Algebra

Polynomial Graphs and their Functions: Part 2

Last time, I discussed how quartics can sometimes look like quadratics and why this happens. Today let’s talk about another case of nested exponents that looks confusing at first. Knowing that $(x^2)^2$ has the appearance of a quadratic, what happens when you square a number, then cube it?

1. $y=(x^2)^3$

And for that matter, what’s the difference between this equation and this one:

2. $y=(x^3)^2$?

Well, logic tells us that both equations will have the same graph because both simplify to $y=x^6$. And indeed, their graphs are the same:

But last time we found that if we squared outside the parentheses, it would like a quadratic. In equation 1, though, the outer part is cubed. How can it still have a parabolic shape? How do we unite the evidence from Part 1 and this new information?

To fully explain it, we need to know about end behavior. End behavior is quite simply the direction in which the two ends of a curve go. For instance, the ends of the graph of $y=x^4-5x^2+4$ both go up:

In contrast, the left end of the graph of $y=x^3-3x^2+3$ goes down, and the right end goes up:

What makes a quartic equation’s ends go in the same direction, and those of a cubic go in different directions? To ask this is to ask the difference between taking a number to the fourth power and to the third power. The answer is basically that with taking a real number to any even power, it is impossible to get a negative number (the negative pairs cancel); however, odd exponents can be negative numbers. This is why in a cubic function, one side goes in the negative direction.

So let’s look at the two equations we had before: $y=(x^2)^3$ and $y=(x^3)^2$. Why are the end behaviors for these two equations the same?

Well, the $x^2$ in equation 1 will always be positive, creating two ends going in the same direction. So the number that is cubed is always positive, and of course if a positive number is cubed the answer is positive as well. The cubing only serves to steepen and flatten the graph as we saw in Part 1; it does not change the end behavior of the graph.

In equation 2, the $x^3$ can be either positive or negative, creating two ends going in opposite directions. However, this quantity is then squared. Now whatever negative numbers that were inside the parentheses have become positive. The squaring does steepen and flatten the graph like in equation 2, but it does change the end behavior of the graph. Now both ends go in the same direction.

Next time, we’ll find out how end behavior and factors are related, and how you can look at a graph of a polynomial and write its equation!