The graph of a family of functions over quadratic extensions of finite fields

Brochero and Teixeira (2023) [4] showed the behavior of the functional graph of $f_a\left(X\right)=X^{q+1}+a{X}^2$ over quadratic extensions of finite fields explicitly for $a\in \{1,-1\}$. In this article, we create a family of functions using repeated iterations of the function $f_a\left(X\right)$ and taking values of $a\in \{1,-1\}$ in each iteration. Let α be an n-sequence of values for a, taken over $\{1,-1\}$, and $f_{\alpha }\left(X\right)$ be the resulting function. We present the form of $f_{\alpha }\left(X\right)$ and use it to derive a closed formula for the number and length of cycles present in the functional graph of $f_{\alpha }\left(X\right)$. We then determine the shape of the trees hanging from each cycle and gather all the results in our main theorem.