When students graph functions on a Cartesian plane, they often make several common mistakes. These mistakes can make it harder to understand how functions work. It’s important to know about these errors so that everyone can create clear and correct graphs.

One big mistake is not labeling the axes correctly. The $x$-axis and $y$-axis need to be clearly labeled. If you forget to do this, it can be confusing to know what the graph is showing. For example, if $y$ is the output and $x$ is the input, you should label them as "Input ($x$)" and "Output ($y$)" to make it clear.

Another common error happens when choosing the scale for the axes. If the scale is inconsistent or not suitable, it can really change the look of the graph. It's important to pick a scale that clearly shows the function over its entire range. For example, when graphing a quadratic function, picking a range that shows its peak point and where it crosses the axes helps a lot. If the intervals are too small or too large, it might misrepresent how the function behaves.

Sometimes, students plot points incorrectly. They might be in a hurry and end up miscalculating or placing points wrong. To avoid this, it helps to calculate important points first, like where the graph crosses the axes and any high or low points. Once you have these points, plot them carefully and connect them with a smooth line if the function is continuous. This makes the graph more accurate.

Also, not thinking about the function's domain and range can lead to big mistakes. Before graphing, students should understand the domain—what $x$ values are allowed—and the range—what $y$ values can come from those $x$ values. If students draw the graph outside this defined area or forget to show possible $y$ values, it can completely misrepresent the function. Some functions might not even have values at specific points, and those should be indicated on the graph.

Another important mistake is misunderstanding a function's behavior at infinity. When graphing polynomial or rational functions, it’s crucial to look at how the graph behaves as $x$ goes very high or very low. If students ignore this, they might guess wrong about how the graph looks far away from the starting point.

Not paying attention to asymptotes can also distort how the graph is viewed. For rational functions, knowing vertical and horizontal asymptotes is really important. Asymptotes show where the function doesn't exist or where it approaches a certain value. This helps explain how the function behaves overall.

Finally, not connecting the graph to its algebraic form can mean missing out on important information. While plotting points is necessary, understanding how changes in the equation affect the graph is just as important. For example, knowing how the numbers in the equation change the graph’s steepness and direction helps with understanding its important features.

In conclusion, to create accurate graphs, avoid these common mistakes: label the axes properly, choose the right scale, plot points carefully, define the domain and range, understand end behavior, note asymptotes, and link algebra with graphing. By mastering these basics, it will be much easier to explore more complex math ideas later on.