By Kylene Arnold Updated Aug 30, 2022

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When you have experimental data points that trace a parabola, scientists and mathematicians often need to reconstruct the exact quadratic function that models the trend. The method below shows how to derive the equation from three known points.

Step‑by‑Step Method

1. Select three points that lie on the same parabola. Example: (1, 5), (2, 11), and (3, 19).
2. Set up the system of equations by substituting each point into the general form f(x)=ax^2+bx+c:
   • For (1, 5): 5 = a(1)² + b(1) + c → a+b+c = 5
   • For (2, 11): 11 = a(2)² + b(2) + c → 4a+2b+c = 11
   • For (3, 19): 19 = a(3)² + b(3) + c → 9a+3b+c = 19
3. Solve the linear system. Subtracting the first equation from the second gives 3a+b = 6. Subtracting the second from the third yields 5a+b = 8. Subtracting these two results gives 2a = 2, so a = 1. Plugging back into 3a+b = 6 gives b = 3. Finally, substitute a and b into a+b+c = 5 to find c = 1.
4. Write the final quadratic function using the solved coefficients: f(x)=x²+3x+1.

Thus, the parabola that passes through (1, 5), (2, 11), and (3, 19) is described by f(x)=x²+3x+1. This systematic approach is foundational in algebra and essential for modeling real‑world data.