Now that we've had a look at the powers of
x, we can start to think about polynomials. First, let's say what
a polynomial is: in words, it is a function that is built by simply
adding together some power functions. For example,
More generally, a polynomial can be written as
The highest power that occurs in the polynomial, in this case
, is called the degree of the polynomial. Degree 2
polynomials are usually called quadratic polynomials and should be quite
familiar; for instance, their graphs are parabolas.
in front of the powers are called
Why are polynomials interesting?
Let's think for a minute about why polynomials are interesting
functions. First of all, they can be computed using only the simplest
of operations: multiplication and addition. This will not be the case
with all the functions we will meet in this course. Of course, in
this age of calculators and computers, that does not seem like as much
of an advantage as it did a hundred years ago.
However, there is another reason why polynomials are interesting
for us which can be explained by an analogy. Animal breeders often
breed a new animal from two different parent so that the
offspring inherits some desirable features from its parents. In the
50s, scientists successfully bred a "liger," an animal whose mother
was a lion and father a tiger. When you saw the liger, you could see
that some parts of it looked like a lion while other parts looked like
Polynomials are somewhat like the liger. You see, polynomials are
built ("bred" if you like) from power functions which, as we have seen, are
relatively simple functions. When you look at a polynomial, you can
still see traces of the power functions which went into its
That's a lot of words! Let's try to illustrate what we mean with
an example. Below is the graph of one of our power functions
. By moving the box around, we can create different
polynomials by adding a quadratic term. In terms of our analogy,
we're building a new animal by adding varying amounts of a tiger to a lion.
Play around with this and
observe what happens.
For your consideration
- What features of the graph remain the same as you vary the
contribution of the quadratic term of the polynomial?
- What features change?
- Describe the behaviour of the graph at the origin as you vary
the quadratic term.
- Describe the behaviour of the graph far away from the origin
as you vary the quadratic term.
- Why do some of the polynomials have "bumps" but not others.
- Remember our observation about power
How can you use this observation
to explain your answers to the questions above?
- Near the origin, power functions with smaller exponents
grow more rapidly.
- Far away from the origin, power functions with
larger exponents grow more rapidly.
Here is another example to consider. You should think about the same