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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. |
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, , x^{3} + 3x^{2} begins to behave like x^{3} 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 = 2^{x} and y = x^{100}. Which one grows faster?
b. Show all points where the graphs of y = 2^{x} and y = x^{100} 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 – x^{2}.
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, .
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