Understanding polynomial graphs can be tough, especially when their shapes change with different degrees.

1. Linear Functions (Degree 1):
   The easiest polynomial is a linear function, which looks like this: $f(x) = mx + c$. Its graph is just a straight line. But, many students get mixed up trying to understand the slope ($m$) and where it crosses the y-axis ($c$). Since it doesn't bend at all, it can feel easier than it really is.

2. Quadratic Functions (Degree 2):
   Quadratic polynomials, shown as $f(x) = ax^2 + bx + c$, are more complicated. Their graphs make a U-shape. Figuring out the highest or lowest point (called the vertex) and the line that splits it in half (axis of symmetry) can be confusing. Also, figuring out if the graph opens up or down based on the value of $a$ adds to the challenge.

3. Cubic Functions (Degree 3):
   Cubic polynomials, explained by $f(x) = ax^3 + bx^2 + cx + d$, are even harder. They can have one or two points where the direction changes, and they might cross the x-axis up to three times. Understanding what happens at twisty points (inflection points) and trying to guess where the graph peaks or dips can be frustrating.

4. Higher-Degree Polynomials:
   When we look at quartic polynomials ($f(x) = ax^4 + \ldots$) and others, things get even trickier. These graphs can sway wildly. This makes it hard to predict how they behave at the ends and how many times they touch the x-axis. Students often find it difficult to draw these graphs without using some tech tools.

To make these tricky polynomial graphs easier, students can use graphing software. Visual tools can help make things clearer. Plus, breaking down polynomials into their key features, like how they behave at the ends, where they touch the axes, and turning points can make learning simpler and help improve guesses about graph shapes.