b). x 0 1 2 3 4 y 16 4 1 1/4 1/16

Notice that as each x increases by 1, each y does NOT decrease by a constant amount. Thus, the data cannot be modeled by a linear function. Instead of a constant amount being added or subtracted, note that each y-value is times the previous y-value. That is, the ratio of consecutive y-values is . So the appropriate model must be exponential.

There are two ways to obtain an exponential model for this data: symbolically and by the calculator using exponential regression.

1). Finding the model symbolically.

The general form of an exponential is . Recall that C is the value of the function when x = 0.
Since the data point (0, 16) is given, C = 16. Since each y-value is times the previous value, this means that a = .

Therefore, the exponential function which models the data is .

2). The equation of the curve can be determined using the calculator exponention regression capability as follows:
Press STAT, then select EDIT. Enter the x values in and the y values in . Check the menu to be sure that there are no functions listed and Plot1 is on (highlighted). Press ZOOM and select 6 to graph the scatterplot in the standard window [-10,10,1] by [-10,10,1]. Note that the data points do not appear to lie in a straight line.

 

Next press STAT, ARROW RIGHT to CALC, the select 0:ExpReg by arrowing down and pressing ENTER. Now press VARS followed by RIGHT ARROW. Then press ENTER three times. Press GRAPH to see the graph, and Y= key to see the equation.

The results show that the equation form is y= a* b^x (same as ), where a=16 and b=.25 or . Thus the result is y = 16*(.25)^x which is identical to . To verify the results visually, plot the scatterplot of the data and simultaneously as shown below.

Therefore, the exponential function which models the data is .