it = [1. Note the position's longitude and latitude and record them in Table 1. Since latitude measures nor'u'soutl'i and longitude measures eastfwest, the latitude coordinate will be 1; and the longitude coordinate wil be 1:. How progres through the davs along the path. lEhoose and record one point from each dav of the storm. Mark each point t = 1r t = 2, etc. Tlack the storm for a total of IE davs so that vou have 5 poinls in the tableF one from each dav of the storm. For example: Pick one of the data points for September 1 This point is t = ID Record the x [longitude] and v [latitude] coordinates in Table 1 Pick one of the data points for September 2. This point is t=1 Record the :I: [longitude} and v [latitude] coordinates in Table 1 Repeat for September 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*** a. Plot x (longitude is the vertical axis) versus t (horizontal axis) (1 point) b. 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 chose appropriate scales and ranges for your axis. (1 point) c. What type of function or regression model do you think would best fit the data based on your graphs? (1 points) What type of function will you be using for x (longitude versus t) 3(-61.0) ??? What type of function will you be using for y (latitude versus t) 3(21) ??? 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 = atz + 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 Calculator at end of portfolio. d. Write your final equations: (2 points) x(t) = . y(t) =Directions for finding a function using data points on the TI 84 Plus To enter values to find the function Hit STAT TI-84 Plus Sher Edition Hit 1: Edit To clear all values in table use arrows to go to column 41 LZ L3 heading, hit CLEAR and then ENTER Enter values in lists as needed. Hit ENTER after each entry Use arrows to move between and within lists To find the regression model Hit STAT Arrow over to CALC . Chose the function you want - there are two screens of 2 choices. Consider: 5 QuadReg - Quadratic at2 +bt + c 6 CubicReg - Cubic at3 +bt2 + ct + d 7 QuartReg - Quartic at4 +bt3 + ct2 + dt + e 0 ExpReg - Exponential abt A PwrReg - Power at TI-84 Plus C Silver Edition TI-84 Plus C Silver Edition TEXAS INSTRUMENTS TEXAS INSTRUMENTS NORMAL FLOAT AUTO REAL DEGREE HIP NORMAL FLOAT AUTO REAL DEGREE HP EDIT CALC TESTS 1:1-Var Stats EDIT CALC TESTS 2:2-Var Stats 7+QuartReg 3:Med-Med 8:LinReg(a+bx) 4:LinReg(ax+b) 9: LnReg 5: QuadReg 0: ExpReg 6: CubicReg A: PurReg 7: QuartReg B: Logistic 8:LinReg(a+bx) C: SinReg 94LnReg D: Manual-Fit Y=mX+b EBQuickPlot&Fit-EQ STAT PLOT