5Question: An environmental educator is analyzing the water flow in a local stream. They model the flow rate \( f(x) \) (in cubic meters per second) as a quadratic polynomial \( f(x) = ax^2 + bx + c \). Given \( f(1) = 6 \), \( f(2) = 11 \), and \( f(3) = 18 \), determine \( f(x+1) \).

["Title: Analyzing Stream Flow: Deriving ( f(x+1) ) From Quadratic Data – A Step-by-Step Environmental Analysis", "Meta Description:\nAn environmental educator models local stream flow using a quadratic function. By analyzing ( f(1) = 6 ), ( f(2) = 11 ), and ( f(3) = 18 ), discover how to determine ( f(x+1) ) and predict future water flow patterns.", "---", "### Introduction", "Understanding water flow dynamics is crucial for environmental stewardship, especially when studying local streams. educators often use mathematical models to analyze stream flow, helping predict changes and inform conservation efforts. In this article, we explore how a quadratic polynomial models stream flow data, walks through solving for the unknown coefficients, and demonstrates how to compute ( f(x+1) )—a powerful tool for forecasting.", "---", "### The Quadratic Model: ( f(x) = ax^2 + bx + c )", "We are given three data points reflecting flow rates at specific positions along the stream:\n- ( f(1) = 6 )\n- ( f(2) = 11 )\n- ( f(3) = 18 )", "Since ( f(x) ) follows a quadratic form, we substitute the points into the model:", "1. ( f(1) = a(1)^2 + b(1) + c = a + b + c = 6 )\n2. ( f(2) = a(2)^2 + b(2) + c = 4a + 2b + c = 11 )\n3. ( f(3) = a(3)^2 + b(3) + c = 9a + 3b + c = 18 )", "This yields the system of equations:\n[\n\begin{cases}\na + b + c = 6 \quad \ ext{(Eq. 1)} \\n4a + 2b + c = 11 \quad \ ext{(Eq. 2)} \\n9a + 3b + c = 18 \quad \ ext{(Eq. 3)}\n\end{cases}\n]", "---", "### Solving the System for ( a ), ( b ), and ( c )", "Subtract Eq. 1 from Eq. 2:\n[\n(4a + 2b + c) - (a + b + c) = 11 - 6 \Rightarrow 3a + b = 5 \quad \ ext{(Eq. 4)}\n]", "Subtract Eq. 2 from Eq. 3:\n[\n(9a + 3b + c) - (4a + 2b + c) = 18 - 11 \Rightarrow 5a + b = 7 \quad \ ext{(Eq. 5)}\n]", "Now subtract Eq. 4 from Eq. 5:\n[\n(5a + b) - (3a + b) = 7 - 5 \Rightarrow 2a = 2 \Rightarrow a = 1\n]", "Substitute ( a = 1 ) into Eq. 4:\n[\n3(1) + b = 5 \Rightarrow b = 2\n]", "Substitute ( a = 1 ), ( b = 2 ) into Eq. 1:\n[\n1 + 2 + c = 6 \Rightarrow c = 3\n]", "Thus, the quadratic model is:\n[\nf(x) = x^2 + 2x + 3\n]", "---", "### Computing ( f(x+1) )", "To predict flow at ( x+1 ), substitute ( x+1 ) into the function:\n[\nf(x+1) = (x+1)^2 + 2(x+1) + 3\n]", "Expand each term:\n[\n(x+1)^2 = x^2 + 2x + 1\n]\n[\n2(x+1) = 2x + 2\n]", "Add all parts:\n[\nf(x+1) = x^2 + 2x + 1 + 2x + 2 + 3 = x^2 + 4x + 6\n]", "---", "### Interpretation and Environmental Application", "The expression ( f(x+1) = x^2 + 4x + 6 ) allows the environmental educator to:\n- Predict stream flow at subsequent monitoring points\n- Compare seasonal or rainfall-induced changes by evaluating ( f ) at scaled inputs\n- Plan conservation responses using forward-flow projections", "This method exemplifies how algebraic modeling supports real-world ecological analysis.", "---", "### Conclusion", "Using the given data and solving the quadratic system, we derived ( f(x) = x^2 + 2x + 3 ), then computed ( f(x+1) = x^2 + 4x + 6 ). Such models empower educators to analyze stream dynamics accurately, contributing to thoughtful environmental management and sustainable water resource planning.", "---", "Keywords: quadratic polynomial, environmental modeling, stream flow analysis, ( f(x) = ax^2 + bx + c ), environmental education, mathematical modeling, water resource prediction, ( f(x+1) ), local stream analysis", "For more insights, explore how dynamic environmental models transform field data into actionable science."]