Question: An ichthyologist models fish population growth with a quadratic function $ h(x) $ satisfying $ h(1) = 4 $, $ h(2) = 11 $, and $ h(3) = 22 $. Find $ h(4) $.

["Title: Predicting Fish Populations: How an Ichthyologist Models Growth Using Quadratic Functions", "Meta Description: Discover how an ichthyologist uses a quadratic function to model fish population growth. Using real data points, learn how to determine the unknown population at $ h(4) $ with precise algebraic modeling.", "---", "### Introduction: Modeling Nature’s Resilience with Quadratics", "In the study of fish populations, recording and predicting growth trends is essential for conserving marine ecosystems and managing fisheries sustainably. While fish populations often follow complex patterns influenced by environmental factors, ichthyologists frequently use mathematical models—especially quadratic functions—to approximate growth behavior in controlled or stable conditions.", "Consider a practical scenario: an ichthyologist studying the population of a common reef fish models its growth over time with a quadratic function $ h(x) = ax^2 + bx + c $, where $ x $ represents time in months. This model is ideal for smooth yet accelerating changes in population, common in healthy, undisturbed habitats.", "Today, we explore how to determine the value of $ h(4) $ when given three key data points:\n- $ h(1) = 4 $\n- $ h(2) = 11 $\n- $ h(3) = 22 $", "By solving for the coefficients $ a $, $ b $, and $ c $, we’ll uncover the future population at month $ x = 4 $ and gain insight into ecological forecasting.", "---", "### Setting Up the Quadratic Model", "We suppose the population growth follows $ h(x) = ax^2 + bx + c $. Using the given values, we form a system of equations:", "1. $ h(1) = a(1)^2 + b(1) + c = a + b + c = 4 $\n2. $ h(2) = a(2)^2 + b(2) + c = 4a + 2b + c = 11 $\n3. $ h(3) = a(3)^2 + b(3) + c = 9a + 3b + c = 22 $", "Our goal is to solve this system to find $ a $, $ b $, and $ c $, then compute $ h(4) = 16a + 4b + c $.", "---", "### Solving the System of Equations", "Start with the equations:\n[\n\begin{align}\n(1)&\quad a + b + c = 4 \\n(2)&\quad 4a + 2b + c = 11 \\n(3)&\quad 9a + 3b + c = 22 \\n\end{align}\n]", "Step 1: Subtract equation (1) from (2):\n[\n(4a + 2b + c) - (a + b + c) = 11 - 4 \Rightarrow 3a + b = 7 \quad \ ext{(Equation A)}\n]", "Step 2: Subtract equation (2) from (3):\n[\n(9a + 3b + c) - (4a + 2b + c) = 22 - 11 \Rightarrow 5a + b = 11 \quad \ ext{(Equation B)}\n]", "Step 3: Subtract Equation A from Equation B:\n[\n(5a + b) - (3a + b) = 11 - 7 \Rightarrow 2a = 4 \Rightarrow a = 2\n]", "Step 4: Substitute $ a = 2 $ into Equation A:\n[\n3(2) + b = 7 \Rightarrow 6 + b = 7 \Rightarrow b = 1\n]", "Step 5: Substitute $ a = 2 $, $ b = 1 $ into equation (1):\n[\n2 + 1 + c = 4 \Rightarrow c = 1\n]", "We now have the complete model:\n$$\nh(x) = 2x^2 + x + 1\n$$", "---", "### Calculating $ h(4) $", "To find the predicted population at month 4:\n[\nh(4) = 2(4)^2 + 1(4) + 1 = 2(16) + 4 + 1 = 32 + 4 + 1 = 37\n]", "Thus, the model projects a population of 37 fish at month 4.", "---", "### Why This Model Matters", "This quadratic function captures accelerating growth consistent with healthy fish populations emerging from favorable environmental conditions. Ichthyologists rely on such precise models to forecast impacts of climate change, fishing pressures, and habitat restoration. Accurate predictions like $ h(4) = 37 $ aid in setting sustainable catch limits and protecting biodiversity.", "---", "### Final Thoughts", "Using quadratic functions to model biological systems demonstrates the power of applied mathematics in ecology. By fitting known data, scientists derive meaningful insights that guide conservation efforts. Whether modeling fish populations or plant growth, quadratic models offer a reliable bridge between observation and prediction—empowering smarter environmental stewardship.", "Unlock the future of ocean sustainability—one equation at a time.", "---", "Keywords:\nichthyologist, fish population model, quadratic function, ecological modeling, quadratic growth, h(x), population forecasting, sustainable fisheries, algebraic model, natural population dynamics", "Content Source: Science education in environmental science, applied quadratic functions in biology, ichthyology research methods."]