Powers of x
Powers of x: fractional exponents
UBC Calculus Online Course Notes

Polynomials

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,

\[  f(x) = -2x^4 -x^2 + 3x + 1. 
 \]

More generally, a polynomial can be written as

\[  f(x) = a_nx^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0. 
 \]

The highest power that occurs in the polynomial, in this case $  n 
 $ , 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. The numbers $ a_i $ in front of the powers are called coefficients.



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 a tiger.

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 construction.


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 $  y = 
x^4  $ . 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

  1. What features of the graph remain the same as you vary the contribution of the quadratic term of the polynomial?

  2. What features change?

  3. Describe the behaviour of the graph at the origin as you vary the quadratic term.

  4. Describe the behaviour of the graph far away from the origin as you vary the quadratic term.

  5. Why do some of the polynomials have "bumps" but not others.

  6. Remember our observation about power functions:
    • Near the origin, power functions with smaller exponents grow more rapidly.
    • Far away from the origin, power functions with larger exponents grow more rapidly.
    How can you use this observation to explain your answers to the questions above?

Here is another example to consider. You should think about the same questions.


Local Extrema

In the demonstrations above, you should have noticed that you could create "bumps" in the graphs by varying the polynomial. These bumps actually represent vitally important information about the function. For instance, if you look at the polynomial $  f(x) = 
x^3 - 0.5 x  $ , you will see that there are two bumps. Let's consider the leftmost bump and call that point $  (x_0,y_0)  $ where, of course, $  y_0 = f(x_0) = x_0^3 - 0.5 x_0  $ .


At the top of the bump, $  y_0  $ is larger than the y-coordinate of all neighbouring points on the graph. This means that the value of the function $  f(x_0)  $ is larger than $  f(x)  $ for all points $  x  $ nearby $  x_0 
 $ . We will call $  x_0  $ a local maximum for the function $  f  $ and we will call $  y_0  $ a local maximum value. Of course, the function does eventually become larger than $  y_0  $ , but not until we move far away from the bump.

In the same way, the rightmost bump of this function represents a local minimum . A point that is either a local maximum or minimum is sometimes called a local extremum for convenience.

Now let's consider the polynomial $  f(x) = x^4 - 0.5 x^2 
 $ above. Here you see that there are three local extrema. In addition, the graph never dips below the level of the two local minima; that is, the value of the function is never smaller than the two local minimum values. (Why are these the same?) In this case, we say that these points are absolute minima.

A point being a local maximum of a function is a bit like you being the best tennis player on your floor in residence. While you might not be as good as Martina Hingis or Pete Sampras (they are the absolute maxima), your accomplishment is certainly worthy of note and we would like to seek you out to honour you. In this same way, we will soon learn some powerful techniques for seeking out the local extrema of functions, for these points are often of great interest.