Subsection 3.4.1 Representing all quadratics
One reason that the quadratic family is so important is that the dynamics of every quadratic is captured by the behavior. This is made precise in Checkpoint 3.4.1
Checkpoint 3.4.1
Let \(g(z) = \alpha z^2 + \beta z + \gamma\) and let \(\phi(z) = a z + b\text{.}\) Show that \(f_c\circ\varphi = \varphi\circ g\) when \(a=\alpha\text{,}\) \(b=\beta/2\text{,}\) and
\begin{equation*}
c = \frac{1}{4} \left(4 \alpha \gamma -\beta ^2+2 \beta \right).
\end{equation*}
That is, \(\varphi\) conjugates \(f_c\) to \(g\text{.}\)
Checkpoint 3.4.2
Let \(g(z) = (1+i)z^2 - z + i/4\text{.}\) Find values of \(a\text{,}\) \(b\text{,}\) and \(c\) such that \(\varphi(z)=az+b\) conjugates \(f_c\) to \(g\text{.}\) Use the computer to generate images of the Julia sets of both \(f_c\) and \(g\text{.}\)