An entomologist studying the flight dynamics of bees models their path as a parabola described by \(y = ax^2 + bx + c\). If the path passes through the points (1, 2), (2, 3), and (3, 5), find the coefficients \(a\), \(b\), and \(c\).

["## Unraveling the Parabolic Flight Path: How an Entomologist Models Bee Flight with Quadratic Dynamics", "Bee flight is a marvel of natural precision—efficient, stable, and elegantly coordinated. To better understand the aerodynamic forces shaping their flight, entomologists often model the trajectory of bees as a parabola described by the quadratic equation:\n[\ny = ax^2 + bx + c\n]\nThis mathematical representation allows researchers to decode key flight behaviors such as climbing, cruising, and landing. In a recent study, an entomologist applied this model to track a bee’s vertical path during pollen navigation, arriving at a system of equations that reveals the precise parabolic shape of its flight.", "### Solving for the Parabola: Finding Coefficients (a), (b), and (c)", "To determine the exact quadratic path, the researcher used three known points along the bee’s flight:\n- At (x = 1), (y = 2)\n- At (x = 2), (y = 3)\n- At (x = 3), (y = 5)", "Each point provides an equation when substituted into (y = ax^2 + bx + c):", "1. (a(1)^2 + b(1) + c = 2) → (a + b + c = 2)\n2. (a(2)^2 + b(2) + c = 3) → (4a + 2b + c = 3)\n3. (a(3)^2 + b(3) + c = 5) → (9a + 3b + c = 5)", "We now solve this system step-by-step.", "Step 1: Subtract equations to eliminate (c).\nSubtract Equation (1) from Equation (2):\n[\n(4a + 2b + c) - (a + b + c) = 3 - 2 \Rightarrow 3a + b = 1 \quad \ ext{(Equation A)}\n]", "Subtract Equation (2) from Equation (3):\n[\n(9a + 3b + c) - (4a + 2b + c) = 5 - 3 \Rightarrow 5a + b = 2 \quad \ ext{(Equation B)}\n]", "Step 2: Eliminate (b) by subtracting Equation A from Equation B.\n[\n(5a + b) - (3a + b) = 2 - 1 \Rightarrow 2a = 1 \Rightarrow a = \frac{1}{2}\n]", "Step 3: Substitute (a = \frac{1}{2}) into Equation A.\n[\n3\left(\frac{1}{2}\right) + b = 1 \Rightarrow \frac{3}{2} + b = 1 \Rightarrow b = 1 - \frac{3}{2} = -\frac{1}{2}\n]", "Step 4: Substitute (a = \frac{1}{2}) and (b = -\frac{1}{2}) into Equation (1) to find (c).\n[\n\frac{1}{2} - \frac{1}{2} + c = 2 \Rightarrow 0 + c = 2 \Rightarrow c = 2\n]", "### Final Result", "The quadratic model of the bee’s flight path is:\n[\ny = \frac{1}{2}x^2 - \frac{1}{2}x + 2\n]", "These coefficients—(a = \frac{1}{2}), (b = -\frac{1}{2}), (c = 2)—capture the essence of the bee’s vertical trajectory, enabling deeper insights into flight control and energy efficiency. Understanding these dynamics not only enriches biomimetic design but also supports conservation efforts by illuminating how bees navigate complex environments with precision.", "For entomologists and engineers alike, modeling pollinator flight through parabolic equations like (y = ax^2 + bx + c) brings science closer to nature’s blueprint.", "---", "Keywords: entomologist, bee flight dynamics, parabolic path, quadratic equation, coefficient calculation, quadratic model, pollination behavior, aerodynamics in insects, flight path modeling, entomology research, biomimetics."]