# ASSIGNMENT # 12

Problem # 1

This assignment deals the use of spreadsheets in order to explore various topics in mathematics. Indeed, a spreadsheet has the potential for being a very rich tool for teachers of middle school and high school. The concept of function is typically introduced in an Algebra I course, and students construct graphs of these functions by creating an x-y table (sometimes referred to as a t-table) and plotting points. A spreadsheet is nice in that students may create extensive x-y charts, plot these points and draw the graph, and then use the graph feature on the spreadsheet program to verify their results. We may use EXCEL to input values of x and then generate the corresponding values of y. Furthermore, EXCEL will generate a graph.

Consider the function

 -20 -8000 -19 -6859 -18 -5832 -17 -4913 -16 -4096 -15 -3375 -14 -2744 -13 -2197 -12 -1728 -11 -1331 -10 -1000 -9 -729 -8 -512 -7 -343 -6 -216 -5 -125 -4 -64 -3 -27 -2 -8 -1 -1 0 0 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000 11 1331 12 1728 13 2197 14 2744 15 3375 16 4096 17 4913 18 5832 19 6859 20 8000

In the table above, the left-hand column contains the values of x and the right-hand column contains the corresponding values of y. These y-values are generated by taking each value of x and cubing it. Of course, this list is not complete since it contains only the integers in the interval [-20, 20]; however, it will suffice for the graph.

This graph should provide the students with a sense of what a graph of the cubic function resembles. Moreover, there are many "gaps" between the points and this could lead into a discussion about the domain and the range not being limited to just the integers.

Now consider a smaller domain, say [-5, 5], and each element in the domain has a common difference of .25. This will incorporate some values of x other than just the integers.

 -5 -125 -4.75 -107.172 -4.5 -91.125 -4.25 -76.7656 -4 -64 -3.75 -52.7344 -3.5 -42.875 -3.25 -34.3281 -3 -27 -2.75 -20.7969 -2.5 -15.625 -2.25 -11.3906 -2 -8 -1.75 -5.35938 -1.5 -3.375 -1.25 -1.95312 -1 -1 -0.75 -0.421875 -0.5 -0.125 -0.25 -0.015625 0 0 0.25 0.015625 0.5 0.125 0.75 0.421875 1 1 1.25 1.95312 1.5 3.375 1.75 5.35938 2 8 2.25 11.3906 2.5 15.625 2.75 20.7969 3 27 3.25 34.3281 3.5 42.875 3.75 52.7344 4 64 4.25 76.7656 4.5 91.125 4.75 107.172 5 125

Hopefully, the graph of this data will appear to be more continuous.

Here, we are able to see that the graph of the function has the same general shape, and it appears to be more continuous than the first graph of the cubic function.

CONCLUSIONS

Spreadsheets are nice tools for experimenting with graphing polynomial functions. It reinforces the idea that the equation of a function simply provides a rule, but the actual function is the set of points generated by the rule. Moreover, students tend to choose some of the "easier" values (meaning small integers) of x to create a t-table, whereas a spreadsheet affords them the opportunity to explore rational numbers--large and small.