To assess the task-based performance of images obtained under different focal spot size and acquisition mode on a dual-energy CT scanner. Axial CT image series of the Catphan phantom were obtained using a tube focus at different sizes. Acquisitions were performed in standard single-energy, high resolution (HR) and dual-energy modes. Images were reconstructed using conventional and high definition (HD) kernels. Task-based transfer function at the 50% level (TTF50%) for teflon, delrin, low density polyethylene (LDPE) and acrylic, as well as image noise and noise texture, were assessed across all focal spots and acquisition modes using Noise Power Spectrum (NPS) analysis. A non-prewhitening mathematical observer model was used to calculate detectability index (dNPW'). TTF50% degraded with increasing focal spot size. TTF50% ranged from 0.67mm-1 for teflon to 0.25mm-1 for acrylic. For standard kernel, image noise and NPS-determined average spatial frequency were 8.3 HU and 0.29mm-1, respectively in single-energy, 12.0 HU and 0.37mm-1 in HR, and 7.9 HU and 0.26mm-1 in dual-energy mode. For standard kernel, dNPW' was 61 in single-energy and HR mode and reduced to 56 in dual-energy mode. The task-based image quality assessment metrics have shown that spatial resolution is higher for higher image contrast materials and detectability is higher in the standard single-energy mode compared to HR and dual-energy mode. The results of the current study provide CT operators the required knowledge to characterize their CT system towards the optimization of its clinical performance.