Algebra

# Polynomial Graphs and their Functions: Part 1

In my Algebra II class today, there were only 5 students, so my teacher decided to do a little investigation on polynomial functions and graphs. I found some rather interesting things which I thought I might share about factors and how they affect graphs, and vice versa.

1. Quartic graphs sometimes look like quadratic graphs, and quintics sometimes look like cubics. The way we (as students, at least) often visualize quartic graphs is something like this:

And many quartics do look like this. However, quartics can take on other shapes as well. For instance, consider this extremely simple example:

$y=x^4$

All the difference we can tell between this graph and that of $y=x^2$ is that the lower portion is slightly flatter, and the upper portion slightly steeper. This kind of graph doesn’t just occur in graphs of the form $y=ax^4+b$. And a similar effect occurs with quintics and cubics. So how does this happen?

Well, let’s do a little symbol magic with the equation of that last graph. Knowing the rules of exponents, we can safely say that

$y=(x^2)^2$.

Look at that, we have a quadratic! The only difference is that in this case, the x is squared twice. This appears to cause the flatness and steepness — it’s actually quite logical. If we think about squaring, we can tell that squaring acts on two groups of numbers in two different ways:

Case 1: If the (absolute value of the) number is greater than 1, then its square is greater than itself. For example, the square of 5 is greater than 5: 25 >; 5.

Case 2: If the (absolute value of the) number is less than 1, then its square is less than itself. For example, the square of 0.5 is less than 0.5: 0.25 <; 0.5.

(Of course, if the number is 1, then its square is 1.)

So the steeper portion is Case 1, where the double squaring makes the number even greater. But the flatter portion is Case 2, where the double squaring makes the number even smaller.

Next time, I’ll discuss the next part of the investigation: other types of “combined functions” and their factors.