When would a graphing calculator make an error while plotting a function?

Question

It is possible for a graphing utility/calculator to plot a function incorrectly...

What are some possible errors that may occur?

What will cause those errors?

Answer 1

You can construct a numerically unstable function like this:

y = (sqrt(x² + 9) - 3) / x²

which then goes wrong if you try and plot it near zero using regular floating point arithmetic, eg

http://www.wolframalpha.com/input/?i=y+%3D+%28sqrt%28x%C2%B2+%2B+9%29+-+3%29+%2F+x%C2%B2+from+-0.001+to+0.001

Google "catastrophic cancellation" for more information

Here's another good example https://www.wolframalpha.com/input/?i=plot+2sqrt%7Be%5E%7Bx%7D%2B1%7D-ln%28sqrt%7Be%5E%7Bx%7D%2B1%7D-1%29-ln%28sqrt%7Be%5E%7Bx%7D%2B1%7D%2B1%29+from+-40+to+-30

Answer 2

One example is the following: $$ \sqrt{x-y} = 0 $$ Any point $(\alpha, \alpha)$ should be lying on this curve $\forall ~\alpha \in \mathbb{R}$ , however, when you try to plot it on Desmos, GeoGebra, or Wolfram|Alpha, the graph shows no real points for some reason.