How to derive the transmitted signal from a constellation diagram
This is a basic question about how passband pulse-amplitude-modulation (PAM) works. It has nothing to do specifically with QAM, but it applies to any type of passband PAM (such as PSK, or any other choice of a constellation).

The question is how to obtain a transmit signal from a given point in a constellation diagram. Let's say that at time $t=kT$ we want to transmit a signal corresponding to the point $a_k$ in the constellation diagram, where $a_k$ is a complex number:

$$a_k=\text{Re}(a_k)+j\text{Im}(a_k)=|a_k|e^{j\phi_k}$$

The complex number is multiplied by a (real-valued) transmit pulse $g(t)$ and by a complex carrier, and the transmitted signal is the real part of that complex-valued signal:

$$\begin{align}s_k(t)&amp;=\text{Re}\left(a_kg(t-kT)e^{j\omega_0t}\right)\\&amp;=\text{Re}(a_k)g(t-kT)\cos(\omega_0t)-\text{Im}(a_k)g(t-kT)\sin(\omega_0t)\\&amp;=|a_k|g(t-kT)\cos(\omega_0t+\phi_k)\end{align}\tag{1}$$

In practice you transmit a sequence of symbols, and the corresponding transmitted signal is given by

$$\begin{align}s(t)&amp;=\sum_ks_k(t)\\&amp;=\cos(\omega_0t)\sum_k\text{Re}(a_k)g(t-kT)-\sin(\omega_0t)\sum_k\text{Im}(a_k)g(t-kT)\\&amp;=\sum_k|a_k|g(t-kT)\cos(\omega_ot+\phi_k)\end{align}\tag{2}$$

As you can see from $(1)$ and $(2)$, the transmitted signal can be represented either as two amplitude-modulated orthogonal carriers (sine and cosine), or as an amplitude and phase modulated carrier. In the first case the real and imaginary parts of $a_k$ determine the amplitudes of the two carriers, and in the second case the magnitude $|a_k|$ determines the amplitude of the carrier, and $\phi_k=\arg\{a_k\}$ is the phase of the carrier.