We present a logistic function of a monotonic polynomial with a lower asymptote, allowing additional flexibility beyond the three-parameter logistic model. We develop a maximum marginal likelihood-based approach to estimate the item parameters. The new item response model is demonstrated on math assessment data from a state, and a computationally efficient strategy for choosing the order of the polynomial is demonstrated. Finally, our approach is tested through simulations and compared to response function estimation using smoothed isotonic regression. Results indicate that our approach can result in small gains in item response function recovery and latent trait estimation.