F1 analyse, determine, and apply scatter plots and determine the equations for curves of best fit, using appropriate technology

 Origionators: Alec, Drew

When you are given a set of data, it can be arranged in a table of values. The set of numbers in the "x" column are the independant values, and the set of numbers in the "y" column are the dependant values.

The data in the table should then be analyzed to determine whether the data will form an exponential function. If the sequence of numbers is a geometric sequence, then it will have a common ratio. The common ratio can be found by dividing a number in the "y" column by the number that comes before it (Ex. y2/y1). Perform the same step for two other numbers in the set of data that are in sequence, and if the quotient equals the quotient that was gathered from the other two numbers, you have found the common ratio.

The equation of the function can then easily be determined. The formula for an exponential equation is:

y=a*B^(b(x+c)) + d  where a=the starting value (y-intercept)

                                           b=the common ration

                                           c=the unit of change in the x-values

                                           d=the horizontal asymptote

*These values can be found by using exponetial regression on the calculator

The data can then be arranged in a scatter plot graph. This can be done by hand, or on the calculator. The independant values should be plotted on the x-axis, and the dependant on the y-axis. The points should then be connected, or the line of best fit should be determined. The line of best fit on a scatter plot graph should start at the y-intercept (a-value) and have points evenly disrributed on either side of the line to be as accurate as possible. The line of best fit can be drawn manually, or a much easier and most likely more accurate method would be to use your calculator.

Example:

(y2/y1)=(2/1)=2                 Equation:     y=a*B^(b(x+c)) + d                

(y3/y2)=(4/2)=2                                      a=1

(y4/y3)=(8/4)=2                                      b=2

Common ratio=2  (r=2)                        c,d=none

                                                                y=2^x

Graph:

 Example:

{1,3,9,27,81...}

(y2/y1)=(3/1)=3

(y3/y2)=(9/3)=3                           

(y4/y3)=(27/9)=3

(y5/y4)=(81/27)=3

Common Ratio=3

y=a*B^(b(x+c)) + d 

a=1  

b=3

c,d=none

y=1*3^x

y=3^x

Then the points should be plotted on a scatter plot graph by hand, or using a graphing calculator or other graphing technology. The graph should look something like this: