Natural genetic promoters are regulated by multiple cis and trans regulatory factors. For quantitative studies of these promoters, the concentration of only a single factor is typically varied to obtain the dose response or transfer function of the promoters with respect to the factor. Such design of experiments has limited our ability to understand quantitative, combinatorial interactions between multiple regulatory factors at promoters. This limitation is primarily due to the intractable number of experimental combinations that arise from multifactorial design of experiments. To overcome this major limitation, we integrate impact printing and cell-free systems to enable multi-dimensional studies of genetic promoters. We first present a gradient printing system which comprises parallel piezoelectric cantilever beams as a scalable actuator array to generate droplets with tunable volumes in the range of 100 pL-10 nL, which facilitates highly accurate direct dilutions in the range of 1-10 000-fold in a 1 μL drop. Next, we apply this technology to study interactions between three regulatory factors at a synthetic genetic promoter. Finally, a mathematical model of gene regulatory modules is established using the multi-parametric and multi-dimensional data. Our work creates a new frontier in the use of cell-free systems and droplet printing for multi-dimensional studies of synthetic genetic constructs.