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It is well-known that income can correlate with the academic performance of K-12 students in the United States (U.S.). However, the mathematical relationship between income and K-12 performance, and how it varies across states, remains poorly understood. To help fill this gap, this study examines the relationship between K-12 student performance scores (defined as the percentage of students meeting or exceeding grade-level expectations) and median household income, across more than 12,200 public school districts in 42 US states. The study focuses on performance in English Language Arts (ELA) and Mathematics in 3rd and 8th grades during the 2018-2019 school year. A number of different mathematical functions are explored to quantitatively characterize this relationship, and the best fitting functions are determined statistically. It was found that in about half of the states, the proficiency rate increases linearly with the median household income, while in the rest of the states the increase is characterized by a saturating function. Further, the results reveal that less affluent states exhibit a steeper increase in performance with income compared to wealthier states. Additionally, grade-level and subject comparisons highlight disparities, including a pronounced decline in math performance from 3rd to 8th grade in most districts. These findings underscore the correlations between socioeconomic factors and educational outcomes and the variations between subjects, grade levels, as well as locations. By expanding our understanding of these relationships, this research offers potentially useful mathematical methodologies for developing evidence-based, quantitative approaches to studying educational equity.