This paper suggests a novel one-dimensional (1D) map to address the limitations of traditional chaotic 1D maps. In contrast to traditional 1D maps, the proposed map has three control parameters a, $\mu$, and c, allowing it to exhibit chaotic behavior over a wide range of values. The dynamic behavior of the new 1D map was analyzed using well-known numerical methods, including the bifurcation diagram and Lyapunov exponent. Both tests showed their complex and diverse behavior. In addition, a novel image encryption scheme was devised using the new function as its pseudorandom number generator. Rigorous statistical testing was applied to the proposed encryption algorithm. The mean square error (MSE) and peak signal-to-noise ratio (PSNR) results, in addition to subjecting 28 images to number of pixels change rate (NPCR) and unified average changing intensity (UACI) tests demonstrated the robustness of the system. The results of this study demonstrate the effectiveness of the new 1D map for use in secure image cryptography applications, providing a more robust and secure alternative to traditional chaotic 1D maps.