Strain engineering, which aims to tune the bandgap of a semiconductor by the application of strain, has emerged as an interesting way to control the electrical and optical properties of two-dimensional (2D) materials. Apart from the changes in the intrinsic properties of 2D materials, the application of strain can also be used to modify the characteristics of devices based on them. In this work, we study flexible and transparent photodetectors based on single-layer MoS2 under the application of biaxial strain. We find that by controlling the level of strain, we can tune the photoresponsivity (by 2-3 orders of magnitude), the response time (from < 80 ms to 1.5 s), and the spectral bandwidth (with a gauge factor of 135 meV/% or 58 nm/%) of the device.