Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
The Definition
In each of the preceding exercises, you dealt with polynomial functions. On sheet 3, we dealt with
power functions, a special type of polynomial. On sheets 1 and 2, we dealt with quadratic functions.
We want to consider polynomials in general. Therefore, it is time that we formally define them.
A POLYNOMIAL FUNCTION
A polynomial function, in the variable x, may be written as:
p( x)  an x n  an 1 x n 1  an  2 x n  2   a2 x 2  a1 x1  a0 x 0 , with an  0 ,
where n is a non-negative integer (called the degree of the polynomial) and each
coefficient, ai is a real number for i = 0, 1, ..., n. Otherwise, the polynomial may
be equal to 0, and in this case, we say the degree is undefined.
Note that this function contains only the operations of addition, subtraction, and
multiplication (integer exponents are considered repeated multiplication).
We want to examine the graph of a polynomial function as more and more factors are multiplied
together. Use your calculator and sketch a quick graph of each of the following functions. Use a
meaningful window for each. Compress vertically to fit in the graph. Give some sense of the yscale, but precision isn't necessary. However, do mark all intercepts carefully.
y1 = (x – 1)
y2 = (x – 1)(x + 2)
5
y3 = (x – 1)(x + 2)(x – 3)
5
Poly 4.1
5
Rev. S11
Mathematical Investigations III
Name:
y4 = (x – 1)(x + 2)(x – 3)(x + 4)
y5 = (x – 1)(x + 2)(x – 3)(x + 4)(x – 5)
y6 = (x – 1)(x + 2)(x – 3)(x + 4)(x – 5)(x + 6)
5
5
5
Write a few sentences describing what you observed about the sequence of functions you just
graphed.
What is happening to the degree of the polynomial as you progressed from y1 to y6 ?
How does the degree affect the graph? Make at least three observations.
Poly 4.2
Rev. S11