Populations can go extinct when their environments deteriorate, but evolutionary rescue occurs when a shrinking population adapts to the new environmental conditions. The emergence of resistance from a drug sensitive bacterial population under treatment can be regarded as an instance of evolutionary rescue. Understanding evolutionary rescue in a particular context such as drug resistance requires knowledge of how the environment changes and how selection coefficients change as a result. In this study, we propose a model for evolutionary rescue under three different scenarios of environmental change: abrupt change, periodic fluctuation and gradual decay. The model makes use of the notion of reaction norms to describe fitness values that depend on both genotype and environmental state. We find that although drug sensitive bacterial populations may be large, allowing them to generate resistant mutants frequently, a harsh abrupt change due to the drug usually drives them extinct. Evolutionary rescue occurs far more frequently under the milder forms of environmental change we investigated. Rescue is favoured when the absolute fitnesses of individuals remain sufficiently high over the range of environment qualities experienced by the population. The minimum environment quality, which is inversely related to drug dose in the antibiotic context, is thus an important factor. Interestingly, in the periodic fluctuation model, the inter-dose period is less influential in promoting rescue through resistance unless the minimum environment quality is in a particular range. We also investigated fitness trade-offs across environments including the case of a resistant allele not subject to any trade-off (a "superbug"). This fitness trade-off affects the probability of rescue in decaying environments, but surprisingly has only a weak effect in the periodic fluctuation scenario. Finally, we use the model to show how niche construction, whereby organisms are the source of environmental change, produces more complex dynamics.