Honors Algebra 3-4 Mr. Felling Expanded Topic - Curve Fitting Extra Credit Project

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1 Honors Algebra 3-4 Mr. Felling Expanded Topic - Curve Fitting Extra Credit Project 'Real-world' math problems often state things like 'The demand function for a product is 2.3 p ( e ) D = where D is how many units are sold if price is P. Have you ever wondered where specific model equations, such as these, come from? In a situation like this, data is collected on how many units are sold at different prices, creating a list of data points. Someone then finds an equation that matches the observed data. This process is called 'curve fitting'...fitting a curve (equation) to the data observed. Although curve fitting can be an advanced topic, we already have some procedures and tools that we can use to do some basic curve fitting. In this project, we'll discuss three methods we can use to find an equation model to match data: Single data-point manual curve fitting Using regression features of graphing calculators Visual curve fitting using graphing software such as GeoGebra. Each method will be discussed with examples with steps you can try. All methods share this same overall procedure: 1. Data is collected showing the relationship between two variables. Data point pairs (at least one, and often many) are obtained. 2. We use our knowledge of the general shapes of equation curves to select a possible equation. This equation has 'parameters' (constants) which can be varied to create a specific equation matching our data. 3. We use one of the three methods discussed below to determine the best parameters which make our equation match the data as closely as possible. Method 1 - Single data-point, manual curve fitting This method is most appropriate when you only have a single data point, but you are very confident that your selected equation will closely match the data. Problems such as radioactive isotope quantities (which lose mass in a well-known manner, with half the mass gone in a 'half-life') or problems involving interest earned by an investment with continuous compounding, represent situations that are well understood, and for which equations are well-known to match the situation. Consider the following example: An investment provides interest compounded continuously. Find an equation model for the balance in the account as a function of time if the balance doubles in 15 years, and $10,000 is initially invested. We don't know the annual interest rate of this account, but we do know that investments which compound rt continuously are defined by the general equation: A = Pe, where A is the Amount (balance), P is the Principal (starting balance), r is the annual interest rate and t is time invested in years. We have a single data point which relates time to amount (t,a). At t=15, A is twice the initial investment, or $20,000. So our single data point is (15, 20000).

2 In the manual method of curve fitting, we substitute the values of a data point into the general equation and solve for any missing parameters (in this case, the interest rate, r). general equation: A = Pe substituting what we know: = 10000e r(15) divide both sides by 10000: r(15) 2 = e use inverse of e (ln): ln 2 = ln e r(15) ln 2 = r(15) divide both sides by 15: ln 2 = r 15 compute r: r We can now create a general model for this investment by putting our calculated parameter, r, into the general equation: ( ) = 10000e A t t rt We've solved problems like these in our textbook. These problems are examples of the first method of curve fitting: using a well known general equation and a single data point to find a specific equation. But in some situations we are not confident of the exact nature of the equation we should use for modeling or we have multiple data points. In these cases, we need other methods to determine an equation model. Method 2 - Using regression features of graphing calculators This method is appropriate when we have multiple points of data and we are less sure which general form of equation will best fit the data. Modern graphing calculators have 'regression' features which allow us to select an equation type and allow the calculator to vary the parameters to find the constants which make the equation most closely fit the entered data. The study of how this regression is accomplished is a subject for more advanced math classes, but in this section we will learn to use this feature on our calculators for curve fitting. Suppose we have the following data for how housing prices vary as a function of square footage: Square-footage Price (thousands of dollars) A plot of the data points is shown at right. What equation model might be appropriate for this data? Linear model: the data is nearly linear, perhaps a linear model would be appropriate. Logarithmic model: the price increases more slowly as size increases which suggests perhaps a logarithmic model might even more closely fit the data.

3 We'll try each of these and let our calculator's regression features find the best fit equations. Regardless of which type of equation we want to try to fit, we first need to enter the data points into the calculator. Data points for curve fitting are entered into our calculator's Lists. Follow the procedure below to complete this example using your calculator: 1) Turn on your calculator and make sure your lists are cleared of data: Press 'STAT' Select 'Edit...' Each list is a column (L1, L2, etc.) If any list has data in it, clear it by using arrow keys to move up the column title (e.g. L1), then pressing 'CLEAR' then 'ENTER'. 2) Enter the square footage data into List 1 (L1): Use arrow keys to move cursor to first entry in the L1 column. Enter '800' then press 'ENTER'. 800 is entered, and cursor moves down in column. Enter the rest of the data points for the square footage into the L1 column. 3) Enter the price data into List 2 (L2): Scroll back up to the top of the list and over to first entry in L2. Enter '110' then press 'ENTER'. 110 is entered, and cursor moves down in column. Enter the rest of the data points for the price into the L2 column. At this point, you should have two columns of data, L1 and L2 filled in with 800 next to 110, 1000 next to 140, etc. Linear Regression Curve Fit: Now we will tell the calculator which general equation model to use and have it find the parameters for best fit using that equation. We'll start by modeling using a straight line. The best fit line is determined by 'linear regression'. The calculator has a linear regression function, called LinReg(ax+b). To use it, we need to tell it which lists to use for the x and y data values, and also where to put the resulting equation. The calculator uses one of the Y1= positions to store the result, so first we'll clear these out, then perform the linear regression: 4) Clear out y= slots: Press 'Y=' and clear out any slots currently not empty by using arrow keys to a line, then pressing 'CLEAR'. 5) Select Linear Regression: Press 'STAT' Right arrow over to the CALC menu Arrow down to 4: LinReg(ax+b) 6) Set up linear regression function: Display should read: LinReg(ax+b) Press 2nd '1' to get 'L1' (telling calculator to use L1 for x values) Press ',' (comma) key Press 2nd '2' to get 'L2' (telling calculator to use L2 for y values) Press ',' (comma) key Press 'VARS', right arrow to 'Y-VARS' menu, select 'Function...', 'ENTER', then select 'Y1' (this tells calculator to put resulting equation in Y1= slot).

4 7) Run the linear regression function: Display should read: LinReg(ax+b) L1, L2, Y1 This runs the linear regression. If all is well, the calculator should return: LinReg y=ax+b a= b= This means the calculator has determined that the line y = ( ) x is the best fit line for this data. This line is plotted along with the data point below: Logarithmic Curve Fit: The linear model above fits the data reasonably well, but it is clear that the data is not perfectly linear. Perhaps a different equation model would fit the data more closely. As square footage increases, the change in price is not exactly linear, but the increase in price is smaller as square footage is larger. We know that logarithmic curves have this trait: y increases more slowly as x increases. Perhaps a logarithmic curve would fit the data more closely than a line. We already have our data in lists L1 and L2, so we don't need to repeat those steps, but we'll repeat the steps for regression, this time using our calculators LnReg function (natural logarithmic regression): 8) Clear out y= slots: Press 'Y=' and clear out the Y1 slot from the linear regression steps by using arrow keys to Y1, then pressing 'CLEAR'. 9) Select Natural Log Regression: Press 'STAT' Right arrow over to the CALC menu Arrow down to 9: LnReg 10) Set up LnReg function: Display should read: LnReg Press 2nd '1' to get 'L1' (telling calculator to use L1 for x values) Press ',' (comma) key Press 2nd '2' to get 'L2' (telling calculator to use L2 for y values)

5 Press ',' (comma) key Press 'VARS', right arrow to 'Y-VARS' menu, select 'Function...', 'ENTER', then select 'Y1' (this tells calculator to put resulting equation in Y1= slot). 11) Run the LnReg function: Display should read: LnReg L1, L2, Y1 This runs the natural log regression. If all is well, the calculator should return: LnReg y=a+blnx a= b= This means the calculator has determined that the logarithmic curve y = ( ) + ( ) ln x is the best fit log curve for this data. This curve is plotted along with the data point below: As we suspected, this logarithmic model much more closely matches the data points than the linear model. We can also have our calculator plot the original data points as well as the best fit line it has calculated for us. To do this, we'll use the STAT PLOT feature: 12) Set the Window to an appropriate region: Press 'WINDOW' Set Xmin to 0 Set Xmax to 4000 Set Ymin to 0 Set Ymax to ) Turn on STAT PLOT Press 'Y=' Use up arrow to move up to Plot1 and highlight Plot1. Use down arrow to move down to Y1 line. The Plot1 should remain highlighted. 12) Graph using 'ZoomStat'

6 Press 'ZOOM' Select 9: ZoomStat This should plot the original data points along with the best fit natural log curve the calculator found. If you press 'STAT' and arrow to the CALC menu and look at the entries, you'll find that your calculator can curve fit to a number of different types of general equations including lines, natural logarithmic, exponential, quadratic, cubic and quartic functions, logistic equations and more. Method 3 Visual curve fitting using graphing software such as GeoGebra (This method requires installing free downloadable GeoGebra software to your computer. It is an interesting and powerful option for curve fitting, but you can complete this project without using GeoGebra using only your calculator if desired. Please read the explanation, even if you are unable to install the software and try it yourself.) Our calculator contains powerful regression functions which allow us to fit curves using many standard equation forms, although we are limited to only those general equations for which the calculator has regression functions. For unlimited curve fitting potential, we can use graphing software such as GeoGebra to visually match curves of any kind to data points. Below is a step-by-step procedure showing how to use GeoGebra for curve fitting. GeoGebra is a free, open-source program available at the website It is a very powerful math modeling program. At the website, there are free downloadable versions for many platforms including Windows, Mac and Linux, as well as Web Applets which do not require installation. The install version is quick to download and install (less than 10 minutes) and does not require registration (although it does require that your computer have the Java Runtime Environment installed.) GeoGebra allows us to enter data points which are automatically plotted on an x-y plot. We can then enter an equation of any form (the only restriction is the equation must be a function). GeoGebra has a tool called a 'slider' which allows a constant in an equation to be changed rapidly and the resulting change in the curve is immediately visible. The general procedure for using GeoGebra for curve fitting is: 1) Configure GeoGebra to display an x-y plot with axes. 2) Enter individual data points into GeoGebra, which GeoGebra displays on the plot. 3) Select a possible model equation. 4) Create 'sliders' for all parameters (constants) in the equation. 5) Enter the model equation using the sliders for the constants. GeoGebra automatically plots the equation. 6) Move the sliders and see how the curve changes, adjusting until the curve visually matches the data points. 7) Read the positions of the sliders to find the specific modeling equation.

7 For a specific example, let's find an equation to model the following data: x y This data can be plotted as follows: If we search in libraries of functions, we might find that the curves of this sort can be generated from polynomials with higher order terms such as: y = ax + bx + cx + dx + ex + f. The following procedure will allow us to find such an equation using GeoGebra which will model this data. 1) Open GeoGebra and display the x-y grid: Click 'View', 'Axes' to turn on x and y axes. Click 'View', 'Grid' to turn on grid. 2) Enter data points: In the Input bar at the bottom of the GeoGebra window, enter a data point as a coordinate (x,y) pair. Include the parentheses. The first data point is entered as (0,1.6). Press 'Enter'. GeoGebra will add this point on the x-y plot.

8 Enter the other data points from the table above. 3) Hide the letter labels for the data points: Click on the 'select arrow' button on the menu bar at upper left and drag a box around all of the data points on the grid. Position the cursor over the box just drawn and right-click. Under the BASIC tab, uncheck the 'Show Label' box and click Close in lower right. 4) Scale, move the display for good view of the points: Click the right-most tool bar button (looks like an x-y axes). Using this tool, you can click on the origin and move the center around the display space. Click the small pull-down arrow at lower right of the 'move' button to pull down a menu which allows you to select zoom in, zoom out buttons. Use the zoom in/out and move buttons to arrange the display as needed for good display of the data points. 5) Add a 'slider' for each constant in our general equation: Click on the button 2nd from right in toolbar to select the 'slider' tool. Click anywhere on the grid area (suggest upper right). Click 'apply' to create a slider called a. Repeat this process to create sliders for constants, b, c, d, e and f. 6) Enter our general modeling equation using the slider parameters: Next, we want to enter a model equation in that uses these a,b,c, etc. slider values as changeable parameters. Let's try to model this data using the polynomial ax + bx + cx + dx + ex + f In the Input box at the bottom, enter the following equation: y=a*x^5+b*x^4+c*x^3+d*x^2+e*x+f GeoGebra should display an initial equation with all constants set to 1 (the starting values of the sliders.) 7) Drag the slider positions and watch the effect on the plotted curve. Adjust until the curve closely matches the data points. This is a trial and error process. Try setting all but one slider to zero and seeing what effect each slider has. Try adjusting one slider for best match, then next slider for best improvement, etc. and repeating with first slider...'narrowing in' to a solution. If nothing is working, try moving one of the larger effect sliders (a or b in this case) a large amount and then adjust the other sliders to compensate back to a good match. If a slider ends up near zero, try setting it to exactly zero. This will simplify the final model. 8) When you have a close enough match, click on the 'select' arrow (left icon in the tool bar at top) and click on any slider to change its value. The model curve changes as you change the value. Use the sliders to try and manually adjust the curve to match the data points. In this case, one possible model has sliders set to: a= -3, b=1.2, c=1.9, d=0, e= -0.2, f=1.6 This suggests a possible modeling equation is: y = x + x + x x Although this process is trial and error, it has the flexibility to try any equation conceivable. Remember that the more sliders (constants) are included in your model, the more difficult it is to visually find a good matching solution.

9 The Project: Find a set of data that interests you and use the curve fitting techniques above to find an equation that models your data. Step 1: Locate a set of data First, you need to identify a set of data to model with an equation. Your data will need to be a comparison of two numerical quantities, (an 'x' and a 'y'), and it can be anything that has some sort of relationship so that if you graph it, is has a reasonably well-behaved curve. Data Set Requirements: - Your data set must include two variables/quantities. - Your data set must include enough points to see a reasonable 'shape' when plotted. Here are some ideas of the kinds of things you could explore: - Financial: Price of gold vs. time, cost of gasoline vs. year, etc. - Weather: Average daily temperature vs. latitude, Number of hurricanes vs. ocean temperature, etc. - Social Issues: Percentage unemployment vs. age of worker, Rates of domestic violence vs. unemployment rate, etc. - Health/Medical: Diabetes vs. obesity, Retired persons' average cost of medicines vs. year. How to find your data: - Internet: use a search engine to begin exploring. One idea is to do a google image search for something like 'interest rate graph'. If you find graphs, you can take data points from the graphs, or find the websites that contain the graphs...they often have interesting data. - Collect some data yourself: maybe there is a shadow in your room on the floor caused by the sun shining in through a window, and it moves across the floor throughout the day. On a weekend, you could measure the distance the edge of the shadow is from the wall every 30 minutes so you have data of the distance vs. time. - Examples from mrfelling website: There is a link on this project's page to a page with some example datasets that you can use if you are having trouble finding your own data. You can try to curve fit any data, so find something you are interested in exploring! Step 2: Use the regression features of your calculator, or GeoGebra, or both, to find a model equation that closely fits your data set. Use the techniques in this project description to create a model equation. Step 3: Write up your results in a report to hand in. Report Requirements: 8 1/2x11" paper report (no poster/display required) must include: 1) A description of how you found or collected your data. 2) A table showing your data. 3) Your model equation (complete equation, including the parameters filled in with the constants you found that makes the equation fit the data.) 4)An x-y plot showing your data points as well as your equation showing that it fits your data. Typed/computer rather than hand-written reports are preferred. Project Grading: - Full credit: all requirements met fully, data set contains many points and shows a relationship, curve fitting procedures produced an equation that closely matches your data set. - 2/3 credit: basic requirements met, but data set is 'sparse' (not very complete or seems made up), and/or curve fitting does not match data well, or graph is inaccurate. - 1/3 credit: not all requirements met or there is little evidence that curve fitting procedures were used to create a model equation.