Fitting Curves to Data Lesson 4.4B

Using the Data Matrix Consider the table of data below. It is the number of Widgets sold per year by Snidly Fizbane's Widget Works (with 1990) being year zero. Place these numbers in the data matrix of your calculator. Plot the points. We seek the function which models this data. This will enable Snidly to project sales for the immediate future and set budgets What type of function does it appear to be? Year Number of Widgets 150 1 175 2 207 3 235 4 260 5 300 6 370

What Kind of Function? It might be linear But it appears to have somewhat of a curve to it Check the successive slopes – place cursor at top of column c3 Enter expression

What Kind of Function? The slopes are not the same It is not linear We will check to see if it is exponential In column 4 have the calculator determine 1.*ln(c2) The text calls this “transforming” the data Enabling us to determine if we have an exponential function

What Kind of Function? Now see if the ln values are equally spaced If graphed would they be linear? Use the c4 – shift(c4) function in column 5 You should find that they are not exactly equal but they are quite close to each other

Plotting the New Data Specify columns to be plotted x values from column 1 y values from column 4 (these are the ln(c2) values) This should appear to be much closer to a straight line

Plotting the New Data Now use the linear regression feature of your calculator Determine the equation of the line for these points x values come from column 1, y values from column 4

Figuring the Original Equation Column 2 had w the number of widgets We took ln(w) to get the y values we plotted That means we have Now we need to solve the equation to solve the above equation for w Hint … raise e to both sides of the equation

Figuring the Original Equation Solving for w Now we end up with an exponential function Now graph the original points (x, widgets)

Figuring the Original Equation The results would be something like this Actually, our calculator could have taken the original points and used exponential regression

Exponential Regression Note the option on the regression menu Check for accuracy

Summary of Steps List the ordered pairs in adjacent columns of the data matrix In a third column have the calculator place   ln(  )  of the y values Plot (x, ln(y)) and note that it is a line Use linear regression with the x column and the ln(y) column. The text may suggest draw a line by eye and determine the equation manually This gives us  ln(y) = m*x + b Solve the above function for y It will be in the form y = A * eB*x This is the exponential function which models the original set of points (x, y)

Assignment Lesson 4.4B Page 183 Exercises 18 – 21 all