Graph analysis can help us understand polynomial functions better, but it isn't always easy. Here are some challenges we might face:

1. Spotting Important Features:

   • Graphs can sometimes trick us. For example, there are tricky behaviors around points where the graph crosses axes and at certain lines we call asymptotes.
   • It can also be hard to tell the difference between the highest and lowest points in a small area compared to the overall graph.

2. Understanding the Ends:

   • Figuring out how a graph behaves at both ends (what we call "end behavior") means we have to look at leading coefficients. If we get this wrong, we might think the function is growing or shrinking differently than it actually is.
   • With higher-degree polynomials, the graph can wobble unexpectedly, making it tougher to guess what will happen.

3. Doing It by Hand:

   • Drawing graphs by hand takes a lot of time and can lead to mistakes.
   • Finding important points on the graph often needs calculus to be exact, which might scare some students away.

Even with these challenges, we can use tools like graphing calculators or software to help us see polynomial functions clearly. These tools can make understanding these tricky parts a lot easier. Plus, if we focus on a step-by-step approach using tests and limits, we can gain a much clearer picture of how polynomials behave.