Finding the Second Differences of a Set of Data

 x 1 3 5 7 9 11 13 y -1 7 23 47 79 119 167

When the x-differences are all constant, then the second differences are useful in telling if a set of data is nearly quadratic or not.

First enter the data by pressing STAT, select 1:EDIT by pressing ENTER. Enter the x values in L1, and the y values in L2 as shown below.

To find the second y-differences, we first find the first differences or y-differences, then the second differences are the differences of the first differences or y-differences.

With the cursor in the L3 column, press the UP ARROW so that L3 is highlighted. Press 2nd LIST[STAT] and RIGHT ARROW to OPS menu.

From the OPS menu select 7: by pressing a 7. now appears at the bottom of the screen beside L3=. Press 2nd L2[2] to type L2, and then press the ) key. Press ENTER and the y-differences will appear in the L3 column.

Examining the y-differences in L3 column, we see that the are not only are the differences not equal, but the are ever increasing in differences. So now we will computer the second differences, that is, the differences of the y-differences. To do this, press the RIGHT ARROW key to move the cursor to column L4, and press the UP ARROW once so the L4 is highlighted. Now repeating some of the steps above, Press 2nd LIST[STAT] and RIGHT ARROW to OPS menu. From the OPS menu select 7: by pressing a 7. now appears at the bottom of the screen beside L4=. Press 2nd L3[3] to type L3, and then press the ) key. Press ENTER and the second differences will appear in the L4 column.

Since the second differences all equal 8, we know that our data is exactly quadratic.

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