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?

YearNumber of Widgets

0 150

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

• The slopes are not the same It is not linear

• We will check to see if it isexponential 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?

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

.14337 5.019y x

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

ln( ) .14337 5.019w x

Figuring the Original Equation

• Solving for w

• Now we end up with an exponential function

• Now graph the original points (x, widgets)

ln( ) 0.14337 5.019

5.019 0.14337

0.14337

ln( ) .14337 5.019

151.26

w x

x

x

w x

e e

w e e

w e

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 Steps1. List the ordered pairs in adjacent columns of

the data matrix 2. In a third column have the calculator place  

ln(  )  of the y values 3. Plot (x, ln(y)) and note that it is a line 4. 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

5. This gives us  ln(y) = m*x + b6. Solve the above function for y7. It will be in the form y = A * eB*x

8. This is the exponential function which models the original set of points (x, y)

Assignment

• Lesson 4.4B

• Page 183

• Exercises 18 – 21 all