Question: A behavioral health researcher studies patient engagement over time and defines a function $ f : \mathbb{R} \to \mathbb{R} $ satisfying $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $. If $ f(1) = 4 $, find $ f(3) $.

["Understanding Patient Engagement Through Behavioral Health Models: Solving a Functional Equation to Inform Long-Term Outcomes", "In behavioral health research, modeling patient engagement over time is crucial for predicting treatment adherence, identifying risk patterns, and improving long-term outcomes. A recent study by a leading behavioral health researcher employs mathematical modeling to analyze how engagement evolves. Inspired by functional equations in applied psychology, the researcher defines a function $ f: \mathbb{R} \ o \mathbb{R} $ that satisfies the equation:\n$$\nf(x + y) + f(x - y) = 2f(x) + 2f(y) \quad \ ext{for all } x, y \in \mathbb{R},\n$$\nwith the condition $ f(1) = 4 $. The goal is to determine $ f(3) $—a critical data point for longitudinal analysis of patient engagement trajectories.", "This functional equation is a well-known form in mathematical analysis: it characterizes quadratic functions. Specifically, any function $ f $ satisfying this identity for all real $ x, y $ must be of the form\n$$\nf(x) = ax^2 + bx + c,\n$$\nthough deeper inspection shows symmetry and growth behavior common in behavioral response models favor simplified quadratic forms.", "We verify this structure step by step.", "Step 1: Plug in $ x = y = 0 $\nSubstituting into the equation:\n$$\nf(0 + 0) + f(0 - 0) = 2f(0) + 2f(0) \Rightarrow 2f(0) = 4f(0) \Rightarrow 2f(0) = 0 \Rightarrow f(0) = 0.\n$$", "So $ c = 0 $ in the quadratic form, and $ f(0) = 0 $.", "Step 2: Assume $ f(x) = ax^2 + bx $\nNow use $ f(1) = 4 $:\n$$\nf(1) = a(1)^2 + b(1) = a + b = 4.\n$$", "We now determine $ a $ and $ b $. But we analyze further to eliminate $ b $.", "Step 3: Use symmetry and plug in $ x = 0 $\nSet $ x = 0 $:\n$$\nf(y) + f(-y) = 2f(0) + 2f(y) = 2f(y) \Rightarrow f(-y) = f(y).\n$$\nThus, $ f $ is even. This implies $ b = 0 $, since odd-powered terms vanish.\nTherefore, $ f(x) = ax^2 $, and from $ a + b = 4 $, with $ b = 0 $, we get $ a = 4 $.", "Hence, $ f(x) = 4x^2 $.", "Step 4: Compute $ f(3) $\n$$\nf(3) = 4 \cdot 3^2 = 4 \cdot 9 = 36.\n$$", "This model reveals that patient engagement measures grow quadratically over time — a pattern suggesting increasing effort and responsiveness in sustained therapeutic engagement. Such mathematical functions help researchers forecast engagement levels and design timely interventions.", "Conclusion:\nBy solving this functional equation, a behavioral health researcher establishes a precise mathematical model $ f(x) = 4x^2 $, directly yielding $ f(3) = 36 $. This approach bridges rigorous science and real-world application, demonstrating how abstract mathematics enhances understanding of complex human behaviors.", "For clinicians and researchers applying behavioral health metrics: functional analysis offers a powerful tool to decode long-term patient dynamics—proving that behind every data point lies a story of change, measurable and meaningful.", "---", "Key Takeaway for Practitioners:\nFunctional equations like this provide a formal framework for modeling engagement trends. When paired with empirical data, they support predictive analytics, helping tailor interventions that improve sustained recovery.", "Keyword-rich summary: behavioral health modeling, functional equation $ f(x+y)+f(x-y)=2f(x)+2f(y) $, real function $ f:\mathbb{R}\ o\mathbb{R} $, patient engagement, mathematical modeling in psychology, foundational quadratic growth in treatment adherence, data-driven therapy outcomes."]