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P. Thompson

MTED 2800/3900

Fall 2004

Graphing Calculator

Functions and Cartesian Graphs

In each of the following, recall that a good explanation is one that provides a strong sense of why things work as they do.

 It will be useful to think of (a) and (b) as sums of functions when trying to describe their behavior and to explain why they behave as they do.
1.          Explain the behavior of the families of functions in (a) and (b) so that an explanation of why the functions in (a) behave as the do for varying values of n is the basis for why the functions in (b) behave as they do for varying values of n.
1. f(x) = x2 + nx
2. g(x) = x3 + nx

2.         We normally think that b and a in "a mod b" stand for whole numbers. 27 mod 3 is 0, because 27 ÷ 3 has remainder 0. 27 mod 5 is 2, because 27 ÷ 5 has remainder 2. But we can think generalize this idea to fractions and irrational numbers, too. The definition of "b mod a" that does this is:

(a mod b) is the remainder obtained when subtracting mb from a, where m is the largest integer less than or equal to .

By this definition, (6.5 mod 2.1) = 0.2, since 3 is the greatest integer less than or equal to , and 6.5 - (3)(2.1) = 0.2. Similarly, (6.5 mod -2.1) = -1.9 because –4 is the largest integer less than or equal to  and 6.5 – (-4)(-2.1) = -1.9 (you should determine this for yourself).

Given that a mod b is defined as above, do this for each function in the following list. (a) describe its behavior; (b) explain its behavior.

3.         Create a polynomial function that crosses the x-axis 7 times in the interval (-5,5) and does not cross the x-axis anywhere else.

4.         It is often said that since, for example, , x3 + 3x2 begins to behave like x3 for large x. When John graphed the two functions, however, he concluded that this couldn't be correct, since the distance between the two graphs "goes to infinity."

a.   Does John have a point?

b. Explain what is going on in a way that removes the paradox.

5. a.    Examine the graphs of y = 2x and y = x100. Which one grows faster?

b.    Show all points where the graphs of y = 2x and y = x100 intersect.

c. What might students be surprised by (or were you surprised by) when considering (a) and (b) together?

6.         John's buddy, Jill, was going to graph this system of equations:

y = 2 - x

y = 2x + 4

y = 20 – x2.

She exclaimed: "This can't be right. y can't be all those things. It can only be one thing." Please discuss what might be bothering Jill and how you can propose to think about this system to avoid Jill's concern.

You may enter function definitions into graphing calculator as y = (some expression). That is, you needn't use function notation. Select "New Math Expression" in the Math menu, or hold down the CTRL key (Windows) or the  key (Mac) while pressing "M", to get a new equation.

Another common definition for (a mod b) is "the smallest non-negative remainder obtainable when subtracting an integral multiple of b from a." This may be expressed symbolically as: For a, b e Reals, .

 Overview Schedule of Assignments Honor Code